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C. Bittner +Chris Biemann +Universit¨at Hamburg, Hamburg, Germany +{debayan.banerjee,mathis.poser,christina.wiethof,eva.bittner,chris.biemann}@uni- +hamburg.de,{varunshankar55,rfpaucar}@gmail.com +Abstract +AI enabled chat bots have recently been put to use to answer +customer service queries, however it is a common feedback +of users that bots lack a personal touch and are often unable to +understand the real intent of the user’s question. To this end, +it is desirable to have human involvement in the customer +servicing process. In this work, we present a system where +a human support agent collaborates in real-time with an AI +agent to satisfactorily answer customer queries. We describe +the user interaction elements of the solution, along with the +machine learning techniques involved in the AI agent. +Introduction +In the pursuit of operational efficiency, companies across +the globe have been deploying automation technology aided +by Artificial Intelligence (AI) for Online Customer Support +(OCS) use cases 1. With the explosive growth of social me- +dia usage, incoming customer queries have grown exponen- +tially and to handle this growth, the use of proper technology +is critical. Some estimates say that by the year 2025, 95% +of all customer interactions will be processed in some form +by AI 2. However, AI in its present state is not advanced +enough to completely replace human agents for most cus- +tomer support scenarios. Additionally, the complete replace- +ment of human workforce by AI is a topic of active ethical +and political debate. For these reasons the development of a +hybrid working environment is required, where both human +agents and AI agents can co-operate to satisfy OCS require- +ments. +In this work we briefly describe a web based user inter- +face that allows a customer to interact with a human sup- +port agent, where the human agent receives helpful sugges- +tions in parallel from an AI agent. In subsequent sections, we +elaborate further on the machine learning techniques used +for the AI agent. +Our present work is a part of a project which aims to find +ways of integrating AI agents into customer support based +workflows, with an aim of reducing workload of human +*These authors contributed equally. +1https://www.gartner.com/smarterwithgartner/4-key-tech- +trends-in-customer-service-to-watch +2https://servion.com/blog/what-emerging-technologies-future- +customer-experience/ +agents. It is one of the primary goals of the project not to +entirely replace the human agent with AI, and instead find +productive means of co-existence of the two. As a part of +this project, an international volunteer-driven organisation, +which organises internships and projects for students across +the globe was involved. In this organisation, prospective stu- +dents participate in text based chat with human agents, and +typically enquire about available opportunities and how to +participate in them. The human agents in turn use their do- +main expertise to provide the necessary information to the +students. +All the students and human agents involved were resi- +dents of Germany and hence the conversations were car- +ried out in the German language. After collecting the con- +versations, an annotation phase was undertaken, where rele- +vant utterances of the conversation were annotated with the +corresponding FAQ IDs. When the conversations originally +took place, there was no singular FAQ database in existence. +For the purpose of this project, such a database was created. +This made it possible to annotate the utterances with relevant +FAQ IDs. +The goal of the dataset is to train an AI agent that can pas- +sively listen to the ongoing conversation and make relevant +suggestions visible only to the human agent, not to the stu- +dent. The human agent may then forward the suggested FAQ +answer to the student, or decide not to do so if the quality of +suggestion is poor. The eventual goal is for the human agent +to spend less time looking for the right answer in a Knowl- +edge Base, and instead offload this task to the AI agent. +Later, a web UI was constructed, as described in the Web +Interface section, that the human agent uses to interact with +the student. The student is not aware of the UI’s existence +and is operating on a separate chat platform. The AI agent +provides timely suggestions in this UI which is visible to the +human agent. +Our scenario differs from conventional Conversational +Question Answering (CQA) or Interactive Information Re- +trieval (IIR) where the user interacts directly with the AI +agent, and the AI agent is responsible for a response at each +turn. In our case, the AI agent is in a passive listening role. +It observes the ongoing conversation between two humans, +and makes suggestions that are only visible to the human +agent. Since the task of the AI agent is not just to suggest +relevant FAQs but also to remain silent when no relevant +1 +arXiv:2301.12158v1 [cs.AI] 28 Jan 2023 + +Figure 1: Screenshot of web based prototype +FAQ is to be suggested, we evaluate both of these aspects in +the evaluation section. +The user interface presented in this work has been pub- +lished before (Poser et al. 2022). The machine learning tech- +niques used to train the AI agent are yet to be published, and +hence a larger focus in this work is on the AI training aspect. +Web Interface +The web-based frontend in Figure 1 is labelled with cer- +tain design features (DF) to be explained shortly. The in- +terface was implemented with Bootstrap and ReactJS while +the backend API is hosted as a Python Flask app. The inter- +face greets humans agents with an avatar named Charlie that +presents a brief usage explanation (DF1). In addition, setting +options for AI support and learning behavior are provided +(DF2). The integrated chat window is based on the open- +source chat framework Rocket Chat. The backend generates +a ranked list of FAQ suggestions based on ML techniques to +be described later. In the frontend, two FAQ items - includ- +ing theme and accuracy in percent - with the highest agree- +ment are displayed (DF3). The discard buttons can be used +to sequentially display four additional FAQ suggestions with +decreasing accuracy. The copy-to-chat buttons insert FAQ +text into the input field of the chat window. Detailed infor- +mation about a respective FAQ can be viewed via the get- +more-info button (DF4). With a counter, points are added +(copy-to-chat) or subtracted (discard), if buttons are clicked +(DF5). A feedback field allows entering search terms to se- +lect and submit a FAQ that matches the interaction (DF6). +Based on customers’ chat messages, exact keyword-based +text matching is performed to automatically record interests +and suggest suitable projects from a database (DF7). +Related Work +The earliest dialogue systems, or chat-bots, were rule based +(Weizenbaum 1966; Colby et al. 1972) and subsequently +corpus based chat-bots were developed (Serban et al. 2015) +. In recent times neural chat-bots are frequently encountered +in day to day customer support scenarios (Ni et al. 2021). +Recently, an interplay of human and AI collaboration in +the process has been explored (Liu et al. 2021). However +current research in this area is focused on the AI bot be- +ing the first line of service, and only in the case of failures +of the bot, a handover is initiated to a human agent, who +plays a secondary role in the process. In contrast, our sce- +nario makes the human agent the first line of support with +the AI agent assisting in parallel. +To train chat-bots, conversational QA datasets such as the +Ubuntu corpus (Lowe et al. 2015), CoQA (Reddy, Chen, +and Manning 2019), DoQA (Campos et al. 2020) and QuAC +(Choi et al. 2018) have made progress in providing the +community with rich grounds for conversational research. +While CoQA relies on passages from broad domains +such as children’s stories and science to retrieve answers, +QuAC relies on Wikipedia articles to create conversations +and answers. DoQA on the other hand, focuses on three +specific domains of cooking, travel and movies from stack- +exchange.com. In scope of how our dataset is modelled, +it is most similar to DoQA, which is a domain specific +conversational dataset which also requires retrieval of the +correct FAQ from a database. CoQA, DoQA and QuAC +datasets are crowd-sourced and collected by the Wizard of +2 + +IntelligentSupportAgent +三 +E +Hi there, I'am Charlie your personal assistant! +Your Personal Settings? +information and knowledge. You can control my settings anytime, To elevate +Do you want my assistance? +on +to learn more about my features. +May I learn from your conversations and interaction with me? +DF2 +On +Happy to work with you! +DF1 +Knowledge +DF3 +Charlie's Suggestions? +Feedback? +FAQ Theme: +Find here the right question, and then press send button. +Copy to chat +Dscardo +DF6 +Get mareinfo +FAQ Theme: +DF4 +Copy ochat +DiacardO +Get moreinfoO +Charlie's Explanations (Get more info) +Ccopyfo chat +Points for Charlie +DF5 +EP's Interest +Projects +Where +DF7 ++ Charlie's Suggestions +Indicate Location (country) +Small text +Copy to chat +When +Discarda +Indicate month +Small teat +Message +What +Copy to chat +Small text +DiscardQ +Indicate what type of projectFigure 2: A sample conversation from the dataset with relevant corresponding FAQ annotation. The text in red is English +translation of the conversation for the purpose of this paper, and not a part of the dataset. +Oz method. On the contrary, our dataset consists of genuine +conversations between two humans whose sole purpose is +to find the best internship possible for the student. During +the conversations, neither of the parties were aware of the +need to form an annotated dataset. Hence, our dataset has +no artificial aspects in the flow of conversation. +The Dortmunder Chat Korpus (Beißwenger et al. 2013) and +The Verbmobil (Wahlster 1993) project provide German +conversational corpus but they do not address the Question +Answering or Information Retrieval domains. +Recently, the GermanQuAD and GermanDPR (M¨oller, +Risch, and Pietsch 2021) projects from DeepSet have +enabled access to Transformer based models trained on +the German text, which we make use of in our evaluation +section, however the dataset they are based on is in the form +of Questions and Answers, and not conversational in nature. +Dataset Creation +To train the AI agent, a conversational dataset had to be con- +structed. For this purpose, the conversations were carried out +on the popular mobile application WhatsApp 3, where both +the human agent and the student were on Whatsapp. The +Web Interface described in the previous section was not in- +cluded in this process. The conversations centered around +topics such as how to register for a project, which projects +are available in a given location, and whether there will be +certifications available at the end etc. The chats were ex- +tracted using the export functionality of WhatsApp. The +3https://play.google.com/store/apps/details?id=com.whatsapp +conversations have been collected over a period of two years, +between 2018 and 2020. In some cases, an individual con- +versation may also span over a duration of several months, +where the student and the human agent re-established con- +tact after a gap of more than a few days. Such information +is visible through the inclusion of the timestamp field in the +dataset for each message that is exchanged. +Relevant consent for releasing their conversations was col- +lected from the participating students and agents. More- +over, the identities of the participants and the organisation +are pseudo-anonymised. Instead of the names of the partici- +pants, they are given a numerical name such as KundeSech- +sundzwanzig, which stands for Customer 26 in German. The +human agent is represented by the term Mitarbeiter which +stands for employee. +A single human agent handled all the 26 conversations +on WhatsApp over a period of time. When the conversa- +tions were carried out between 2018-2020, no single FAQ +database existed at the organisation. The human agent in- +stead used relevant domain expertise and experience within +the organisation, and referred to a set of disjoint sources of +information when the chats took place. Later in 2021, the hu- +man agent and a fellow domain expert colleague compiled a +single FAQ database that covers most of the issues discussed +in the conversations. Specific turns of the conversations were +manually annotated with relevant FAQs by the human agent +and then verified by the domain expert colleague. +Dataset Analysis +Chats and FAQs. As depicted in Figure 4 the 26 collected +conversations vary in length ranging from 22 utterances +3 + +Mitarbeiter : Hey! ich bin Mitarbeiter. Du hast dich bei +FAQ 1 +uns angemeldet und ich wurde gerne mit dir daruber +sprechen / telefonieren :). Wann hattest du denn dafur +"Question":"wann kann ich ein projekt machen?" +Zeit? +Employee : Hey!I am an employee here.You have +"When can Idoaproject?" +registered with us and I would like to talk to you about it +"Answer":"projekte sind jederzeitmoglich", +KundeVierzehn : Guten Morgen, Ich interessiere mich +"Projects can be done atany time” +sehr fur Projekte in der Turkei. Wenn der Start im Januar +moglich ist. +Customer Fourteen: Good Morning,Iam very interested +in projects in Turkey. If the start is possible in January. +FAQ55 +Mitarbeiter/Employee : https://.org/opportunity/984743 +"Question": "was mache ich nun nachdem ich mich +https://.org/opportunity/1002581 +beworbenhabe?" +KundeVierzehn : Hallo Mitarbeiter, ich habe mich gerade +'what do I doafter I haveapplied?" +beworben. +Customer Fourteen: Hello Employee, I just applied. +"Answer":"prozess:bewerben-kontaktmit +auslandspartner-akzeptiert-vertrag und gebuhr +Mitarbeiter : Ah super! Ich kummere mich darum dass du +approved", +schnell kontaktiert wirst :) +"process:apply-contactwithforeign +Employee: Ah great! I will make sure that you will be +contacted quickly :) +partner -accepted-contract and fee -approved'to 607 utterances, with an average of 239 utterances per +conversation. The entire set of conversations consists of +6,219 utterances. 20.9 % of the utterances are annotated +with the relevant FAQ ID. A significant portion of the +dataset consists of chit-chat or other non-specific topics +where no suggestion is supposed to be made by the AI agent +to the human agent. +Since certain topics in the chat are discussed more often +than others, as seen in Figure 3, the distribution of relevant +annotated FAQ IDs also is imbalanced with FAQ ID 71 +being the most frequent. FAQ 71 pertains to the procedure +of registering online for projects. +We have split the dataset into train, dev and test splits in +roughly 70:10:20 ratios. The train, dev and test splits have +17, 3 and 6 conversations, respectively, consisting of 3,693 , +891 and 1,635 utterances. +Experimental Setting +Task Definition +We define the task with the following inputs: current utter- +ance uk, the set of FAQs F, and the history of utterances so +far {u1, u2, ...., uk−1}. The task for the model is to rank the +correct FAQ item from F to the top. If for a given utterance +no FAQ is appropriate, the model must produce as the top- +ranked output a special class that denotes absence of FAQ +suggestion. We hereby call this class no-suggestion. +Models +As baselines we use the following settings: +dumb In this setting, the system produces 10 suggestions, +with class no-suggestion at the top and FAQ IDs 1 to 9 +as the subsequently ranked suggestions as output. +random In this setting, the system produces at random 10 +classes as output without repetition. The output may contain +one of the FAQ IDs or the no-suggestion class. +Additionally. +we +employed +BM25 +(Robertson +and +Zaragoza 2009) based text search ranking as a baseline +method. In this method we searched the input query string +against the FAQ database and used the ranked list of results. +To produce strong performance, we employ Dense Pas- +sage Retrieval (Karpukhin et al. 2020) techniques . As a +baseline, we use fb-multiset-english, which is a set of en- +coders 4 that were pre-trained on English Natural Questions +(Kwiatkowski et al. 2019), TriviaQA (Joshi et al. 2017), We- +bQuestions (Berant et al. 2013), and CuratedTREC (Baudiˇs +and ˇSediv´y 2015). +Finally, we use pre-trained context and query encoders +for the German language provided by DeepSet +5 and +fine-tune them on our dataset for 100 epochs with a learning +rate of 1e-05 with the Adam optimizer. We use random +sampling for choosing negative examples during training. +We choose the best performing model based on mrr@10 +on the dev split. We used deepset-german encoders, which +come comes from DeepSet and is trained on GermanQuAD +4facebook/dpr-ctx encoder-multiset-base +5https://www.deepset.ai/germanquad +Figure 3: Distribution of conversation topics in the dataset. +Figure 4: The length of each conversation +(M¨oller, Risch, and Pietsch 2021) dataset. +For query, we concatenate 4 consecutive utterances of +conversation and consider it the input to the model. For con- +text, we concatenate the question and answer for each FAQ +and make the DPR model consider these as the passages +database from which it has to rank the best possible FAQ. +Evaluation Metrics +As our metric, we choose the Mean Reciprocal Rank +(MRR). For each query candidate, the model produces an +MRR, which is the reciprocal of the position of the correct +FAQ in the ranked list. We consider only the top 10 candi- +dates, and hence, if the correct candidate is not in the top 10, +we consider the MRR as 0. We compute the eventual MRR +by taking a mean of the MRR of each query sample in the +test set. +4 + +payment +project planning +organisation +insurance +conditions +browser +benefits +location +project +certificate +breach of contract +price +postprocessing +time +supervisor +scholarship +preparation +application +0 +100 +200 +300 +400Conversation ID +0 +100 +200 +300 +400 +500 +600 +TurnsWe evaluate separate MRRs for those utterances which have +empty FAQ suggestions as gold annotation, and the ones +which have non-empty FAQ gold suggestions. As explained +before, the task of the AI agent is not just to recommend the +right FAQ when needed, but it must also remain silent when +no FAQ is suitable. We measure the ability of AI agent on +both these tasks in Table 1. +Experimental Setup +Since a large percentage of the utterances (79.1%) belongs +to the no-suggestion class we experiment with differ- +ent mixture of faq classes and the no-suggestion class. +During preparation of train and dev sets to be fed to the +model, we calibrate the ratio of no-suggestion utter- +ances differently as follows: +mean In this setting, we compute the mean of the frequency +of the faq classes and include these many samples of ran- +domly chosen no-suggestion utterances as input. +highest-freq In this setting, we find the most frequent faq +class and include the same number of no-suggestion +class samples. +sum In this setting, the number of samples of the utterances +in no-suggestion class is equal to the sum of the num- +ber of utterances in all the faq classes combined. +original In this setting we consider all utterances as in- +put which leads to roughly 80:20 class imbalance of +no-suggestion class and the faq classes. +It must be noted that in all the above settings, we +always include every faq class utterance. For input +to the model we concatenate 4 consecutive utterances +{uk−3, uk−2, uk−1, uk} for each utterance uk. When con- +catenating the utterances, we also append the sender name +to the beginning of each utterance. +Model/Setting +no-suggestion +faq +dumb +1.0 +0.02 +random +0.04 +0.06 +BM25 +0 +0.27 +fb-multiset-english +mean +0.12 +0.40 +highest-freq +0.35 +0.48 +sum +0.81 +0.44 +original +0.96 +0.33 +deepset-german +mean +0.12 +0.58 +highest-freq +0.42 +0.57 +sum +0.84 +0.50 +original +0.95 +0.38 +Table 1: MRR@10 values for different models and settings +on test split of dataset +Results +We first analyse the baseline results from Table 1 +: +The +dumb +setting +achieves +perfect +MRR +in +the +no-suggestion category since in this setting the AI +agent chooses ’silence’ as the top ranked candidate for all +turns. However it produces extremely poor results for turns +that do require suggestions, since there is no intelligence or +logic built in to his setting when fetching FAQ items. This +also highlights why we need to evaluate our system on two +different classes. If we had computed a singular MRR score +for all turns, a model which remains silent all the time would +score high accuracy. The random setting achieves poor per- +formance in both categories. The BM25 setting produces 0 +MRR in no-suggestion class because there is no way +to ask a text search method to not return any results. It al- +ways fetches some set of results, and in effect, is unable to +produce silence as output. +The Deep Passage Retrieval approaches using the +deepset-germandpr set of models perform the best, +which comes as no surprise since these encoders were pre- +trained on German QA datasets, and further fine-tuned on +our dataset. In comparison fb-multiset-english per- +forms worse since the encoders are not aware of the German +language. We find that among the different settings of vary- +ing proportions of the inclusion of no-suggestion class +in the input, the sum setting produces a balanced perfor- +mance in the two categories of no-suggestion and faq. +Another notable point in the table is the performance of the +dumb model which always produces no-suggestion as +output hence achieving perfect MRR@10 of 1.0 in the rel- +evant samples, but it produces the worst results in the faq +classes, hence rendering it of little use to human agent. We +observe that as no-suggestion class performance im- +proves, faq class performance drops. This brings forth in- +teresting questions on how to calibrate the performance of +the model to reach a sweet spot for the human agent. An +MRR of 0.5 or greater for the faq classes means that the +right FAQ is generally either in the first or in the second +position, which is a positive contribution to lessen the hu- +man agent’s workload, since most user interface implemen- +tations for our scenario would display the top 3 FAQs to hu- +man agent together. It is, however, more important for the +no-suggestion MRR to be closer to 1.0, since the si- +lence class being ranked second still produces suggestions +that the human agent has to process, increasing noise for the +human agent. +Human Evaluation +To evaluate the usability aspects of the prototype and its in- +fluence on the task, we conducted interviews with 18 human +agents after usage. Additionally, we inspected their usage +behavior via screen recordings to supplement the qualita- +tive results. Overall, human agents indicated that they would +continue to use the prototype and highlighted that it is partic- +ularly helpful for agents who do not have much experience +in handling customers. During customer interactions, agents +sent on average 16 (SD: 5; Median: 14) messages during the +customer interaction. 17 agents used the FAQ answer sug- +gestions via the copy-to-chat-button at least three times. On +average, agents edited two (SD: 2; Median: 2) of the sug- +gested responses in the input field before sending them. +Overall, an average of six (SD: 2.5; Median: 7) sugges- +tions were used, whereby the detailed version via get-more- +info button (Mean: 3.7; SD: 2.6; Median: 4.5) was used more +5 + +frequently than the short version (Mean: 2.6; SD: 2.4; Me- +dian: 2). To receive alternative FAQ answer suggestions, the +discard-button was clicked on average 15 times (SD: 10.8; +Median: 15). The display of two suggestions and the op- +tion for additional explanatory information via the get-more- +info-button were perceived as helpful “so that you can think +in which direction you might go” (agent1). Agents experi- +enced relief through displayed suggestions and the majority +saved time making decisions, especially by using the copy- +to-chat-button: “ I just had to copy them, which affected the +speed” (agent14). 16 agents utilized the feedback function +on average four times, while nine people successfully pro- +vided feedback. However, agents expressed the need for an +adaptation of the feedback function, as it was unclear. Con- +cerning the recommendation of projects, the pressure to re- +call knowledge or search in parallel to the customer inter- +action was reduced as relevant information was presented. +Thereby, it “took out the uncomfortable part of working with +such a consultation, which is looking up stuff ” (agent16) +Limitations +The current solution suffers from the following limitations: +1) The web interface was developed for internal evaluation +purposes and is not available for general public use. 2) The +collection of the dataset suffers from class imbalance and +bias issues, since only a single person was involved in col- +lecting the conversations. 3) The feedback function of the UI +did not work as expected by the human agents. The human +agents expected the feedback regarding wrong suggestions +to be immediately learnt by the system, however during the +evaluation phase we did not re-train our models, or perform +on-line learning from the provided feedback. +Conclusion and Future Work +In this work we present a web interface for demonstrating +hybrid human-AI collaborative system that can handle cus- +tomer support queries. We show through machine based and +human based evaluations, that with the limited and imbal- +anced data we collected, we found appropriate methods to +train an AI agent that is able to provide appropriate assis- +tance to its human counterpart, which is the goal of our re- +search. +For future work, we wish to implement active on-line +learning from the human agent’s usage of the feedback fea- +ture in the UI. 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Recent Advances in Deep Learning Based Dialogue +Systems: A Systematic Survey. +Poser, M.; Wiethof, C.; Banerjee, D.; Shankar Subramanian, +V.; Paucar, R.; and Bittner, E. A. C. 2022. Let’s Team Up +with AI! Toward a Hybrid Intelligence System for Online +Customer Service. In Drechsler, A.; Gerber, A.; and Hevner, +A., eds., The Transdisciplinary Reach of Design Science Re- +search, 142–153. Cham: Springer International Publishing. +ISBN 978-3-031-06516-3. +Reddy, S.; Chen, D.; and Manning, C. D. 2019. +CoQA: +A Conversational Question Answering Challenge. Trans- +actions of the Association for Computational Linguistics, 7: +249–266. +Robertson, S.; and Zaragoza, H. 2009. +The Probabilistic +Relevance Framework: BM25 and Beyond. +Foundations +and Trends® in Information Retrieval, 3(4): 333–389. +Serban, I. V.; Lowe, R.; Henderson, P.; Charlin, L.; and +Pineau, J. 2015. A Survey of Available Corpora for Building +Data-Driven Dialogue Systems. +Wahlster, W. 1993. Verbmobil: Translation of Face-To-Face +Dialogs. In Proceedings of Machine Translation Summit IV, +127–136. Kobe, Japan. +Weizenbaum, J. 1966. ELIZA—a Computer Program for the +Study of Natural Language Communication between Man +and Machine. Commun. ACM, 9(1): 36–45. +7 + diff --git a/-dFLT4oBgHgl3EQfvC_J/content/tmp_files/load_file.txt b/-dFLT4oBgHgl3EQfvC_J/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8d10f1de42bb9e48e81dbbcd45e8a643174f897c --- /dev/null +++ b/-dFLT4oBgHgl3EQfvC_J/content/tmp_files/load_file.txt @@ -0,0 +1,563 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf,len=562 +page_content='The AAAI 2023 Workshop on Representation Learning for Responsible Human-Centric AI (R2HCAI) A system for Human-AI collaboration for Online Customer Support Debayan Banerjee* Mathis Poser* Christina Wiethof* Varun Shankar Subramanian Richard Paucar Eva A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Bittner Chris Biemann Universit¨at Hamburg, Hamburg, Germany {debayan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='banerjee,mathis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='poser,christina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='wiethof,eva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='bittner,chris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='biemann}@uni- hamburg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='de,{varunshankar55,rfpaucar}@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='com Abstract AI enabled chat bots have recently been put to use to answer customer service queries, however it is a common feedback of users that bots lack a personal touch and are often unable to understand the real intent of the user’s question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' To this end, it is desirable to have human involvement in the customer servicing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' In this work, we present a system where a human support agent collaborates in real-time with an AI agent to satisfactorily answer customer queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' We describe the user interaction elements of the solution, along with the machine learning techniques involved in the AI agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Introduction In the pursuit of operational efficiency, companies across the globe have been deploying automation technology aided by Artificial Intelligence (AI) for Online Customer Support (OCS) use cases 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' With the explosive growth of social me- dia usage, incoming customer queries have grown exponen- tially and to handle this growth, the use of proper technology is critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Some estimates say that by the year 2025, 95% of all customer interactions will be processed in some form by AI 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' However, AI in its present state is not advanced enough to completely replace human agents for most cus- tomer support scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Additionally, the complete replace- ment of human workforce by AI is a topic of active ethical and political debate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' For these reasons the development of a hybrid working environment is required, where both human agents and AI agents can co-operate to satisfy OCS require- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' In this work we briefly describe a web based user inter- face that allows a customer to interact with a human sup- port agent, where the human agent receives helpful sugges- tions in parallel from an AI agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' In subsequent sections, we elaborate further on the machine learning techniques used for the AI agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Our present work is a part of a project which aims to find ways of integrating AI agents into customer support based workflows, with an aim of reducing workload of human These authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' 1https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='gartner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='com/smarterwithgartner/4-key-tech- trends-in-customer-service-to-watch 2https://servion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='com/blog/what-emerging-technologies-future- customer-experience/ agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' It is one of the primary goals of the project not to entirely replace the human agent with AI, and instead find productive means of co-existence of the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' As a part of this project, an international volunteer-driven organisation, which organises internships and projects for students across the globe was involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' In this organisation, prospective stu- dents participate in text based chat with human agents, and typically enquire about available opportunities and how to participate in them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' The human agents in turn use their do- main expertise to provide the necessary information to the students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' All the students and human agents involved were resi- dents of Germany and hence the conversations were car- ried out in the German language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' After collecting the con- versations, an annotation phase was undertaken, where rele- vant utterances of the conversation were annotated with the corresponding FAQ IDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' When the conversations originally took place, there was no singular FAQ database in existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' For the purpose of this project, such a database was created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' This made it possible to annotate the utterances with relevant FAQ IDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' The goal of the dataset is to train an AI agent that can pas- sively listen to the ongoing conversation and make relevant suggestions visible only to the human agent, not to the stu- dent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' The human agent may then forward the suggested FAQ answer to the student, or decide not to do so if the quality of suggestion is poor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' The eventual goal is for the human agent to spend less time looking for the right answer in a Knowl- edge Base, and instead offload this task to the AI agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Later, a web UI was constructed, as described in the Web Interface section, that the human agent uses to interact with the student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' The student is not aware of the UI’s existence and is operating on a separate chat platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' The AI agent provides timely suggestions in this UI which is visible to the human agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Our scenario differs from conventional Conversational Question Answering (CQA) or Interactive Information Re- trieval (IIR) where the user interacts directly with the AI agent, and the AI agent is responsible for a response at each turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' In our case, the AI agent is in a passive listening role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' It observes the ongoing conversation between two humans, and makes suggestions that are only visible to the human agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Since the task of the AI agent is not just to suggest relevant FAQs but also to remain silent when no relevant 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='12158v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='AI] 28 Jan 2023 Figure 1: Screenshot of web based prototype FAQ is to be suggested, we evaluate both of these aspects in the evaluation section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' The user interface presented in this work has been pub- lished before (Poser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' The machine learning tech- niques used to train the AI agent are yet to be published, and hence a larger focus in this work is on the AI training aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Web Interface The web-based frontend in Figure 1 is labelled with cer- tain design features (DF) to be explained shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' The in- terface was implemented with Bootstrap and ReactJS while the backend API is hosted as a Python Flask app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' The inter- face greets humans agents with an avatar named Charlie that presents a brief usage explanation (DF1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' In addition, setting options for AI support and learning behavior are provided (DF2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' The integrated chat window is based on the open- source chat framework Rocket Chat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' The backend generates a ranked list of FAQ suggestions based on ML techniques to be described later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' In the frontend, two FAQ items - includ- ing theme and accuracy in percent - with the highest agree- ment are displayed (DF3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' The discard buttons can be used to sequentially display four additional FAQ suggestions with decreasing accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' The copy-to-chat buttons insert FAQ text into the input field of the chat window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Detailed infor- mation about a respective FAQ can be viewed via the get- more-info button (DF4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' With a counter, points are added (copy-to-chat) or subtracted (discard), if buttons are clicked (DF5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' A feedback field allows entering search terms to se- lect and submit a FAQ that matches the interaction (DF6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Based on customers’ chat messages, exact keyword-based text matching is performed to automatically record interests and suggest suitable projects from a database (DF7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Related Work The earliest dialogue systems, or chat-bots, were rule based (Weizenbaum 1966;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Colby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' 1972) and subsequently corpus based chat-bots were developed (Serban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' 2015) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' In recent times neural chat-bots are frequently encountered in day to day customer support scenarios (Ni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Recently, an interplay of human and AI collaboration in the process has been explored (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' However current research in this area is focused on the AI bot be- ing the first line of service, and only in the case of failures of the bot, a handover is initiated to a human agent, who plays a secondary role in the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' In contrast, our sce- nario makes the human agent the first line of support with the AI agent assisting in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' To train chat-bots, conversational QA datasets such as the Ubuntu corpus (Lowe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' 2015), CoQA (Reddy, Chen, and Manning 2019), DoQA (Campos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' 2020) and QuAC (Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' 2018) have made progress in providing the community with rich grounds for conversational research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' While CoQA relies on passages from broad domains such as children’s stories and science to retrieve answers, QuAC relies on Wikipedia articles to create conversations and answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' DoQA on the other hand, focuses on three specific domains of cooking, travel and movies from stack- exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' In scope of how our dataset is modelled, it is most similar to DoQA, which is a domain specific conversational dataset which also requires retrieval of the correct FAQ from a database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=" CoQA, DoQA and QuAC datasets are crowd-sourced and collected by the Wizard of 2 IntelligentSupportAgent 三 E Hi there, I'am Charlie your personal assistant!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Your Personal Settings?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' information and knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' You can control my settings anytime, To elevate Do you want my assistance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' on to learn more about my features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' May I learn from your conversations and interaction with me?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' DF2 On Happy to work with you!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=" DF1 Knowledge DF3 Charlie's Suggestions?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Feedback?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' FAQ Theme: Find here the right question, and then press send button.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='Copy to chat ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='Dscardo ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='DF6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='Get mareinfo ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='FAQ Theme: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='DF4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='Copy ochat ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='DiacardO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='Get moreinfoO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content="Charlie's Explanations (Get more info) " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='Ccopyfo chat ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='Points for Charlie ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='DF5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content="EP's Interest " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='Projects ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='Where ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='DF7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content="+ Charlie's Suggestions " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='Indicate Location (country) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='Small text ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='Copy to chat ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='When ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='Discarda ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='Indicate month ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='Small teat ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='Message ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='What ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='Copy to chat ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='Small text ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='DiscardQ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='Indicate what type of projectFigure 2: A sample conversation from the dataset with relevant corresponding FAQ annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' The text in red is English translation of the conversation for the purpose of this paper, and not a part of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Oz method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' On the contrary, our dataset consists of genuine conversations between two humans whose sole purpose is to find the best internship possible for the student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' During the conversations, neither of the parties were aware of the need to form an annotated dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Hence, our dataset has no artificial aspects in the flow of conversation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' The Dortmunder Chat Korpus (Beißwenger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' 2013) and The Verbmobil (Wahlster 1993) project provide German conversational corpus but they do not address the Question Answering or Information Retrieval domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Recently, the GermanQuAD and GermanDPR (M¨oller, Risch, and Pietsch 2021) projects from DeepSet have enabled access to Transformer based models trained on the German text, which we make use of in our evaluation section, however the dataset they are based on is in the form of Questions and Answers, and not conversational in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Dataset Creation To train the AI agent, a conversational dataset had to be con- structed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' For this purpose, the conversations were carried out on the popular mobile application WhatsApp 3, where both the human agent and the student were on Whatsapp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' The Web Interface described in the previous section was not in- cluded in this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' The conversations centered around topics such as how to register for a project, which projects are available in a given location, and whether there will be certifications available at the end etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' The chats were ex- tracted using the export functionality of WhatsApp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' The 3https://play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='com/store/apps/details?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='id=com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='whatsapp conversations have been collected over a period of two years, between 2018 and 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' In some cases, an individual con- versation may also span over a duration of several months, where the student and the human agent re-established con- tact after a gap of more than a few days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Such information is visible through the inclusion of the timestamp field in the dataset for each message that is exchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Relevant consent for releasing their conversations was col- lected from the participating students and agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' More- over, the identities of the participants and the organisation are pseudo-anonymised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Instead of the names of the partici- pants, they are given a numerical name such as KundeSech- sundzwanzig, which stands for Customer 26 in German.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' The human agent is represented by the term Mitarbeiter which stands for employee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' A single human agent handled all the 26 conversations on WhatsApp over a period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' When the conversa- tions were carried out between 2018-2020, no single FAQ database existed at the organisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' The human agent in- stead used relevant domain expertise and experience within the organisation, and referred to a set of disjoint sources of information when the chats took place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Later in 2021, the hu- man agent and a fellow domain expert colleague compiled a single FAQ database that covers most of the issues discussed in the conversations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Specific turns of the conversations were manually annotated with relevant FAQs by the human agent and then verified by the domain expert colleague.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Dataset Analysis Chats and FAQs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' As depicted in Figure 4 the 26 collected conversations vary in length ranging from 22 utterances 3 Mitarbeiter : Hey!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' ich bin Mitarbeiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Du hast dich bei FAQ 1 uns angemeldet und ich wurde gerne mit dir daruber sprechen / telefonieren :).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Wann hattest du denn dafur "Question":"wann kann ich ein projekt machen?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='" Zeit?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Employee : Hey!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='I am an employee here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='You have "When can Idoaproject?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='" registered with us and I would like to talk to you about it "Answer":"projekte sind jederzeitmoglich", KundeVierzehn : Guten Morgen, Ich interessiere mich "Projects can be done atany time” sehr fur Projekte in der Turkei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Wenn der Start im Januar moglich ist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Customer Fourteen: Good Morning,Iam very interested in projects in Turkey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' If the start is possible in January.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' FAQ55 Mitarbeiter/Employee : https://.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='org/opportunity/984743 "Question": "was mache ich nun nachdem ich mich https://.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='org/opportunity/1002581 beworbenhabe?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='" KundeVierzehn : Hallo Mitarbeiter, ich habe mich gerade \'what do I doafter I haveapplied?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='" beworben.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Customer Fourteen: Hello Employee, I just applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' "Answer":"prozess:bewerben-kontaktmit auslandspartner-akzeptiert-vertrag und gebuhr Mitarbeiter : Ah super!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Ich kummere mich darum dass du approved", schnell kontaktiert wirst :) "process:apply-contactwithforeign Employee: Ah great!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=" I will make sure that you will be contacted quickly :) partner -accepted-contract and fee -approved'to 607 utterances, with an average of 239 utterances per conversation." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' The entire set of conversations consists of 6,219 utterances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='9 % of the utterances are annotated with the relevant FAQ ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' A significant portion of the dataset consists of chit-chat or other non-specific topics where no suggestion is supposed to be made by the AI agent to the human agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Since certain topics in the chat are discussed more often than others, as seen in Figure 3, the distribution of relevant annotated FAQ IDs also is imbalanced with FAQ ID 71 being the most frequent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' FAQ 71 pertains to the procedure of registering online for projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' We have split the dataset into train, dev and test splits in roughly 70:10:20 ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' The train, dev and test splits have 17, 3 and 6 conversations, respectively, consisting of 3,693 , 891 and 1,635 utterances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Experimental Setting Task Definition We define the task with the following inputs: current utter- ance uk, the set of FAQs F, and the history of utterances so far {u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='., uk−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' The task for the model is to rank the correct FAQ item from F to the top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' If for a given utterance no FAQ is appropriate, the model must produce as the top- ranked output a special class that denotes absence of FAQ suggestion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' We hereby call this class no-suggestion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Models As baselines we use the following settings: dumb In this setting, the system produces 10 suggestions, with class no-suggestion at the top and FAQ IDs 1 to 9 as the subsequently ranked suggestions as output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' random In this setting, the system produces at random 10 classes as output without repetition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' The output may contain one of the FAQ IDs or the no-suggestion class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Additionally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' we employed BM25 (Robertson and Zaragoza 2009) based text search ranking as a baseline method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' In this method we searched the input query string against the FAQ database and used the ranked list of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' To produce strong performance, we employ Dense Pas- sage Retrieval (Karpukhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' 2020) techniques .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' As a baseline, we use fb-multiset-english, which is a set of en- coders 4 that were pre-trained on English Natural Questions (Kwiatkowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' 2019), TriviaQA (Joshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' 2017), We- bQuestions (Berant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' 2013), and CuratedTREC (Baudiˇs and ˇSediv´y 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Finally, we use pre-trained context and query encoders for the German language provided by DeepSet 5 and fine-tune them on our dataset for 100 epochs with a learning rate of 1e-05 with the Adam optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' We use random sampling for choosing negative examples during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' We choose the best performing model based on mrr@10 on the dev split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' We used deepset-german encoders, which come comes from DeepSet and is trained on GermanQuAD 4facebook/dpr-ctx encoder-multiset-base 5https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='deepset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='ai/germanquad Figure 3: Distribution of conversation topics in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Figure 4: The length of each conversation (M¨oller, Risch, and Pietsch 2021) dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' For query, we concatenate 4 consecutive utterances of conversation and consider it the input to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' For con- text, we concatenate the question and answer for each FAQ and make the DPR model consider these as the passages database from which it has to rank the best possible FAQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Evaluation Metrics As our metric, we choose the Mean Reciprocal Rank (MRR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' For each query candidate, the model produces an MRR, which is the reciprocal of the position of the correct FAQ in the ranked list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' We consider only the top 10 candi- dates, and hence, if the correct candidate is not in the top 10, we consider the MRR as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' We compute the eventual MRR by taking a mean of the MRR of each query sample in the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' 4 payment project planning organisation insurance conditions browser benefits location project certificate breach of contract price postprocessing time supervisor scholarship preparation application 0 100 200 300 400Conversation ID 0 100 200 300 400 500 600 TurnsWe evaluate separate MRRs for those utterances which have empty FAQ suggestions as gold annotation, and the ones which have non-empty FAQ gold suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' As explained before, the task of the AI agent is not just to recommend the right FAQ when needed, but it must also remain silent when no FAQ is suitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' We measure the ability of AI agent on both these tasks in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Experimental Setup Since a large percentage of the utterances (79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='1%) belongs to the no-suggestion class we experiment with differ- ent mixture of faq classes and the no-suggestion class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' During preparation of train and dev sets to be fed to the model, we calibrate the ratio of no-suggestion utter- ances differently as follows: mean In this setting, we compute the mean of the frequency of the faq classes and include these many samples of ran- domly chosen no-suggestion utterances as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' highest-freq In this setting, we find the most frequent faq class and include the same number of no-suggestion class samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' sum In this setting, the number of samples of the utterances in no-suggestion class is equal to the sum of the num- ber of utterances in all the faq classes combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' original In this setting we consider all utterances as in- put which leads to roughly 80:20 class imbalance of no-suggestion class and the faq classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' It must be noted that in all the above settings, we always include every faq class utterance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' For input to the model we concatenate 4 consecutive utterances {uk−3, uk−2, uk−1, uk} for each utterance uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' When con- catenating the utterances, we also append the sender name to the beginning of each utterance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Model/Setting no-suggestion faq dumb 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='02 random 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='06 BM25 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='27 fb-multiset-english mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='40 highest-freq 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='48 sum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='44 original 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='33 deepset-german mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='58 highest-freq 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='57 sum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='50 original 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='38 Table 1: MRR@10 values for different models and settings on test split of dataset Results We first analyse the baseline results from Table 1 : The dumb setting achieves perfect MRR in the no-suggestion category since in this setting the AI agent chooses ’silence’ as the top ranked candidate for all turns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' However it produces extremely poor results for turns that do require suggestions, since there is no intelligence or logic built in to his setting when fetching FAQ items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' This also highlights why we need to evaluate our system on two different classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' If we had computed a singular MRR score for all turns, a model which remains silent all the time would score high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' The random setting achieves poor per- formance in both categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' The BM25 setting produces 0 MRR in no-suggestion class because there is no way to ask a text search method to not return any results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' It al- ways fetches some set of results, and in effect, is unable to produce silence as output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' The Deep Passage Retrieval approaches using the deepset-germandpr set of models perform the best, which comes as no surprise since these encoders were pre- trained on German QA datasets, and further fine-tuned on our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' In comparison fb-multiset-english per- forms worse since the encoders are not aware of the German language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' We find that among the different settings of vary- ing proportions of the inclusion of no-suggestion class in the input, the sum setting produces a balanced perfor- mance in the two categories of no-suggestion and faq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Another notable point in the table is the performance of the dumb model which always produces no-suggestion as output hence achieving perfect MRR@10 of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='0 in the rel- evant samples, but it produces the worst results in the faq classes, hence rendering it of little use to human agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' We observe that as no-suggestion class performance im- proves, faq class performance drops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' This brings forth in- teresting questions on how to calibrate the performance of the model to reach a sweet spot for the human agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' An MRR of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='5 or greater for the faq classes means that the right FAQ is generally either in the first or in the second position, which is a positive contribution to lessen the hu- man agent’s workload, since most user interface implemen- tations for our scenario would display the top 3 FAQs to hu- man agent together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' It is, however, more important for the no-suggestion MRR to be closer to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='0, since the si- lence class being ranked second still produces suggestions that the human agent has to process, increasing noise for the human agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Human Evaluation To evaluate the usability aspects of the prototype and its in- fluence on the task, we conducted interviews with 18 human agents after usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Additionally, we inspected their usage behavior via screen recordings to supplement the qualita- tive results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Overall, human agents indicated that they would continue to use the prototype and highlighted that it is partic- ularly helpful for agents who do not have much experience in handling customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' During customer interactions, agents sent on average 16 (SD: 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Median: 14) messages during the customer interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' 17 agents used the FAQ answer sug- gestions via the copy-to-chat-button at least three times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' On average, agents edited two (SD: 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Median: 2) of the sug- gested responses in the input field before sending them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Overall, an average of six (SD: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Median: 7) sugges- tions were used, whereby the detailed version via get-more- info button (Mean: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' SD: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Median: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='5) was used more 5 frequently than the short version (Mean: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' SD: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Me- dian: 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' To receive alternative FAQ answer suggestions, the discard-button was clicked on average 15 times (SD: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content='8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Median: 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' The display of two suggestions and the op- tion for additional explanatory information via the get-more- info-button were perceived as helpful “so that you can think in which direction you might go” (agent1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Agents experi- enced relief through displayed suggestions and the majority saved time making decisions, especially by using the copy- to-chat-button: “ I just had to copy them, which affected the speed” (agent14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' 16 agents utilized the feedback function on average four times, while nine people successfully pro- vided feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' However, agents expressed the need for an adaptation of the feedback function, as it was unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Con- cerning the recommendation of projects, the pressure to re- call knowledge or search in parallel to the customer inter- action was reduced as relevant information was presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Thereby, it “took out the uncomfortable part of working with such a consultation, which is looking up stuff ” (agent16) Limitations The current solution suffers from the following limitations: 1) The web interface was developed for internal evaluation purposes and is not available for general public use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' 2) The collection of the dataset suffers from class imbalance and bias issues, since only a single person was involved in col- lecting the conversations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' 3) The feedback function of the UI did not work as expected by the human agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' The human agents expected the feedback regarding wrong suggestions to be immediately learnt by the system, however during the evaluation phase we did not re-train our models, or perform on-line learning from the provided feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Conclusion and Future Work In this work we present a web interface for demonstrating hybrid human-AI collaborative system that can handle cus- tomer support queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' We show through machine based and human based evaluations, that with the limited and imbal- anced data we collected, we found appropriate methods to train an AI agent that is able to provide appropriate assis- tance to its human counterpart, which is the goal of our re- search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' For future work, we wish to implement active on-line learning from the human agent’s usage of the feedback fea- ture in the UI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' We would also like to collect a larger and more balanced dataset for future iterations of the AI agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' Acknolwedgements The research was financed with funding provided by the German Federal Ministry of Education and Research and the European Social Fund under the ”Future of work” program (INSTANT, 02L18A111).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=' References Baudiˇs, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFLT4oBgHgl3EQfvC_J/content/2301.12158v1.pdf'} +page_content=';' metadata={'source': 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manuscript no. 43751corr +©ESO 2023 +January 9, 2023 +Framework for the architecture of exoplanetary systems +I. Four classes of planetary system architecture⋆ +Lokesh Mishra1, 2 , Yann Alibert1 , Stéphane Udry2 +, and Christoph Mordasini1 +1 Institute of Physics, University of Bern, Gesellschaftsstrasse 6, 3012 Bern, Switzerland +e-mail: exomishra@gmail.com +2 Geneva Observatory, University of Geneva, Chemin Pegasi 51b, 1290 Versoix, Switzerland +Received 10 04 2022; accepted 05 12 2022 +ABSTRACT +We present a novel, model-independent framework for studying the architecture of an exoplanetary system at the system level. This +framework allows us to characterise, quantify, and classify the architecture of an individual planetary system. Our aim in this en- +deavour is to generate a uniform systematic method to study the arrangement and distribution of various planetary quantities within a +single planetary system. We propose that the space of planetary system architectures be partitioned into four classes: similar, mixed, +anti-ordered, and ordered. A central aim of this paper is to introduce these four architecture classes. We applied our framework to +observed and synthetic multi-planetary systems, thereby studying their architectures of mass, radius, density, core mass, and the core +water mass fraction. We explored the relationships between a system’s (mass) architecture and other properties. Our work suggests +that: (a) similar architectures are the most common outcome of planet formation; (b) internal structure and composition of planets +shows a strong link with their system architecture; (c) most systems inherit their mass architecture from their core mass architecture; +(d) most planets that started inside the ice line and formed in-situ are found in systems with a similar architecture; and (e) most +anti-ordered systems are expected to be rich in wet planets, while most observed mass ordered systems are expected to have many dry +planets. We find, in good agreement with theory, that observations are generally biased towards the discovery of systems whose den- +sity architectures are similar, mixed, or anti-ordered. This study probes novel questions and new parameter spaces for understanding +theory and observations. Future studies may utilise our framework to not only constrain the knowledge of individual planets, but also +the multi-faceted architecture of an entire planetary system. We also speculate on the role of system architectures in hosting habitable +worlds. +Key words. Planetary systems – Planets and satellites: detection – Planets and satellites: formation – Planets and satellites: physical +evolution +1. Introduction +Over the last 25 years, our knowledge of exoplanetary astro- +physics has improved dramatically. While the first decade was +marked by sensational discoveries of individual exoplanets (e.g. +Vidal-Madjar et al. 2003; Santos et al. 2004; Bouchy et al. 2005; +Udry et al. 2007; Kalas et al. 2008; Charbonneau et al. 2009; +Snellen et al. 2010), we are now in an age of population-level ex- +oplanetary statistics (for a recent review, see Zhu & Dong 2021). +We now know that (statistically) almost every star hosts a planet +and one in two Solar-like stars host a rocky planet in their habit- +able zone (Hsu et al. 2019; Bryson et al. 2021). Moreover, many +exoplanet-hosting stars have multiple planets orbiting them. +The arrangement of multiple planets and the collective dis- +tribution of their physical properties around host star(s) char- +acterises the architecture of a planetary system (Mishra et al. +2021). Exoplanets in some multi-planetary systems are thought +to behave like ‘peas in a pod’ (Lissauer et al. 2011; Ciardi et al. +2013; Millholland et al. 2017; Weiss et al. 2018). The peas in a +pod trend consists of the following correlations: size, whereby +adjacent exoplanets are either similar or ordered in size (i.e. the +outer planet is larger); mass, whereby adjacent exoplanets are ei- +⋆ Catalogue of observed planetary systems used in this work is avail- +able online at https://cdsarc.cds.unistra.fr/cgi-bin/qcat? +J/A+A/. +ther similar or ordered in mass; spacing, whereby for a system +with three or more planets, the spacing between an adjacent pair +of exoplanets is similar to the spacing between the next consec- +utive pair; packing, whereby smaller planets tend to be packed +together closely and larger planets are in wider orbital configu- +rations. +While the statistical method used by Weiss et al. (2018) has +been debated (Zhu 2020; Murchikova & Tremaine 2020; Weiss +& Petigura 2020), support for the astrophysical nature of the +peas in a pod correlations (as opposed to emerging from detec- +tion biases) has emerged from theoretical studies and numerical +simulations (Adams 2019; Adams et al. 2020; He et al. 2019; +He et al. 2021; Mulders et al. 2020). In particular, Mishra et al. +(2021) reproduced the observations from Weiss et al. (2018) us- +ing a model of planet formation and evolution (the Bern Model +Emsenhuber et al. (2021a,b)) and a model for the detection bi- +ases of a Kepler-like transit survey (using KOBE). We showed +that when nature’s underlying exoplanetary population (consist- +ing of detected and undetected exoplanets) resembles peas in a +pod, then a population of transiting exoplanets will have correla- +tions that are consistent with those found by Weiss et al. (2018). +In addition, Mishra et al. (2021) suggested that the four trends +are not independent of each other. The size correlations seem to +emerge from the mass correlations, while the mass and packing +Article number, page 1 of 28 +arXiv:2301.02374v1 [astro-ph.EP] 6 Jan 2023 + +A&A proofs: manuscript no. 43751corr +trends could combine to give rise to the spacing trend. The peas +in a pod trends are amenable to a unification. +Most of the current studies on this topic utilise statistical +correlation coefficients at the population level, that is, the cor- +relation is measured for adjacent planetary pairs from several +planetary systems. While useful in terms of testing the existence +(or otherwise) of architecture trends, these coefficients may have +limited utility for analysing the architecture of a single planetary +system. Being statistical in nature, a reliable estimate of these +coefficients requires large datasets - which seems difficult for a +single system. Although there are some planetary system-level +studies (Kipping 2018; Alibert 2019; Mishra et al. 2019; Gilbert +& Fabrycky 2020; Bashi & Zucker 2021, discussed in Sect. 3.1), +the current literature lacks a prescription for uniformly assessing +the multi-faceted architectures of several quantities (e.g. mass +architecture, radius architecture, or eccentricity architecture) for +a single planetary system. +We seek a framework that allows us to characterise the ar- +chitecture of an individual planetary system. Our motivations +for developing such a framework arise from questions related +to: formation, such as the extent to which a system’s architec- +ture is shaped by initial conditions (i.e. the environment in and +around the star and protoplanetary disk formation regions) (Jin +& Li 2014; Safsten et al. 2020); evolution, the role of physical +processes such as orbital migration or giant impacts in shaping +the final architecture of planetary systems (Mulders et al. 2020); +identification, which particular stars host planets that resemble +peas in a pod, and, in particular, whether the planets in systems +like TOI-178 (Leleu et al. 2021), Trappist-1 (Agol et al. 2021), +or 55 Cancri (Bourrier et al. 2018) show mass/size similarities; +other architectures, we know that there are many planetary sys- +tems that do not follow the peas in a pod architecture (e.g. the +Solar System). Overall, it is not obvious how the architecture of +any individual planetary system should be uniformly assessed. +In this series of papers, we propose a framework for examin- +ing the architecture of planetary systems at the system level. The +philosophy behind system level analysis is to consider the en- +tire planetary system as a single unit of a physical system. This +framework allows us to not only quantify, compare, and inves- +tigate a system’s architecture, but also offers some unexpected +benefits. As it turns out, the framework allows for a conceptu- +ally intuitive partitioning of the space of possible architectures. +We label the four classes of planetary system architectures as: +similar, ordered, anti-ordered, and mixed. In this way, our work +extends the trends initiated by the notion of peas in a pod ar- +chitecture. Furthermore, we verify the unification of the peas +in a pod correlations proposed in Mishra et al. (2021). We find +that, Similar architectures are the most common type of plane- +tary system architectures and their high occurrence explains why +the intra-system radius uniformity was already observable from +the first four months of Kepler data (Lissauer et al. 2011). +Our framework engenders novel questions. For instance, +if nature produces distinct classes of architecture in multi- +planetary systems, then what is the frequency or occurrence rates +of these architecture classes? How does the occurrence of an ar- +chitecture class depend on stellar and protoplanetary disk en- +vironment? How does the architecture of a system evolve over +time? What is the role of stellar evolution, protoplanetary disk +interactions, and planet formation in shaping the final architec- +ture? How is a planet’s internal composition related to the sys- +tem’s architecture? Or does the ability of a planet to host life +depends on the architecture of the planetary system? In this se- +ries of papers, we explore these questions. Although the num- +ber of multi-planetary systems is low today, this may change +in the next few decades. Thanks to large survey missions such +as PLATO (Rauer et al. 2014), GAIA (Gaia Collaboration et al. +2016), TESS (Ricker et al. 2015), LIFE (Quanz et al. 2022), and +others, the growing number of known multi-planetary systems +will allow for a better understanding to emerge. We hope our +work encourages observers to dedicate more observation time +to detecting planets within a known planetary system, that is, in +finding multi-planetary systems. +The architecture classification scheme proposed in this pa- +per is a model-independent framework. To demonstrate our clas- +sification framework and explore its consequences, we applied +our framework to simulated planetary systems. To illustrate our +framework on real systems, we also applied our framework to +observed exoplanetary systems. We emphasise that while the re- +sults emerging from the application of our framework on these +datasets may suffer from some limitations (arising from theoret- +ical modelling or detection biases for observed systems); how- +ever, the concept of our architecture classification scheme, being +model-independent, does not share these limitations. In this pa- +per, we present the catalogues of planetary systems we apply our +framework to in Sect. 2, along with a newly curated catalogue of +observed exoplanetary systems and simulated planetary systems, +using the Bern Model. We introduce our framework in Sect. 3. +In Sect. 4, the characteristics of the architecture classes are dis- +cussed. We explore the link between the internal composition +of planets and the system architecture class in Sect. 5. Then, in +Sect. 6, we speculate on how habitability could depend on the +architecture of planetary systems. Our conclusions are given in +Sect. 7. +In a companion paper, we investigate the formation path- +ways, i.e. the role of initial conditions and physical processes +in shaping the final architecture (Mishra et al. (2023) referred to +as Paper II). Our work demonstrates that the processes of planet +formation and evolution are imprinted on the entire system-level +architecture. We find that protoplanetary disks with low solid- +mass give rise to planetary systems endowed with a mass similar- +ity. On the other hand, massive disks and high metallicity often +lead to mass Ordered, Anti-Ordered, or Mixed system architec- +tures. Planet-planet and planet-disk interactions play a decisive +role in shaping these three architectures. +2. Catalogues +2.1. Theoretical dataset: Bern Model +In this series of works, we demonstrate our architecture frame- +work by analysing the architecture of synthetic planetary sys- +tems. These systems were numerically computed using the Gen- +eration III Bern Model of planet formation and evolution (Em- +senhuber et al. 2021a,b) that is based on the core-accretion +paradigm of planet formation (Pollack et al. 1996; Alibert et al. +2004, 2005). The model follows the growth of protoplanetary +embryos embedded in a protoplanetary disk of gas and solids +around a solar-type star. A diverse range of physical processes +are simultaneously occurring and coherently computed in this +1D star-disk-embryo system. These include: stellar and disk +physics (evolution of and interaction between star and viscous +disk, condensation of volatile and refractory species, etc.), plane- +tary formation physics (accretion of planetesimals and gases, in- +ternal structure calculations, etc.), and additional physics (orbital +and tidal migration, planet-planet N-body interactions, planet- +disk interactions, atmospheric escape, deuterium fusion, etc.). +We describe these physical processes in Sect. A and a descriptive +summary of these processes is provided in Mishra et al. (2021, +Article number, page 2 of 28 + +L. Mishra et al.: Architecture Framework I – Four classes of planetary system architecture +Fig. 1. Mass-distance diagram. This figure shows the masses and the +distances of planets in all catalogues used in this study. Shaded regions +show the parameter space spanned by synthetic planets observed via +radial velocity surveys (Bern RV Multis), transit surveys (Bern KOBE +Multis), and ongoing missions (Bern Compact Multis). The parameter +space for Bern KOBE Multis has been mapped from its original radius- +period plane. +in particular, Fig. 1 and Sections 2, 3, and Appendix A). More +details can also be found in (Emsenhuber et al. 2021a,b). +We synthesised 1000 planetary systems, each starting with +100 lunar mass protoplanetary embryos, wherein the following +initial conditions were varied: mass of protoplanetary gas disk, +photo-evaporation rate, dust-to-gas ratio, disk inner edge, and +the starting location of embryos. In Fig. 1, we show all synthetic +planets on the mass-distance diagram. For each synthetic plane- +tary system failed embryos, objects with mass less than 0.1M⊕, +were removed from further analysis1. +Three observationally motivated catalogues were prepared +from the synthetic dataset. This allowed us to facilitate a compar- +ison of the architecture from observed planetary systems with the +synthetic planetary systems and to make predictions. The param- +eter space spanned by the planets in these catalogues is shown in +Fig. 1. These catalogues are as follows: +Bern RV Multis: We assume a radial velocity (RV) survey +which can find planets with periods ≤ 15 yr and semi-amplitude +KRV ≥ 20 cm/s. These numbers are motivated by (a) long- +running RV surveys such as the HARPS survey (Mayor et al. +2003, 2011) and the California Legacy Survey (Rosenthal et al. +2021b; Fulton et al. 2021); (b) current precision achieved by +ESPRESSO (Lillo-Box et al. 2021; Netto et al. 2021); and (c) +making predictions for future RV surveys. Such RV detectable +synthetic planetary systems with four or more planets form the +1 As long as the mass threshold for failed embryos is kept under 0.1M⊕, +the results presented in this paper are not sensitive to the threshold limit. +We removed these small objects since they (a) failed to grow as massive +planets, (b) are insignificant to the dynamical evolution of the system, +and (c) are currently unobservable in exoplanetary systems. All results +arising from the Bern RV Multis, Bern KOBE Multis, and Bern Com- +pact Multis are insensitive to these failed embryos. +Bern RV Multis catalogue, which includes 3 828 planets around +565 stars. +Bern KOBE Multis: We assume a Kepler-like transit survey +which continuously observes 2 × 105 stars for 3.5 yr (Thompson +et al. 2018). A planet which transits three or more times and pro- +duces a transit S/N of 7.1 or more is considered detectable. The +reliability and completeness of such a survey is replicated and +those synthetic planets which would have been vetted as ‘plan- +etary candidates’ by the Kepler Robovetter (Thompson et al. +2018), are kept. Such transiting synthetic planetary systems with +four or more planets form the Bern KOBE Multis catalogue. +KOBE was developed and introduced in Mishra et al. (2021). +There are 6 715 planets around 1283 stars in this catalogue. +Bern Compact Multis: Ongoing transit missions such as +CHEOPS and TESS have been successful in characterising com- +pact multi-planetary systems, such as TOI-178 (Leleu et al. +2021) and TOI-561 (Lacedelli et al. 2021). Inspired by these dis- +coveries, we investigated the architecture of compact planetary +systems simulated by the Bern Model. Our aim is to understand +the architecture and make predictions for such systems based on +the core-accretion paradigm (Pollack et al. 1996; Alibert et al. +2004, 2005). All planets with periods of ≤ 100 d and masses of +≥ 0.1 M⊕ were retained. Synthetic planetary systems, in this pa- +rameter space, with four or more planets form the Bern Compact +Multis catalogue, with 2 412 planets around 400 stars included. +2.2. Observational dataset: A new catalogue +To demonstrate our framework on observed exoplanetary sys- +tems, we have curated a new catalogue of known multi-planetary +systems2. A salient feature of this catalogue (and the philosophy +behind this work) is its focus on considering planetary systems +as a single unit of a physical system. Unlike focussing on in- +dividual exoplanets or a single detection technique, our aim is +to study the planetary system as a whole. There are two serious +challenges to this endeavour. Firstly, the biases present in de- +tection methods tend to prevent a complete, reliable picture of +an exoplanetary system from emerging (either via undetected or +mischaracterised planets). Secondly, detecting planets on long +orbital periods requires long-term, repeated observations, which +is considerably challenging. We hope that upcoming missions +and future surveys can mitigate these difficulties. +We included a planetary system in our catalogue if: (a) it has +at least four known planets and (b) masses are available for at +least four planets. For example, Kepler-33, a five planet system, +is included because mass measurements are available for four +of its planets3. The criterion of requiring minimum four plan- +ets emerges due to (a) the requirement for enough planets for +adequately characterising the architecture and (b) because for +systems with lower number of planets, it is perhaps difficult to +uniformly assess whether the low multiplicity is an outcome of +natural processes or detection biases. To keep the comparison be- +tween observations and theory uniform, all catalogues in this se- +ries of works only consider planetary systems with four or more +planets. The architecture framework can, however, handle two- +or three-planet systems as well. To make this catalogue useful +to the wider community and enable future studies, we gathered +several key stellar and exoplanetary properties. For host stars, we +report the mass, radius, luminosity, effective temperature, metal- +licity, age, and distance, along with their identification numbers +2 The catalogue was last updated in April 2021. +3 For this study, the distinction between mass and minimum mass is +ignored. +Article number, page 3 of 28 + +105 +Bern +Bern +Bern +Bern Model +Compact +KOBE +RV +Observations +Multis +Multis +Multis +Solar System +104 +103. +Mass [M] +102 += 20 cm/s +KRV +101 +100 +10-1 +0 +100 +2 +5 +10-2 +7 +10-2 +10-1 +100 +101 +102 +103 +SMA[AU]A&A proofs: manuscript no. 43751corr +Table 1. Observed multi-planetary systems: There are 41 planetary systems with 194 planets in this catalogue. Only the first five rows are shown +here. The entire table is available online. Online version includes additional identification columns: KIC ID, TIC ID, and GAIA ID. Missing +information is indicated by ‘–’. References for individual systems are given in appendix Sect. B. +Stellar parameters +Hostname +Multiplicity +M⋆[M⊙] +R⋆[R⊙] +L⋆[L⊙] +Teff[K] +[Fe/H] +Age [Gyr] +Distance [pc] +Sun +8 +1 +1 +1 +5, 772 +0 +04.6 ± 0.1 +0 +Trappist-1 +7 +0.1 ± 0.002 +0.1 ± 0.001 +5.53e − 04 +2, 566 ± 026 ++0.04 ± 0.08 +07.6 ± 2.2 +012.0 +TOI-178 +6 +0.7 ± 0.03 +0.7 ± 0.01 +0.1 ± 01.08 +4, 316 ± 070 +−0.23 ± 0.05 +07.1 ± 6.1 +062.7 +HD 10180 +6 +1.1 ± 0.05 +1.1 ± 0.04 +1.5 ± 00.02 +5, 911 ± 019 ++0.08 ± 0.01 +04.3 ± 0.5 +039.0 +HD 219134 +6 +0.8 ± 0.03 +0.8 ± 0.005 +0.3 ± 00.01 +4, 700 ± 020 ++0.11 ± 0.04 +11.0 ± 2.2 +006.5 +Planetary parameters +Hostname +Planet +Mp[M⊕] +Rp[R⊕] +ap[AU] +e +i [◦] +min. Mp +Sun +� +j,s,u,n +m,v,e,m, +� +��������� +... +00.815 ± 00.000, +00.055 ± 00.000,��������� +��������� +... +0.949 ± 0.000, +0.383 ± 0.000,��������� +��������� +... +0.723 ± −, +0.387 ± −,��������� +��������� +... +0.01 ± −, +0.21 ± −,��������� +��������� +... +3.39 ± −, +7.00 ± −,��������� +� +F,F,F,F,... +� +Trappist-1 +� +f,g,h +b,c,d,e, +� +��������� +... +01.308 ± 00.056, +01.374 ± 00.069,��������� +��������� +... +1.097 ± 0.014, +1.116 ± 0.014,��������� +��������� +... +0.016 ± 0.000, +0.012 ± 0.000,��������� +��������� +... +0.01 ± 0.00, +0.01 ± 0.00,��������� +��������� +... +89.78 ± 0.12, +89.73 ± 0.17,��������� +� +F,F,F,F,... +� +TOI-178 +� +e,f,g +b,c,d, +� +��������� +... +04.770 ± 00.680, +01.500 ± 00.440,��������� +��������� +... +1.669 ± 0.114, +1.152 ± 0.073,��������� +��������� +... +0.037 ± 0.001, +0.026 ± 0.001,��������� +��������� +... +− ± −, +− ± −,��������� +��������� +... +88.40 ± 1.60, +88.80 ± 1.30,��������� +� +F,F,F,F,... +� +HD 10180 +� +f,g,h +c,d,e, +� +��������� +... +12.014 ± 00.699, +13.222 ± 00.445,��������� +��������� +... +− ± −, +− ± −,��������� +��������� +... +0.129 ± 0.002, +0.064 ± 0.001,��������� +��������� +... +0.13 ± 0.05, +0.07 ± 0.03,��������� +��������� +... +− ± −, +− ± −,��������� +� +T,T,T,T,... +� +HD 219134 +� +d,g,h +b,c,f, +� +��������� +... +04.230 ± 00.200, +04.620 ± 00.140,��������� +��������� +... +1.458 ± 0.048, +1.544 ± 0.059,��������� +��������� +... +0.065 ± −, +0.039 ± −,��������� +��������� +... +0.06 ± 0.04, +0.00 ± −, ��������� +��������� +... +87.38 ± 0.10, +85.19 ± 0.13,��������� +� +F,F,T,T,... +� +(when available) in the Kepler Input Catalogue (KIC), TESS In- +put Catalogue (TIC), and GAIA ID. For planets, we report mass +or minimum mass, radius, semi-major axis, eccentricity, and in- +clination. In a conservative approach, errors (reported when pos- +sible) are the maximum of the upper and lower error bounds +available in the literature. When multiple publications reported +planetary parameters, a more recent publication was preferred. +When a single publication reported parameters for all planets in +a system, then such a consistent set of solution was given pref- +erence (e.g. GJ 676 A or Kepler-11). For stellar parameters, if +a star was included in KIC, then the values from Berger et al. +(2020) are reported. Most other stellar parameters come from +the TIC (Stassun et al. 2019) or from individual publications. +There are 41 planetary systems that meet our criteria and de- +fine our multi-planetary system catalogue (Table 1). With a total +of 194 planets in our catalogue, the number of planetary systems +with four, five, six, seven, and eight planets is 24, 7, 8, 1, and 1. +In this paper, we present the observed planetary systems as they +are known today and we do not correct the observations for any +detection biases. Instead, to assist in making comparisons with +the theory, detection biases will be placed on simulated plane- +tary systems (Sect. 2.1). Figure 1 shows the mass of observed +exoplanets as a function of their semi-major axis. +While our observed multi-planetary systems catalogue en- +genders system-level studies, its current form poses several tech- +nical difficulties. Foremost, the number of observations is only +forty-one. Secondly, multiple detection methods, such as radial +velocity or transits (etc.) were employed to observe these plan- +etary systems. Each observation technique suffers from certain +limitations and detection biases. This implies that the observed +systems in our catalogue do not constitute a homogeneous and +complete set of observations. These two limitations of the ob- +servations catalogue prohibit us from deducing any statistically +strong result. Nevertheless, we used the observed systems for +(a) exemplifying system-level approach to real planetary systems +and (b) using our framework on observations to explore trends +in the architecture of observed systems. +Our results from the observed catalogue may be affected by +another source of difficulty. There are two systems in our cata- +logue that host some planets without known mass measurements +(Kepler-33 b and Kepler-80 f and g). Since these two systems +have at least four planets with known masses, they have been +included in our study. However, this does not impact the results +of the present study in a drastic way. All three planets in these +systems without mass measurements are either the innermost +and/or the outermost planets in their respective systems. There- +fore, the missing measurements do not have a strong influence +on the characterisable mass architecture. The missing measure- +ment may have a strong effect if any planet with unknown mass +was in between two planets with known masses. +3. Characterizing architecture: A new framework +3.1. Literature review +We review some approaches from other studies that have tried to +capture planetary system-level properties in this section. Kipping +(2018) investigated similarity and ordering (of planetary sizes) at +the level of an individual system. Using an entropy based frame- +work on Kepler systems, he concludes that initial conditions +are inferable from the present-day architecture. As we go on to +show in this series, our work not only supports this conclusion, +but additionally demonstrates the possible links between initial +conditions and final architecture. Although the above-mentioned +study considers a similar problem to the one we deal with here, +our frameworks differ considerably. Built on step-functions and +combinatorics, the aforementioned framework does not take into +account the magnitude of variation. +Alibert (2019) proposed a concept of distance between two +planetary systems. The Alibert distance captures inter-system +differences, whereas our framework quantifies intra-system sim- +ilarities. The Alibert distance is useful to quantify the similarity +(or dissimilarity) between two planetary systems and in unsuper- +vised machine-learning algorithms to find clusters in the space +of planetary systems. Bashi & Zucker (2021) recently proposed +Article number, page 4 of 28 + +L. Mishra et al.: Architecture Framework I – Four classes of planetary system architecture +another concept for distance based on a statistical distance. The +‘weighted’ energy distance is the distance between two plane- +tary systems, with each planet represented on the log-period and +log-radius plane, utilising planetary masses (from a mass-radius +relationship) as weights. As with the Alibert distance, the Bashi- +Zucker distance requires two planetary systems and thus it is +not suitable for characterising the global architecture for a single +planetary system. +Gilbert & Fabrycky (2020) proposed seven parameters for +quantifying the global structure of planetary systems: dynam- +ical mass (ratio of mass in planets to stellar mass), mass par- +titioning (normalised mass disequilibrium), mass monotonicity +(weighted Spearman correlation coefficient), characteristic spac- +ing (average mutual Hill radii), gap complexity, flatness, and +multiplicity (n). Of these measurements, mass partitioning and +mass monotonicity have close parallels with our framework. The +input information required to compute mass partitioning, and +monotonicity is exactly the same as the input information for +our architecture framework, namely, a set of planetary masses. +However, we find that the output displays a curious mix of con- +cepts. +Mass partitioning is zero for a system in which all planets +have the same mass. When one planet has some mass and all +other planets have negligible mass, the mass partitioning for this +system is unity. While this parameter captures the two extreme +cases, it is difficult to interpret and employ this measure in cases +other than these two extremes. Behaving similarly to a correla- +tion coefficient, mass monotonicity has a range of [-1,1]. It is de- +fined as the Spearman correlation coefficient (between mass and +distance) multiplied by the mass partitioning (which is weighted +by n−1). Although the work of Gilbert & Fabrycky (2020) stud- +ies the architecture of planetary systems at the system-level, we +seek a framework which can also be used with planetary prop- +erties other than mass, such as radius, bulk density, water mass +fraction, eccentricities, and so on. +Millholland et al. (2017) and Wang (2017) showed that the +peas in a pod pattern reported by Ciardi et al. (2013); Weiss +et al. (2018) also extends to planetary masses. Millholland et al. +(2017), using planetary masses derived from transit-timing vari- +ations, studied the clustering of planets in the mass-radius plane +and found that the sum of distances (in the log mass-size space) +between adjacent planets of real systems is much smaller than +a bootstrapped randomised population. Based on a set of 29 RV +observed systems, Wang (2017) infer two types of planetary sys- +tems. Planetary systems with masses of ≲ 30M⊕ show intra- +system mass uniformity, while systems with masses ≳ 100M⊕ +do not follow the peas in a pod pattern – indicating that there are +only two possibilities for the architecture structure. As we show +in this series of works, their hypothesis of only two architecture +types is too simple and cannot capture the richness of physics. +3.2. Concept +With our framework, we initially aimed to capture the key aspect +about the peas in a pod architecture trends. These trends are cor- +relations between adjacent planets or between consecutive pairs +of adjacent planets. We want to capture these ideas at the level +of a single planetary system through a unified framework. We +do this by studying how a quantity, qi, (such as mass, size, or +period ratio) varies for all planets within a system. Here, i in- +dexes the planets within a system. For all quantities, we adopt +an ‘inside-out’ convention, namely, we start with the innermost +planet (qi=1) and go to the next adjacent planet (qi=2), and so +on. By comparing how qi varies for each planet inside-out, we +Distance from star +Quantity (e.g. Mass) +Similar +Anti-Ordered +Ordered +Mixed +Fig. 2. Classes of architecture. This schematic diagram shows the four +architecture classes: similar, anti-ordered, mixed, and ordered. Depend- +ing on how a quantity (e.g. mass or size) varies from one planet to an- +other, the architecture of a system can be identified. +are actually estimating how qi varies with distance from the host +star. +In comparing a quantity, qi, with distance, four kind of vari- +ations emerge. In one scenario, a quantity could show little to no +variation. In another case, the value of a quantity may increase +with increasing distance or, conversely, the quantity could de- +crease from one planet to another. Finally, it is also possible for +a quantity to not have any clear variations from one planet to +another. We identify these four scenarios as the four classes of +architectures that can exist at the level of a single planetary sys- +tem. This idea is depicted in Fig. 2. +Mishra et al. (2021) suggested that the mass correlations +could originate from planet-formation physics and the correla- +tions of size and spacing could be derivative. Therefore, we first +apply our framework using planetary masses (except in Sects. +5 and 6). As depicted in Fig. 2, when the masses of all planets +within a system are similar to each other, we label the architec- +ture of such systems as ‘similar’. This architecture class corre- +sponds to the peas in a pod architecture reported in observations +(Weiss et al. 2018; Millholland et al. 2017). When the masses of +planets tend to increase inside-out, the architecture of such sys- +tems is labelled ‘ordered’. If the planetary mass tends to decrease +from the inner planet to the outer, we label the architecture of +these systems as ‘anti-ordered’. Finally, if a large increasing and +decreasing variation in the planetary masses is present, we label +the architecture of such systems as ‘mixed’. The mixed architec- +ture class is also useful in capturing all other architecture pat- +terns which do not fall under the other three architecture classes. +Kipping (2018), for example, has analysed some interesting re- +peating patterns. By introducing these architecture classes, our +framework organises the possibilities for system architecture. +Article number, page 5 of 28 + +A&A proofs: manuscript no. 43751corr +One might wonder, at this point, why introduce such a con- +cept and the ensuing mathematical machinery? While part of this +work began as an inspired exploration to categorise our under- +standing of system architecture, it turns out that there are good +physical reasons to pursue this process. As is shown in this and +a companion paper, planetary systems that have the same archi- +tecture tend to have a host of other properties in common, such +as internal structures (core-mass, ice-mass) distributions. Most +importantly, systems with a common architecture tend to have +same formation pathways, initial conditions, and evolutionary +histories. Practically, this means that a quick glance at a system’s +architecture may reveal a lot more about its formation scenario. +Our architecture classification framework utilises two quan- +tities – the coefficient of similarity and the coefficient of vari- +ation, introduced in Sects. 3.3 and 3.4, respectively. These two +coefficients allow us to quantify the conceptual ideas we have +presented above. Together, these coefficients define a new space +of possibilities for system architectures. In Sect. 3.5, we iden- +tify the regions of this architecture space that correspond to the +four architecture classes introduced above. As this framework +deals with the architecture of multi-planetary system, systems +with only one planet are not studied within this framework. +3.3. Coefficient of similarity +The term ‘coefficient of similarity’ is commonly used in the +fields studying statistics of ecology and genetics (Gower 1971; +Dalirsefat et al. 2009). We borrow the term but develop our +own concept and definition.Let q be a planetary quantity such +as mass, size, period ratios of adjacent planets, bulk density, ec- +centricity, and so on4. The value of this quantity for the ith planet +in a system is denoted by qi. The coefficient of similarity, CS , +measures how q changes from one planet to another, inside-out. +For a system with n planets, it is defined as: +CS (q) = +1 +n − 1 +i=n−1 +� +i=1 +� +log qi+1 +qi +� +. +(1) +There is a clear physical interpretation for CS (q): the coefficient +of similarity measures the average order of magnitude variation +in the quantity q from one planet to another. The definition of +the coefficient of similarity allows us to map the architecture of +a planetary system on a one dimensional axis. When CS (q) ≈ +0, then the system’s architecture could imply a similarity in q. +When CS (q) is positive, then planets within a system are ordered +in q. Conversely, CS (q) being negative, implies that the planets +are anti-ordered. +We have developed a mathematical formalism to study the +sensitivity of the coefficient of similarity. In Appendix C, we de- +rive the limiting values of the coefficient of similarity and present +the results here. For example, when the qi values for all planets +in a system are within 10% of each other, then the maximum +possible value of CS (q) is 0.09 (see Eq. C.10). For maximum +tolerances of 20%, 40%, 60%, and 80%, the maximum possi- +ble value of CS (q) are 0.18, 0.37, 0.60, and 0.95 respectively. +In Fig. C.1, we show the dependence of the maxCS (q) on t. +The coefficient of similarity cannot distinguish between two +classes of architecture: similar and mixed. Systems which show +similarity will have CS (q) ≈ 0. However, system with mixed ar- +chitecture have large increasing and decreasing variations, such +4 For quantities which admit zero as a possible value, the coefficient of +similarity may become ill-defined. This is a coordinate singularity and +can be dealt with an appropriate treatment (see Eq. 4 Sect. 5.4). +that the log of ratios qi+1 +qi cancels itself out. Such systems will also +have CS (q) ≈ 0. We propose the coefficient of variation to distin- +guish these two architecture classes. The coefficient of similarity +depends on the actual order in which planets exist (inside-out) in +a system. As we go on to show, the coefficient of variation does +not depend on the ordering of planets in a system. +3.4. Coefficient of variation +The coefficient of variation, CV, is a standard descriptive statistic +used to measure the magnitude of variation in a set of numbers +(Katsnelson & Kotz 1957; Sharma et al. 2010; Abdi 2010). It is +defined as the ratio of the standard deviation with the mean: +CV(q) = σ(q) +¯q . +(2) +The coefficient of variation is a positive quantity. When all +qi have the same value then CV(q) = 0. Planetary systems con- +sisting of planets that have a small (or large) variability in their +qi values will have a small (or large) value of the coefficient of +variation. Now, the distinction between systems showing simi- +larity and mixed architecture is clear. While similar systems will +have a low value of the coefficient of variation, mixed systems +will have a high value of coefficient of variation. +Since this coefficient is a well known statistical measure, +there are some derivations for its limit. A classical result from +Katsnelson & Kotz (1957) shows that, for a set of n numbers, +the maximum value of the coefficient of variation is +√ +n − 1. +However, this result is only a particular case in our setup. In Ap- +pendix C, we develop a mathematical formalism to understand +the limits of the coefficient of variation and present the results +here. When the qi values for all planets in a system are within +10%, 30%, 50%, 70%, and 90% of each other, the absolute the- +oretical upper limit of CV(q) is 0.10, 0.31, 0.58, 0.98, and 2.06 +respectively. Figure C.1 shows how this upper limit varies with +the maximum tolerance, t, for a system. +3.5. Classifying the architectures of planetary systems +We are interested in obtaining a mapping from the scale- +invariant coefficients to an architecture class. In Appendix D, +we present some considerations that motivate the selection of +boundaries between the four classes. The selected boundaries +were additionally tested on thousands of mock planetary systems +to check their ability to correctly classify the four architecture +classes. We propose the following boundaries for identifying the +architecture class based on planetary masses. +Architecture class +Condition +Anti-ordered +CS (M) < −0.2 +Ordered +CS (M) > +0.2 +Similar +|CS (M)| ≤ 0.2 and CV(M) ≤ +√ +n − 1 +2 +Mixed +|CS (M)| ≤ 0.2 and CV(M) > +√ +n − 1 +2 +(3) +A natural (and welcome) outcome of these criteria is that a +two-planet system can never have a mixed class architecture. The +boundary between similar and mixed class is half the maximum +possible value of the coefficient of variation. For the solar sys- +tem, CS (M) = 0.36 and CV(M) = 1.85. This framework robustly +Article number, page 6 of 28 + +L. Mishra et al.: Architecture Framework I – Four classes of planetary system architecture +Fig. 3. New parameter space: Architectures of planetary systems. Both panels shows the coefficient of similarity (mass) as a function of the +coefficient of variation (mass). The shaded regions show the allowed parameter space for planetary systems. The white gaps (between two shaded +regions) mark the mathematically forbidden regions of this architecture space. Different parts of this parameter space are identified with four +architecture classes, in accordance with Eq. 3. Each point corresponds to an individual planetary system. For visual clarity, the shaded and +unshaded regions are drawn only for systems hosting up to fifteen planets. Left: Planetary systems from the Bern model and observations. Right: +Synthetically observed systems depicting the detection biases of radial velocity and transit surveys. +identifies the architecture of the solar system as ordered5. This +classification is in line with the historic understanding of the so- +lar system architecture: small rocky planets on the inside and +giant planets on the outside. If, however, Neptune were replaced +with an Earth-like planet, the architecture of the solar system +would be classified as mixed. Considering only the inner four +planets of the solar system, CS (M) = 0.10 and CV(M) = 0.85, +would make the architecture of the inner solar system belong to +the similar class. The architecture of the outer four giants in the +solar system is anti-ordered and we have CS (M) = −0.42 and +CV(M) = 1.11. +Figure 3 shows the CS (M) versus CV(M) space for plane- +tary systems from several catalogues. The Bern model planetary +systems occupy all four regions of this architecture space. Ob- +served planetary systems, however, span only a limited region +of this parameter space, given the low multiplicity of observed +planetary systems. The architecture space spanned by the ob- +served planetary systems (shaded contour) is in agreement with +the synthetically observed planetary systems from Bern Com- +pact Multis, Bern KOBE Multis, and Bern RV Multis. +The architecture for the systems in the synthetically observed +catalogue was calculated based only on the planets that were de- +tected (for RV/KOBE) or included (for Bern Compact Multis) in +the above-mentioned catalogue. It is theoretically possible for a +single Bern model system to exhibit different architectures de- +pending on the planets which are detected or included. The re- +verse is also true – the architecture of an observed planetary sys- +tem may change if new planets are discovered or old controver- +sial candidates are rejected. While the ground truth architecture +for observations seems elusive, a comparison with synthetic ob- +5 Even if the masses of each solar system planet were randomly varied +within 85% of their original values, the emerging architecture is still +ordered. With 1M trials, varying the masses randomly within 90% of +their original values lead to ordered (for ≈ 99.45% trials), mixed (for +≈ 0.55% trials), and similar (for ≈ 0.001% trials) architectures. +servations can bring forth patterns which are unexpected. With +this in mind, we consider the following example. +Detection biases, in both radial velocities and transits, gen- +erally disfavour the discovery of less-massive and small planets +at larger distances. This implies that anti-ordered architectures +are difficult to detect. In fact, we have no known example of a +planetary system showing anti-ordered architecture in our obser- +vations catalogue. This is surprising for two major reasons: (a) +theory suggests their existence: there are several synthetic plan- +etary systems from the Bern Model whose architecture is anti- +ordered; (b) theory suggests their discovery: all three syntheti- +cally observed catalogues contain some (albeit few) anti-ordered +planetary systems. Since the number of systems in our catalogue +is too low, we refrain from making any conclusions and, instead, +we await the discovery of anti-ordered architectures in the future. +However, if such architectures are not found despite considerable +efforts, this result will become a strong indicator for shaping our +understanding of planet formation. +Another aspect of this new architecture space is the underly- +ing mathematical structure6. In Fig. 3, the shaded areas shown +regions where a planetary system, with n ∈ [2, 15] planets, is al- +lowed. A system with two planets, for example, can only occupy +the shaded region labelled ‘n = 2’. All non-shaded regions (in +white – except the shaded regions for 16 or more planets which is +not drawn), on this architecture space, is mathematically forbid- +den. These are parts of the architecture parameter space that no +planetary system, irrespective of its configuration, can occupy. +6 Visualizing this structure is easy (not shown). (a) Construct mock +planetary systems with masses, for each mock planet, randomly drawn +from a uniform distribution with suitable limits. (b) It is suggested to +vary the number of planets in these mock systems randomly. (c) Calcu- +late the CS (M) and the CV(M) using equations 1 and 2. (d) Plot CS (M) +versus CV(M) for this mock population. For large number of systems +the plot should be symmetric about CS (M) = 0. +Article number, page 7 of 28 + +Coefficient of Similarity (Mass)[unitless] +Bern Model +Observations +Solar System +2 +4 +u +0 +1 +2 +? +4 +Coefficient of Variation (Mass) [unitless]Coefficient of Similarity (Mass)[unitless] +Observations Contour +BernRVMultis +Bern KOBE Multis +Bern Compact Multis +4 +0 +1 +2 +3 +4 +Coefficient of Variation (Mass) [unitless]A&A proofs: manuscript no. 43751corr +This strong result stems from the mathematical limits that were +derived for this work (see Sects. 3.3, 3.4, and appendix C). +For clarity and future convenience, we introduced some ter- +minology to the method. When the architecture framework (i.e. +CS and CV) is applied on planetary bulk masses, the resulting in- +formation tells us the mass architecture of a system, namely, the +arrangement and distribution of masses in said system. Similarly, +when this framework is applied on radii, it gives us the radius ar- +chitecture (arrangement and distribution of radii) for the system +(Sect. 5.1). Similarly, we can obtain the bulk-density architecture +(Sect. 5.2), core-mass architecture (Sect. 5.3), water mass frac- +tion architecture (Sect. 5.4), period-ratio or spacing architecture, +eccentricity-architecture, and so on. In this series of papers, we +identify a system’s architecture based on its bulk mass ar- +chitecture. Thus, when a system is said to be similar, we are +referring to the similarity in terms of the mass architecture. +4. Characteristics of architecture classes +4.1. General comments +In earlier studies on the peas in a pod architecture, the strength of +population-level (i.e. across many planetary systems) trends was +quantified using Pearson correlations coefficient (Weiss et al. +2018; Zhu 2020; Chevance et al. 2021; Millholland & Winn +2021; Mishra et al. 2021). The correlation coefficients were cal- +culated using planetary quantities in the log space (i.e. by first +taking the log10 of all quantities). This resulted in higher values +of the correlation coefficient since quantities have limited range +to perambulate in the log space. Consider planetary masses. We +calculated the correlation coefficient between the mass of adja- +cent inner and outer planets in the Bern model population (see +Fig. 7 in Mishra et al. 2021). The value of the coefficient is 0.66 +in the log space and 0.16 in the linear space. This highlights that +the planetary masses are more closely clustered in log than in +linear space. +We tested the same correlation for all systems in each ar- +chitecture class. We expect planetary masses in mixed, ordered, +and anti-ordered systems should (by definition) have low cor- +relations. On the other hand, similar class architecture should +exhibit a strong correlation. Surprisingly, in log space all archi- +tecture classes show strong correlations. The coefficient value +is 0.67 for similar class, 0.69 for mixed class, 0.50 for ordered +class, and 0.58 for anti-ordered class architectures. However, in +the linear space the coefficient values reflects our expectation: +0.61 for the similar class, 0.20 for the mixed class, 0.16 for or- +dered class, and 0.05 for anti-ordered class. This underscores +that strong correlations in the log space may not be indicative of +substantive architecture trends. It also shows that our framework +is capable of identifying systems in which the ’peas in a pod’ +architecture is discernible even in the linear space. +For all 41 observed planetary systems in our catalogue, we +report their architecture classes in Table 2. Figure 6 shows the +architecture of all observed multi-planetary systems in our cat- +alogue. The systems are sorted by their coefficient of similarity +values. The figure also shows the four classes of architecture for +a few randomly selected synthetic planetary systems. To under- +stand the characteristics of the different architectures, we study +the distribution of planetary masses, radii, and semi-major axes +as well as the multiplicity distributions. For planetary systems +across all catalogues, this is shown in Fig. 7. We describe the +characteristics of different architectures in the following subsec- +tions. The discussion in the next subsection involves results de- +rived from both observed and synthetic planetary systems. In ad- +Table 2. Architecture type of known multi-planetary systems (see Table +1 for catalogue and Fig. 6 for architecture plot). +Hostname +Multiplicity +CS (M) +CV(M) +Architecture Class +Solar System +8 ++0.36 +1.85 +Ordered +Trappist-1 +7 +−0.10 +0.45 +Similar +TOI-178 +6 ++0.08 +0.46 +Similar +HD 10180 +6 ++0.14 +0.66 +Similar +HD 219134 +6 ++0.27 +1.49 +Ordered +HD 34445 +6 ++0.17 +0.84 +Similar +Kepler-11 +6 ++0.22 +1.03 +Ordered +Kepler-20 +6 ++0.00 +0.44 +Similar +Kepler-80 +6 +−0.00 +0.19 +Similar +K2-138 +6 ++0.03 +0.61 +Similar +55 Cnc +5 ++0.52 +1.37 +Ordered +GJ 667 C +5 +−0.02 +0.29 +Similar +HD 158259 +5 ++0.11 +0.29 +Similar +HD 40307 +5 ++0.07 +0.33 +Similar +Kepler-102 +5 ++0.02 +0.41 +Similar +Kepler-33 +5 ++0.46 +0.67 +Ordered +Kepler-62 +5 ++0.15 +0.68 +Similar +HD 20781 +4 ++0.29 +0.59 +Ordered +TOI-561 +4 ++0.33 +0.64 +Ordered +DMPP-1 +4 ++0.29 +0.81 +Ordered +GJ 3293 +4 ++0.27 +0.62 +Ordered +GJ 676 A +4 ++0.90 +0.99 +Ordered +GJ 876 +4 ++0.12 +1.20 +Mixed +HD 141399 +4 ++0.06 +0.40 +Similar +HD 160691 +4 ++0.63 +0.82 +Ordered +HD 20794 +4 ++0.08 +0.25 +Similar +HD 215152 +4 ++0.07 +0.23 +Similar +HR 8799 +4 +−0.07 +0.17 +Similar +K2-266 +4 ++0.03 +0.60 +Similar +K2-285 +4 ++0.01 +0.31 +Similar +Kepler-89 +4 ++0.17 +0.91 +Mixed +Kepler-106 +4 ++0.11 +0.26 +Similar +Kepler-107 +4 ++0.13 +0.42 +Similar +Kepler-223 +4 +−0.06 +0.22 +Similar +Kepler-411 +4 +−0.08 +0.34 +Similar +Kepler-48 +4 ++0.74 +1.64 +Ordered +Kepler-65 +4 ++0.68 +1.63 +Ordered +Kepler-79 +4 +−0.10 +0.24 +Similar +WASP-47 +4 ++0.59 +0.95 +Ordered +tau Cet +4 ++0.12 +0.37 +Similar +HD 164922 +4 ++0.49 +1.29 +Ordered +dition, we present a gallery of mass-distance diagrams showing +the four architecture classes in Appendix E. +Figure. 4 shows the coefficient of similarity of masses as a +function of the total planetary mass in a system for all synthetic +planetary systems from the Bern model. This figure shows sev- +eral key aspects. Firstly, it illustrates the four architecture classes +as separate clouds of scattered points strengthening the proposed +four classes of planetary system architecture. Secondly, it shows +that the architecture framework is scale-invariant, that is, the +system architecture is sensitive only to the relative distribution +of a quantity – and not its absolute value. For example, while +most similar system have ⪅ 100M⊕ mass in their planets (sug- +gesting a lack of giant planets), there are some similar systems +with mass values of ≈ 2000M⊕ for their planets and host giant +planets. Likewise, most ordered systems host giant planets and +have ⪆ 2000M⊕ mass in their planets, there is an ordered sys- +tems without any giant planets. Also, it illustrates that the coeffi- +cient of similarity partitions planetary systems into three groups: +anti-ordered, similar and mixed in one group, and ordered. This +demonstrates that the coefficient of variation is necessary to dis- +tinguish between the similar and mixed systems. Finally, the di- +agram shows that the architecture class of a system has strong +links with the total mass of planets in the system. This hints that +there must be general patterns in the formation pathways of sys- +Article number, page 8 of 28 + +L. Mishra et al.: Architecture Framework I – Four classes of planetary system architecture +10 +0 +10 +1 +10 +2 +10 +3 +10 +4 +Total Mass in Planets [M +] +3 +2 +1 +0 +1 +2 +3 +Coefficient of Similarity (Mass) [unitless] +Similar +Anti-Ordered +Mixed +Ordered +Fig. 4. Four classes of system architecture. The diagram shows the coef- +ficient of similarity for a system as a function of the sum of mass of each +planet in a system. Dashed horizontal lines correspond to CS = ±0.2. +This diagram emphasises the four classes of planetary system architec- +ture, namely: anti-ordered, similar, mixed, and ordered. It also shows +that the coefficient of similarity can not distinguish between similar and +mixed architectures. +tems of the same architecture. This topic is discussed in Paper II, +from this series. +4.2. Frequency of architecture +The frequency of each architecture class across all catalogues is +shown in Fig. 5. Similar systems are the most common archi- +tecture classes emerging from simulations, with a frequency of +≈ 80.2%. About ≈ 8% of synthetic systems show mixed and +anti-ordered architectures. Ordered architecture is a rare out- +come in simulations (≈ 1.5%). In observations, similar class is +the most common architecture (≈ 59%). Fifteen observed exo- +planetary systems (out of forty-one) are part of the ordered ar- +chitecture class (≈ 37%). About ≈ 5% of observed planetary +systems show mixed architecture. There are no known examples +of observed system with anti-ordered architecture. +Comparing the frequency of architecture classes for ob- +served systems with synthetically observed systems brings out +some peculiar features. Firstly, theoretical catalogues seem to +suggest that observations should find more similar systems and +fewer ordered systems. The frequency of similar (ordered) sys- +tems in our observed catalogue is significantly lower (higher). +Secondly, while the frequency of mixed systems seems to be in +agreement with synthetic observations, this agreement is not sta- +tistically significant. +These discrepancies probably arise from the incompleteness +prevalent in our observations catalogue. Transit surveys are con- +ducted in a manner which allows the completeness and reliability +of these survey to be estimated. The completeness of RV surveys, +on the other hand, is very difficult to estimate. Further, the obser- +vation techniques used to find the exoplanets in our observations +catalogue are heterogeneous, consisting of RV, transits, transit- +Similar Anti-Ordered Ordered +Mixed +0 +20 +40 +60 +80 +100 +Frequency [%] +Bern Model +Bern KOBE Multis +Bern RV Multis +Bern Compact Multis +Observations +Fig. 5. Frequency diagram for the architecture classes. Currently, there +are no known examples of observed planetary systems with anti-ordered +architecture. The length of error bars visualises the total number of sys- +tems in each bin as: 100/ +√ +bin counts. +timing variations, and direct imaging; this complicates the es- +timation of completeness. The PLATO mission is an upcoming +space mission that is equipped to allow for statistical estimates +of cosmic occurrence rates of planetary system architecture in +our galaxy (Rauer et al. 2014). If more exoplanetary systems are +uniformly detected and characterised, then it would be possible +to estimate the occurrence rate of the different classes of system +architecture. While such a result would constitute an important +knowledge about our Universe, it could also become an excel- +lent way of constraining our knowledge of initial conditions for +planetary formation and the physical processes which shape the +system architecture. The frequency of architecture class in sim- +ulations is a direct consequence of the initial conditions and the +physical processes modelled in the Bern model. +4.3. Architecture class: similar +Planetary systems have a similar architecture when all planets in +the system have masses that are approximately similar to each +other. These planetary systems are the archetypical examples of +the peas in a pod trend. There are several well-known planetary +systems exhibiting similar architecture, such as Trappist-1 (Agol +et al. 2021), TOI-178 (Leleu et al. 2021), Kepler-20 (Buchhave +et al. 2016), and so on. This architecture is the most common +outcome of planetary formation and is also the most frequent +architecture class in our observed catalogue. +Similar systems in the Bern model are composed of several +low-mass planets. They tend to have limited diversity in plan- +etary masses when compared with the observed systems. The +mass distribution, for similar systems in the Bern model, shows +that there are many low-mass (< 1M⊕) planets in these systems. +This peak is missing in observations as well as synthetic observa- +tions as low mass exoplanets are difficult to observe. This could, +however, be remedied in future as current radial velocity spectro- +Article number, page 9 of 28 + +A&A proofs: manuscript no. 43751corr +0.10 +0.10 +0.08 +0.07 +0.06 +0.02 +0.00 +0.00 +0.01 +0.02 +0.03 +0.03 +0.06 +0.07 +0.07 +0.08 +0.08 +0.11 +0.11 +0.12 +0.13 +0.14 +0.15 +0.17 +0.22 +0.27 +0.27 +0.29 +0.29 +0.33 +0.36 +0.46 +0.49 +0.52 +0.59 +0.63 +0.68 +0.74 +0.90 +0.12 +0.17 +Coefficient of Similarity (Mass) [unitless] +10 +3 +10 +2 +10 +1 +10 +0 +10 +1 +10 +2 +SMA [AU] +Trappist-1 +Kepler-79 +Kepler-411 +HR 8799 +Kepler-223 +GJ 667 C +Kepler-80 +Kepler-20 +K2-285 +Kepler-102 +K2-138 +K2-266 +HD 141399 +HD 215152 +HD 40307 +HD 20794 +TOI-178 +Kepler-106 +HD 158259 +tau Cet +Kepler-107 +HD 10180 +Kepler-62 +HD 34445 +Kepler-11 +GJ 3293 +HD 219134 +DMPP-1 +HD 20781 +TOI-561 +Sun +Kepler-33 +HD 164922 +55 Cnc +WASP-47 +HD 160691 +Kepler-65 +Kepler-48 +GJ 676 A +GJ 876 +Kepler-89 +1 M +50 M +1 MJ +10 MJ +Similar +Ordered +Mixed +10 +2 +10 +3 +Teq[K] +0.80 +0.63 +0.55 +0.49 +0.28 +0.24 +0.22 +0.11 +0.07 +0.05 +0.02 +0.01 +0.00 +0.01 +0.23 +0.56 +0.86 +1.02 +1.09 +1.28 +2.16 +0.14 +0.13 +0.13 +0.12 +0.11 +0.06 +0.06 +Coefficient of Similarity (Mass) [unitless] +10 +2 +10 +1 +10 +0 +10 +1 +10 +2 +10 +3 +SMA [AU] +673 +4 +816 +879 +131 +141 +274 +396 +946 +397 +453 +871 +461 +683 +790 +778 +893 +965 +912 +153 +110 +254 +522 +959 +828 +911 +402 +914 +System ID +1 M +50 M +1 MJ +10 MJ +Anti-Ordered +Similar +Ordered +Mixed +10 +2 +10 +3 +Teq[K] +Fig. 6. Architecture plot showing the architecture of observed (left) and randomly selected synthetic planetary systems (right). Each row is for one +planetary system and the circles in that row represent planets. The area of the circle encodes planetary mass, and the colour shows the equilibrium +temperature. The coefficient of similarity for each system is shown on the right y-axis. The x-axis shows the semi-major axis, which is different +for the two panels. +graphs reach the ≈ 20 cm/s precision necessary for discovering +exoplanets in the super-Earths and Earths mass range (Lillo-Box +et al. 2021; Netto et al. 2021). The radius distribution of similar +systems implies that these systems are prominently composed of +rocky planets, super-Earths and sub-Neptunes7. +7 Throughout this paper, we use planetary classes (e.g. rocky, super- +Earths, etc.) from the radius based classification of Kopparapu et al. +(2018) +Article number, page 10 of 28 + +L. Mishra et al.: Architecture Framework I – Four classes of planetary system architecture +10 +1 +100 +101 +102 +103 +104 +105 +Mass [M +] +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +Density +Bern Model +Similar +Anti-Ordered +Mixed +Ordered +0.5 +1.0 +2.0 +4.0 +8.0 +16.0 +Radius [R +] +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Density +Bern Model +Similar +Anti-Ordered +Mixed +Ordered +10 +2 +10 +1 +100 +101 +102 +103 +SMA [AU] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Density +Bern Model +Similar +Anti-Ordered +Mixed +Ordered +0 +5 +10 +15 +20 +25 +30 +Multiplicity +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +Density +Bern Model +Similar +Anti-Ordered +Mixed +Ordered +10 +1 +100 +101 +102 +103 +104 +Mass [M +] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +Density +Bern RV Multis +Similar +Anti-Ordered +Mixed +Ordered +0.5 +1.0 +2.0 +4.0 +8.0 +16.0 +Radius [R +] +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +Density +Bern RV Multis +Similar +Anti-Ordered +Mixed +Ordered +10 +2 +10 +1 +10 +1 +100 +100 +100 +101 +SMA [AU] +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +Density +Bern RV Multis +Similar +Anti-Ordered +Mixed +Ordered +4 +6 +8 +10 +12 +14 +Multiplicity +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +Density +Bern RV Multis +Similar +Anti-Ordered +Mixed +Ordered +10 +1 +100 +101 +102 +103 +104 +Mass [M +] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +Density +Bern KOBE Multis +Similar +Anti-Ordered +Mixed +Ordered +0.5 +1.0 +2.0 +4.0 +8.0 +16.0 +Radius [R +] +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +Density +Bern KOBE Multis +Similar +Anti-Ordered +Mixed +Ordered +10 +2 +10 +1 +10 +1 +100 +100 +100 +101 +SMA [AU] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +Density +Bern KOBE Multis +Similar +Anti-Ordered +Mixed +Ordered +4 +6 +8 +10 +12 +14 +Multiplicity +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +Density +Bern KOBE Multis +Similar +Anti-Ordered +Mixed +Ordered +10 +1 +100 +101 +102 +103 +104 +Mass [M +] +0.0 +0.2 +0.4 +0.6 +0.8 +Density +Bern Compact Multis +Similar +Anti-Ordered +Mixed +Ordered +0.5 +1.0 +2.0 +4.0 +8.0 +16.0 +Radius [R +] +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +Density +Bern Compact Multis +Similar +Anti-Ordered +Mixed +Ordered +10 +2 +10 +1 +10 +1 +100 +100 +100 +101 +SMA [AU] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Density +Bern Compact Multis +Similar +Anti-Ordered +Mixed +Ordered +4 +6 +8 +10 +12 +14 +Multiplicity +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +Density +Bern Compact Multis +Similar +Anti-Ordered +Mixed +Ordered +10 +1 +100 +101 +102 +103 +104 +105 +Mass [M +] +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +Density +Observations +Similar +Mixed +Ordered +0.5 +1.0 +2.0 +4.0 +8.0 +16.0 +Radius [R +] +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +Density +Observations +Similar +Mixed +Ordered +10 +2 +10 +1 +100 +101 +102 +103 +SMA [AU] +0.0 +0.2 +0.4 +0.6 +0.8 +Density +Observations +Similar +Mixed +Ordered +4 +6 +8 +10 +12 +14 +Multiplicity +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Density +Observations +Similar +Ordered +Mixed +Fig. 7. Characteristics of the architecture classes. These plots show the distribution of various quantities (columns) as function of different cata- +logues (rows). Left to right: Distributions of mass, radius, distance, and multiplicity in the following catalogues (top to bottom): Bern model, Bern +RV Multis, Bern KOBE Multis, Bern Compact Multis, and observations. All catalogues are described in Sect. 2. Some notable features from these +plots are discussed in Sect. 4. All individual distributions are normalised such that the area under each curve sums to unity. The dotted vertical line +in the radius distributions marks 1.75R⊕ – approximately, the location of the well-known gap in the radius distribution (Fulton et al. 2017). Since +there are only two mixed systems with the same multiplicity (n = 4) in our observations catalogue, a vertical line replaces the density kernel. The +Gaussian density kernels in all other cases were estimated using Scott’s rule (Scott 2015). +Article number, page 11 of 28 + +A&A proofs: manuscript no. 43751corr +The Bern RV Multis show a bimodal planetary distance dis- +tribution for similar systems (as well as for mixed and ordered). +The approximate location of the gap is 0.28 au or 55 d (for a solar +mass star). This bi-modality is not visible in our observed cata- +logue. Planets in similar and mixed systems in the Bern Model +also show a dip around this location. In the Bern Model, in- +wardly migrating giant planets (≳ 100 M⊕) tend to stop around +0.4 au or 100 d. Inside this region, low-mass planets are popu- +lous. We attribute this bi-modality to these two populations of +planets. This bi-modality probably arises because planets switch +their orbital migration from type I to type II depending on their +masses (Emsenhuber et al. 2021a). This bi-modality cannot be +seen in Bern Compact Multis because we only include planets +with periods less than 100d. For Bern KOBE Multis, the com- +pleteness of the Kepler mission for large distant planets is poor +(see Fig. C.2 in Mishra et al. 2021). However, a dip at this loca- +tion in Bern KOBE Multis is visible. It would be interesting to +see if such a bi-modality is also present in the Kepler catalogue. +We tested the significance of this bi-modality with Hartigan’s dip +test (Hartigan & Hartigan 1985). The dip test is suggestive of the +bi-modality for the Bern RV Multis and Bern KOBE Multis (p- +value < 0.05) and insignificant for the other catalogues. +A system’s architecture is sensitive only to the relative distri- +bution of a quantity (such as mass) amongst its planets and not +the absolute distribution. HR 8799 offers an example (Marois +et al. 2008) as a relatively young system with four directly im- +aged giant planets. Our framework identifies the architecture +of this systems as similar. Most observed similar systems are +composed of low-mass planets (≲ 100M⊕), making HR 8799 a +unique exception. This shows that the architecture framework is +sensitive only to the relative variations in the mass. Additionally, +there are only two systems (out of 1000) in our simulated cata- +logue where a similar architecture arises from only giant planets. +Even then, these two synthetic systems have only two giant plan- +ets much closer to the star than the HR 8799 planets. The Bern +Model does not produce many HR 8799-like systems. This sug- +gests that a system with similar architecture made up of only +giant planets is probably rare. One possibility could be that sys- +tems (e.g. HR 8799) with such architecture are probably diffi- +cult to form via core accretion pathway (Konopacky & Barman +2018). Such systems may require additional formation mecha- +nisms such as protoplanetary disk instabilities (Schib et al. 2021; +Boley et al. 2010; Kratter et al. 2010). +4.4. Architecture class: mixed +Planetary systems where the planetary masses (inside-out) show +broad increasing and decreasing variations have mixed archi- +tecture. GJ 876 and Kepler-89 host planetary systems with a +mixed class architecture. GJ 876 is an M dwarf low luminous +(≈ 0.01 L⊙) star hosting four planets with masses between +8 − 888M⊕. The outer three planets are in a Laplace mean- +motion resonance (Millholland et al. 2018). Kepler-89, on the +other hand, is an early F, highly luminous (≈ 3.5 L⊙) star. It hosts +a compact four planet system with masses between 10 − 100M⊕. +Despite the starkly different stellar properties, the architecture +of these two systems is analogous: CS (M) = 0.12 and 0.17, +CV(M) = 1.2 and 0.9, respectively. While the coefficient of simi- +larity is low for both systems, the coefficient of variation is larger +than +√ +3/2, which helps us identify the architecture of these sys- +tems as mixed class. Indeed, Fig. 6 indicates that this identifica- +tion is correct. +The frequency of this architecture class in the Bern model +is ≈ 8.2%. The Bern model’s synthetic mixed architecture +planetary systems (Fig. 6 right) tend to have numerous Earth- +mass planets outside 10 au. This parameter space (mass-distance +plane, Fig. 1), however, remains inaccessible to most exoplanet +detection techniques. These systems are also composed of super- +Earths, sub-Neptunes, Neptunes, and Jovian planets. The bi- +modality in distance distribution (discussed before) is prominent +for these architectures in Bern RV Multis. We found a Harigan’s +dip statistic of 0.03 and p-value of ∼ 0.2 (Hartigan & Hartigan +1985). +4.5. Architecture class: anti-ordered +Planetary systems where the planetary mass shows an overall +decrease with distance have an anti-ordered architecture. There +are no observed examples of this architecture class in our cata- +logue. The frequency of this architecture class in the Bern model +is ≈ 8.4%. About ≈ 4% of systems in Bern KOBE Multis, +≈ 3.2% of systems in Bern Compact Multis, and ≈ 1.2% of +systems in Bern RV Multis have this architecture. This shows +that it is an observationally challenging system architecture to +detect. However, even if 1% of observed exoplanetary systems +are Anti-Ordered we should already have found about 30-40 +such systems. More work is necessary to identify the handful +of these systems from the already observed systems. Many cur- +rently known single hot Jupiter systems may host additional +small, distant, and as yet undetected planets – revealing these +potentially anti-ordered systems. +Anti-ordered systems in the Bern Model are mostly com- +posed of low mass planets ≲ 5M⊕ and giants ≳ 100M⊕. In +the Bern Model, the radius distribution of this architecture class +peaks for Rocky and Super-Earths planets. It decreases for sub- +Neptunes and Neptunes and then increases again for Jovian plan- +ets. Many of the low-mass planets that make up this architecture +class are outside 10au, making their detection very challenging. +The multiplicity distribution shows that these systems tend to +have fewer planets than similar or mixed architecture. This is +an indication that the formation pathway of these architectures +differs considerably from the other two types of architecture. +Planets from anti-ordered architectures show a weak distance bi- +modality feature (discussed earlier in this work). This is under- +standable since these architectures consist of massive planets in +the inner parts and less massive planets in the outer parts of the +system. The distance bi-modality seems to arise from low mass +planets (migrating via type I) inside 0.28au or 55 days and giant +planets (migrating via type II) outside 0.28au or 55 days. This +adds further strength in attributing the distance bi-modality to +planetary migration. +4.6. Architecture class: ordered +Planetary systems where the planetary masses shows an overall +increase with distance have an ordered architecture. The increas- +ing mass may be monotonic (e.g. TOI-561, HD 20781, DMPP- +1,HD 160691, HD 164922) or non-monotonic (e.g. the Solar +System, Kepler-11, 55 Cnc, Kepler-48, Kepler-65). Ordered ar- +chitecture is a rare outcome for the Bern model. Observations +are generally biased against discovering small and less massive +planets which are farther away from their host star. Such biases, +however, make ordered systems the second most common archi- +tecture class. Fifteen systems in our catalogue exhibit this archi- +Article number, page 12 of 28 + +L. Mishra et al.: Architecture Framework I – Four classes of planetary system architecture +tecture. Unsurprisingly, the most notable known example of this +architecture class is the Solar System. +The mass and radius distributions of ordered architecture in +the Bern Model shows considerable difference from other archi- +tecture. The mass distribution peaks around 1000M⊕. Most of +the Bern model’s ordered systems tend to have at least one gi- +ant planet. These systems are also composed of sub-Neptunes, +Neptunes, and Jovian planets. +5. Internal composition across architecture classes +So far we have seen the new architecture framework (Sect. 3) and +some characteristics of the four classes of architecture (Sect. 4). +In this section, we study the connection between the bulk mass +architecture classes and the internal composition of the planets. +This section demonstrates that the same architecture framework +can be used to study the multi-faceted nature of planetary sys- +tem architecture – from bulk mass architecture to density archi- +tecture. We study several different aspects of the planetary in- +ternal composition: (a) radius architecture (Sect. 5.1); (b) bulk +density architecture (Sect. 5.2); (c) Core/Envelope mass archi- +tecture (Sect. 5.3); and (d) fraction of volatiles and water ice in +core architecture (Sect. 5.4). We explore these connections for +planetary systems in the simulated (Bern model) and syntheti- +cally observed catalogues (Bern RV Multis, Bern KOBE Mul- +tis, Bern Compact Multis). All results in this section are derived +from synthetic planetary systems only. +5.1. Radius architecture +Weiss et al. (2018) showed that the size of adjacent exoplanets +were similar – coining the phrase ‘peas in a pod’ to describe this +architecture. Millholland et al. (2017); Wang (2017) extended +these ideas to planetary masses, showing that the masses of ad- +jacent planets are also correlated. In Mishra et al. (2021), we +suggested that the peas in a pod trends in terms of size effec- +tively emerge from the mass trends. Here, we attempt to set our +assumption on firmer ground. +Figure 8 (top) shows the coefficient of similarity for radii as +a function of the coefficient of similarity of masses, for systems +with two or more planets. This allows us to compare the system- +level radius architecture with the system-level mass architecture. +We easily see that most systems seem to follow a linear rela- +tionship. The Pearson correlation coefficient is 0.89, indicating +a strong positive correlation between the mass and radius archi- +tecture. The coefficient value increases to 0.96, when systems +with only three or more planets are considered. Since the mass- +radius relation is not a bijective function (i.e. one-to-one corre- +spondence), there are some systems that show a strong deviation +from the linear relation. +Figure 8 (bottom) shows the radii architecture for the syn- +thetic planetary systems8. This shows that most systems that +are ordered (or anti-ordered) in mass are also ordered (or anti- +ordered) in terms of radius. The figure also shows that systems +which are similar or mixed in mass architecture have CS (R) ≈ 0. +Systems with mass similarity have lower CV(R) compared to sys- +tems with mass mixture, suggesting that for most systems, the +8 A future study could investigate the boundaries for robust architec- +ture identification, as in Eq. 3, but based on radius instead of mass. +Such a classification is readily applicable since radius measurements +tend to be uniformly available and are better agreed upon amongst sev- +eral observers. Data-driven approaches such as machine learning could +be useful in such an endeavour. +3 +2 +1 +0 +1 +2 +3 +Coefficient of Similarity (Mass) [unitless] +0.8 +0.6 +0.4 +0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +Coefficient of Similarity (Radius) [unitless] +RPearson = 0.89 +RSpearman = 0.94 +Lin. Fit: y = 0.23 × +0.0006 +Bern Model +Observations - 41 systems +Solar System +Fig. 8. Radii architecture. Top: The diagram shows the coefficient of +similarity of radii as a function of the coefficient of similarity of masses, +for synthetic and observed planetary systems. The dashed line shows the +corresponding linear fit. Bottom: Radius architecture of synthetic plane- +tary systems contrasted with the mass architecture. In the bottom panel, +the marker colour and shape indicates the bulk mass architecture of a +system and its position on the diagram suggests its radii architecture. +radius architecture closely follows the mass architecture. At the +planetary level the radius of a planet is correlated with its mass +via the planet’s chemical composition (Lopez & Fortney 2014). +Our architecture framework shows that such relationships also +exist at the system level. A few mass-ordered systems show sim- +ilarities in radius. These few systems have the following com- +mon features: two mass-ordered giant planets with similar sizes +(masses ∼ several MJ’s, and radius ≈ 1RJ). This illustrates that +Article number, page 13 of 28 + +0.6 +Based on Cs(M): +[unitless] +Similar +Anti-Ordered +0.4 +Mixed +Ordered +Coefficient of Similarity (Radius) +0.2 +0.0 +-0.2 +-0.4 +-0.6 +0.0 +0.5 +1.0 +1.5 +2.0 +Coefficient of Variation (Radius) [unitless]A&A proofs: manuscript no. 43751corr +while mass architecture and radius architecture are related, they +are not always identical. +We conclude that the peas in a pod radius correlations gen- +erally arise from the underlying mass architecture. We consider +the mass architecture primal because planets, foremost, accrete +mass from the protoplanetary disk and, consequently, are char- +acterised by a size that is in accordance with their internal struc- +ture. +5.2. Density architecture +Bulk density (or simply density) is a directly measurable quan- +tity which is sensitive to the internal structure of a planet. +This makes density an important characteristic for understand- +ing planetary structure. The density of a planet depends on +many parameters and many physical processes. For example, a +planet’s mass may depend on its accretion history, starting loca- +tion, amount of material in disk, competition with other planets, +and so on. Giant impacts may also affect a planet’s density, as +explained in Bonomo et al. (2019). In this section, we study the +arrangement and distribution of planetary density around their +host star, namely, the density architecture of a system. +Figure 9 (left) shows the density of a planet, simulated via the +Bern model, as a function of its mass and starting location. The +figure also shows the density of solar system planets and few ob- +served exoplanets (from our catalogue). The plot can be roughly +divided into two halves: (a) planets with a mass of < 100 M⊕ and +(b) planets with a mass of > 100 M⊕. In our simulations, most +planets which started inside the ice line tend to have terrestrial +Earth-like densities. These planets are 0.5 − 3R⊕ and ⪅ 10M⊕. +Planets starting around or outside the ice line generally accrete +more volatile rich material and H/He gas. These planets have +lower densities due to their larger sizes. Planet which started +outside the ice line (3-10 au) show a broad diversity in their den- +sities. As they accrete more gases, their density decreases fur- +ther. These planets are roughly 2 − 10 R⊕ and are characterised +by masses that vary by four orders of magnitude. Planets more +massive than 100 M⊕ seem to lie on a single curve. Since the size +of these planets remains the same (≈ 1RJ or 11R⊕,), their densi- +ties increases linearly with their masses. Planets that started in +the outer regions (30-40 au) cluster on the density-mass plane. +These planets have low densities (< 2g/cm3) and low masses +(⪅ 1 M⊕). +The density architecture for simulated systems in the Bern +Model is shown in Fig. 9 (middle). An important relation be- +tween mass architecture and density architecture is seen. Some +systems which are ordered (or anti-ordered) in mass are also or- +dered (or anti-ordered) in density, that is, these systems have +large positive (or negative) CS (ρ). In other words, simulations +suggest that planetary systems can also be ordered or anti- +ordered in density. A system is ordered in density when the in- +ner planets have small densities and the outer planets have larger +densities – and vice-versa for density anti-ordered systems. Sys- +tems with mass architectures of similar and mixed are strongly +clustered around CS (ρ) ≈ 0 and CV(ρ) < 1. The inset shows +that similar mass systems tend to have small CV(ρ), while mixed +mass systems have larger CV(ρ). This implies that some systems +that are similar (or mixed) in mass show some similarity (or mix- +ture) in density. A system with a similar density architecture will +host planets that have approximately similar densities. However, +the region CS (ρ) ≈ CV(ρ) ≈ 0 is empty, indicating the absence +of planetary systems where the density of planets (inside out) +is approximately the same. While there are exceptions, overall, +for many systems, the density architecture seems to follow their +mass architecture. +This approximate link between the mass and density archi- +tecture stems from massive planets (> 100 M⊕) whose densities +increase with their mass (see 9 (left)). Systems which do not host +any massive planet are mostly similar in their mass architecture +and have CS (ρ) ≈ 0. The inset shows that the Aryabhata’s num- +ber increases as a system approaches the CS (ρ) ≈ CV(ρ) ≈ 0 +region (see Paper II for the definition of Aryabhata’s number). If +a system has more surviving planets that started from inside the +ice line, then the densities of these planets will be more similar to +each other. This means that the density architecture of a system +shows some dependence on the starting location of a planet. +We also investigated if the relation between the mass and +density architectures is observable. Figure 9 (right) shows the +density architecture for systems from our synthetically observed +catalogues. Also shown is the density architecture of some ob- +served exoplanetary systems for which the mass and radius mea- +surements were available. The density architecture of syntheti- +cally observed catalogues shows a trend which is quite unlike +Fig. 9 (middle). There is an unexpectedly good agreement be- +tween the synthetically observed systems and the observed plan- +etary systems. We attribute the peculiar shape of this plot to the +difficulty of detecting distant planets. Transit and RV observa- +tions favour the detection of planets within ∼ 1 au. Many close- +in planets tend to have Earth-like densities, while planets far- +ther out have lower densities (due to either their volatile rich or +gaseous composition). Overall, this would lead to an observed +density architecture where inner planets have higher densities +and outer planets have lower densities. A situation such as this +will be characterised by negative CS (ρ), which is readily seen +from Fig. 9 (right). +In summary, many synthetic systems show a relationship be- +tween their mass architectures and their density architectures. +Bern model systems that are ordered or anti-ordered in their +mass also tend to be ordered or anti-ordered in their densities. +The dispersion of planetary bulk densities in similar class sys- +tems is lower than mixed class systems. This relation seems to +emerge from massive planets whose densities increases linearly +with their masses (since they cannot grow their sizes any more). +These relations can be considered as a prediction from this work. +As future observations probe the outer parts of an exoplane- +tary system, we may anticipate the discovery of several systems +whose mass and density architectures are closely linked. +5.3. Core and envelope mass architecture +In this section, we show that (a) most simulated planetary sys- +tems inherit their architecture from the underlying core mass +architecture; (b) the accretion of gases tends to accentuate the +underlying core mass architecture, and (c) the observed mass ar- +chitecture of a planetary system is a gateway to studying the core +mass architecture of the system, since the two are strongly cor- +related. Exceptions to the first two statements tend to arise for +those systems undergoing strong, multi-body dynamical effects +such as planet-planet scattering. +The fraction of mass which is partitioned into a planet’s core +and its envelope is governed by planetary formation physics. The +end result is dictated by an interplay of several concurrent pro- +cesses (see Emsenhuber et al. 2021a; Mordasini et al. 2012b, +for discussion). In the core-accretion scenario, giant planets are +formed when planetary cores can undergo run-away gas accre- +tion (Pollack et al. 1996; Alibert et al. 2004, 2005). Proto-planets +that have failed to trigger runaway gas accretion comprise a di- +Article number, page 14 of 28 + +L. Mishra et al.: Architecture Framework I – Four classes of planetary system architecture +100 +102 +104 +Mass [M +] +10 +2 +10 +1 +100 +101 +102 +Bulk Density [g/cm3] +J +S +Similar +Anti-Ordered +Mixed +Ordered +Observations +Solar System +10 +1 +100 +101 +Embryo Starting Distance [AU] +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +CV ( ) +1.00 +0.75 +0.50 +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +CS ( ) +0 +1 +-0.1 +0 +Similar +Anti-Ordered +Mixed +Ordered +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Aryabhata's Number +0.0 +0.5 +1.0 +1.5 +2.0 +CV ( ) +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +0.2 +0.4 +CS ( ) +Bern RV Multis +Bern KOBE Multis +Bern Compact Multis +Sun +Trappist-1 +TOI-178 +Kepler-11 +K2-138 +TOI-561 +K2-266 +KOI-94 +Kepler-107 +Kepler-223 +Other +Observed +Systems +Fig. 9. Density architecture. Left: Bulk density of simulated and few observed planets as a function of their mass and starting locations (for +synthetic planets). The marker indicates the mass architecture of the system to which a synthetic planet belongs to. Middle: Density architecture, +of synthetic planetary systems, as seen through the coefficient of similarity versus the coefficient of variation plot. The marker shape and colour +indicates their host system mass architecture and the system’s Aryabhata’s number (see Paper II), respectively. Right: Density architecture of +planetary systems from the simulated observed catalogue and few observed planetary systems. +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +CS (Core Mass) +2 +1 +0 +1 +2 +CS (Mass) +n +i +Menv [M +] +Bern Model +y = x +Similar +Anti-Ordered +Mixed +Ordered +100 +101 +102 +103 +104 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +CV (Core Mass) +0 +1 +2 +3 +4 +5 +CV (Mass) +n +i +Menv [M +] +Bern Model +y = x +Similar +Anti-Ordered +Mixed +Ordered +100 +101 +102 +103 +104 +0.5 0.0 +0.5 +1.00 +0.75 +0.50 +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +CS (Mass) +Bern +Compact +Multis +0.5 0.0 +0.5 +CS (Core Mass) +Bern +KOBE +Multis +0.5 0.0 +0.5 +Bern +RV +Multis +Fig. 10. Mass architecture as a function of core-mass architecture. Panels compare the mass architecture with the core-mass architecture via the +coefficient of similarity (left) and coefficient of variation (middle). In the left panel, the points corresponding to similar systems are very tightly +clustered on the y = x line and are not visible due to over-plotting of points from other architectures. This signifies the core-mass architecture is +very strongly correlated with the mass architecture for similar systems. The sum of mass in the envelope of each planet in a system is indicated in +colour. The right panel plots the coefficient of similarity for masses and core masses for systems in the synthetically observed catalogues. +verse group of planets: Earths, Super-Earths, mini-Neptunes, and +Neptunes. +The bifurcation of a planet’s mass into its core and its enve- +lope can probe the formation history. For example, in our sim- +ulations, most giant planets (⪆ 1 MJ) have about 1% of their +mass in their cores and the rest is in their gassy envelope. On +the other hand, low mass planets (⪅ 10 M⊕) hardly accrete any +gaseous envelope. However, the mass in a planet’s core and en- +velope is not an observable. Even for the solar system planets, +internal structure models guide our knowledge of core and enve- +lope masses (see Helled et al. 2020, for a review on Uranus and +Neptune). +As giant planets dominated by their H/He envelopes are rare, +we expect a strong correlation between the mass architecture (i.e. +the arrangement and distribution of planetary masses) and the +core-mass architecture (i.e. the arrangement and distribution of +core-masses) to exist also at the system level. In Fig. 10, we show +the coefficient of similarity and the coefficient of variation of +planetary mass as a function of the coefficient of similarity and +the coefficient of variation of core mass. The colour indicates the +total mass of envelope accreted by all planets in a system. +Comparing the coefficient of similarity for planetary masses +and core masses (Fig. 10, left panel), we observe that a large frac- +tion of systems (> 90%) follow the y=x line. This implies that for +most planetary systems, the arrangement and distribution of core +masses is imprinted on the mass architecture of the system. Sys- +tems which show large deviations from the y=x line have gener- +ally accreted a large amount of gaseous envelope. This suggests +that the formation of one or more giant planet is partly responsi- +ble for the deviations. We also observe another important feature. +Planetary systems that are ordered in mass are also often ordered +in their core-masses. Conversely, mass anti-ordered systems tend +to be anti-ordered in their core masses as well. In addition, or- +dered systems are either on or above the y=x line, whereas anti- +ordered systems are either on or below this line. This suggests +that the accretion of gases generally accentuates the underlying +core mass architecture. +Considering the coefficient of variation for masses and core +masses (Fig. 10, middle), we see that most of the planetary sys- +tems lie either on or above the y=x line. The CV value measures +the amount of variation in a set of numbers. This suggests that +the variation in total masses, for most systems, is either similar +or larger than the variation in the core masses. This is under- +standable, since the amount of gas accreted by a planet shows +a strong correlation with the mass of the planet’s core. How- +ever, there are a handful of systems where the variation in total +Article number, page 15 of 28 + +A&A proofs: manuscript no. 43751corr +mass is less than the variation in core masses. Systems that are +similar in the mass architecture are strongly clustered over the +y=x line. This stems from the low amount of gas (0 − 20M⊕) +accreted by planets in these systems. Figure 10 (middle) shows +that mixed class systems, as opposed to similar systems, form +a separate cluster. Physically, this difference is arising from the +larger amount of gas (50−5 000M⊕) accreted by planets in these +systems. +Here, the question arises as to whether the strong correlation +between mass architecture and core-mass architecture is observ- +able. In Fig. 10 (right), we show CS (M) as function of CS (Mcore) +for the three synthetically observed catalogues. All three cata- +logues probe the inner regions of a planetary system. The figure +shows that the correlation between mass architecture and core +mass architecture is strong in all three catalogues. This suggests +that the observed mass architecture of a planetary system can be +used to study the underlying core-mass architecture of the sys- +tem. This is potentially useful to distinguish among competing +models of planet formation. +5.3.1. Role of embryo starting location +We have seen that the core mass architecture of a system strongly +governs the overall architecture of the system. The arrangement +of planets in a system also reflects the final distances of these +planets. It is, therefore, instructive to understand some key as- +pects which shape these two important properties. The core mass +and the final distance of a planet are strongly influenced by, +among other effects, the distance at which an embryo starts in +our simulations. Figure 11 shows the core mass (left) and the +final distance (right) as a function of the starting distance. In +the Bern model, lunar mass (0.01M⊕) protoplanetary embryos +are initialised with a random starting location between the inner +edge of the disk and 40 au. We also recall that failed embryos +(objects with a total masses < 0.1M⊕) are removed from our +analysis. +Emsenhuber et al. (2021a); Burn et al. (2021) analysed the +nature of planetary migration using migration maps. Both stud- +ies show the existence of so-called convergence zones. Within +these zones, planets can migrate outwards. However, outside this +zone inward migration is prevalent. The existence of such con- +vergence zones suggests that there ought to be regions of planet +over-densities; this are essentially regions where planets are ra- +dially ‘stuck.’ These studies attribute the presence of these zones +to dust opacity transitions and disc structures, finding that these +zones evolve with the disc. For a 0.01M⊙ disc, around a solar +mass star at 1Myr, these zones are: (a) for low-intermediate mass +planets (⪅ 1M⊕) extending from disk inner edge to about 1au and +(b) for intermediate mass planets (1 − 10M⊕) around 2-3 au. +Figure 11 (left) shows that even for embryos that start at +the same initial distance, the mass accreted by a planetary core +can differ by two to three orders of magnitude. These differ- +ences arise from (a) varying solid disc masses; (b) competition +for accretion in the feeding zone (Alibert et al. 2013); (c) dy- +namical state of solids in the disc resulting from planetesimal- +planetesimal, planetesimal-protoplanet, planetesimal-gas disc +interactions, and so on. Nevertheless, the starting distance seems +to play a significant role in this scenario. The ice line seems to +divide the parameter space into two regions: fewer planets inside +the ice line have low mass cores (⪅ 1M⊕), while many planets +outside the ice line have low-mass cores. +Inside the ice line, most planets have cores of 1−10M⊕. Plan- +ets that start very close to the star (⪅ 0.1au) are unable to accrete +a lot of material owing to their small Hill spheres. This explains +their small cores masses. Inside the ice line, planets belonging to +systems of mixed, anti-ordered, and ordered architecture tend to +have more massive cores than planets belonging to similar sys- +tems. Around the ice line, planets show a large variety of core +masses ranging from 0.1M⊕ to 100M⊕. Outside the ice line we +see the same trend as before: planets that are in similar systems, +for the same starting location, usually have less massive cores +than planets which belong to systems of other architectures. +The final distance of a planet depends on several factors such +as: (a) migration type (type I or type II), (b) planet’s mass, (c) +local disc properties, and (d) multi-body effects such as N-body +scattering. The joint distribution of a planet’s final and starting +locations shows an intriguing trend. Generally, for many plan- +ets, the final distance strongly correlates with their starting loca- +tion. Orbital migration allows planets to move (mostly) inwards +– positioning many planets below the y=x line. N-body effects +(such as planet-planet scattering or outward migration) may scat- +ter some planets further away from their host star. These planets +are located above the y=x line. Curiously, many planets which +end up farther away than their starting location were initialised +around the ice line and are mostly low massive (⪅ 20M⊕). We +attribute this over-density to the outward migration convergence +zone around the ice line discussed above. +Another important finding is that planets inside the ice line +in similar systems probably formed in situ. Fig. 11 (right) shows +that most planets, inside the ice line, which did not migrate in- +wards are part of similar architecture systems. Conversely, most +of the planets which have migrated inwards seem to belong to +systems that have mixed, anti-ordered, and ordered architectures. +Outside the ice line, many planets have migrated inwards. Most +planets starting around 20 au (or more) accrete little mass in their +cores and show little radial displacement (Hansen & Murray +2012; Chiang & Laughlin 2013). The properties of these em- +bryos may draw some influence from our modelling choice as +well. The N-body integrator in this model is used for 20 Myr. +Longer integration times may allow some embryos to have more +massive cores via giant impacts. +5.4. Core water-ice mass fraction architecture +Our model calculates the internal structure of a planetary core +(for details see Emsenhuber et al. 2021a; Mordasini et al. 2012a). +We solved 1D differential equations demanding mass conser- +vation and hydrostatic equilibrium, with a modified polytrope +equation serving as the equation of state (Seager et al. 2007). +The chemical composition of each planetary core is also tracked. +This is accomplished by tracking the chemical makeup of the +accreted planetesimals and other colliding planets. The underly- +ing chemical models have thirty-two refractory and eight volatile +species (Thiabaud et al. 2014; Marboeuf et al. 2014a,b). These +different chemical species are grouped into three different ma- +terials which make the planet’s core, in our model: (a) iron, (b) +silicates, and (c) ice. All refractory species (except iron) make +up the silicate mantle and all volatile species contribute to ice. +Since H2O constitutes 60% of all ice by mass, we label this +latter component as water ice. The water mass fraction ( fice) of +each planetary core is computed. +We assume that inside the H2O ice line, only refractory ele- +ments contribute to the solid phase of a planetesimal. Outside +this evolving ice line, due to their condensation, volatile ele- +ments also contribute to the solid phase of a planetesimal. Fig- +ure 12 (left) shows the water mass fraction of a planet’s core as +a function of its initial location. Most planets which start inside +the ice line have little to no volatiles in their cores. A jump in +Article number, page 16 of 28 + +L. Mishra et al.: Architecture Framework I – Four classes of planetary system architecture +Fig. 11. Role of starting location. Plot shows the planetary core mass (left) and final distance (right) versus the starting distance. The marker style +indicates the architecture of the system to which the planet belongs. The vertical grey shaded region indicates the evolving locations of the ice line +(Burn et al. 2019). The dotted line in the right panel shows the y=x line. +0.0 +0.5 +1.0 +1.5 +2.0 +CV (fice) +1 +0 +1 +2 +3 +4 +5 +CS (fice) +Bern Model +Similar +Anti-Ordered +Mixed +Ordered +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Aryabhata's Number +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +fice +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +Density +"Dry" +"Moist" +"Wet" +Bern Model +Similar +Anti-Ordered +Mixed +Ordered +Fig. 12. Planetary core water-ice mass fraction. Left: Core water mass fraction of a planet as a function of its starting location. The architecture +of the system to which a planet belongs to is shown by marker characteristics. The vertical shaded regions shows the location of the ice line. +Middle: Water mass fraction architecture seen through the coefficient of similarity versus the coefficient of variation plot. The shape of the marker +shows a system’s mass architecture, and the colour depicts its Aryabhata’s number (see Paper II for definition). Right: Distribution of fice across +architecture classes. Depending on fice, planets are labelled as ‘dry’, ‘moist’, or ‘wet’. +fice is seen around the ice line. Outside the ice line, most plan- +ets have at-least 40% fice in their cores. This suggests that the +history of formation and evolution of a planet is imprinted on its +water mass fraction. +We are interested in studying the ice mass fraction architec- +ture of a planetary system. However, we cannot directly apply +our framework (Eqs. 1, 2) because the water mass fraction is +a quantity that admits 0 as a value. While this can lead to ill- +defined numbers, this issue has a simple remedy. For quantities +that can be 0, we propose the following modification to Eq. 1: +CS (q) = lim +ϵ→0 +1 +n − 1 +i=n−1 +� +i=1 +� +log qi+1 + ϵ +qi + ϵ +� +. +(4) +Numerically, we calculated the coefficient of similarity with +ϵ = 10−10. We verified this step by calculating the coefficient +of similarity for quantities which do not admit zero (such as +masses). In a bootstrapped numerical experiment of 10,000 tri- +als, the coefficient of similarity for mass was calculated using +both equations (1 and 4). The relative difference between the +two outcomes ranged between 10−12 to 10−10. +The ice mass fraction architecture of Bern Model systems is +shown in Fig. 12 (middle). A prominent feature from this fig- +ure is that most systems have CS ( fice) either close to 0 or posi- +tive. A system with CS ( fice) ≈ 0 and low CV(fice) will be com- +posed of planets whose core water mass fraction is similar to +one another. A system with positive CS ( fice) will be composed +of planets whose core water mass fraction increases inside out. +Article number, page 17 of 28 + +Bern Model +102 +Similar +Anti-Ordered +Mixed +Ordered +10 +10 +10-1 +10-1 +100 +101 +Embryo Starting Distance [AU]10 +Bern Model +Similar +Anti-Ordered +[AU] +102 +Mixed +Ordered +Planet Final Distance [ +101 +RV +Multis +KOBE +100 +Multis +Compact +Multis +10- +10- +10-1 +100 +101 +Embryo Starting Distance [AU]0.6 +Bern Model +Similar +0.5 +Anti-Ordered +Mixed +"Wet" +Ordered +0.4 +0.3 +0.2 +"Moist' +0.1 +0.0 +10-1 +100 +101 +Embryo Starting Distance [AU]A&A proofs: manuscript no. 43751corr +Figure 11 (right) tells us that many planets that started outside +the ice line, and are water rich have not suffered any major ra- +dial displacement. Thus, a positive CS ( fice) should be a default +scenario for most planetary systems. About 74% systems in the +Bern model have CS ( fice) > 0.1. Almost 97% of systems have +CS ( fice) > 0. We propose the ‘Aryabhata formation scenario’ to +explain the ‘non-default’ systems. This scenario and the related +quantity ‘Aryabhata’s Number’ are described in Paper II. +5.5. Frequency of dry, moist, and wet planets +We are interested in exploring the link between the water mass +fraction architecture and the mass architecture of a system. To +this end, we divide planets into three categories based on their +water mass fraction. A planet is called ‘dry’ if fice ≤ 1%, ‘moist’ +if fice ∈ (1, 40]%, and ‘wet’ if fice > 40%. These labels serve to +simplify our analysis and allows us to see general trends between +system architecture and planetary composition. The distribution +of water mass fraction across systems of different architecture +classes is shown in Fig. 12 (right). While all three planet classes +are present in all four architecture classes, there are some ob- +servable trends. +Figure 12 (right) shows that similar architectures host many +of the dry planets produced in the Bern model and anti-ordered +architectures are mostly composed of wet planets. This tells us +that many of the planets that start inside the ice line become part +of similar architecture systems. Conversely, systems with anti- +ordered architecture are mostly composed of planets that started +outside the ice line. Mixed architecture systems are generally +composed of more planets that started outside the ice line than +inside, as compared to similar architecture systems. Moist plan- +ets are present in all architecture classes. We quantify the fre- +quency of dry, moist, and wet planets as a function of mass archi- +tecture class (similar, mixed, ordered, or anti-ordered), metallic- +ity (low or high), and source catalogues (Bern model, Bern Com- +pact Multis, Bern KOBE Multis, and Bern RV Multis). Figure 13 +shows the planets per star (i.e. the number of each planet type di- +vided by the number of stars) across these forty sub-categories. +Overall, compared to synthetically observed catalogues, +Bern model simulations demonstrate more wet planets. This +is understandable since we are looking at the entire underly- +ing population, which includes planets from the outer parts of +these systems. Likewise, synthetically observed catalogues tend +to have more dry planets. Systems around low-metallicity stars +(regardless of the catalogue) generally tend to have a higher +frequency of dry planets as opposed to systems around high- +metallicity stars. The frequency of wet planets shows a notice- +able increase for systems around high-metallicity stars. Amongst +the different catalogues, Bern Compact Multis have the highest +frequency of dry planets, followed by Bern KOBE Multis, and +Bern RV Multis. Low-metallicity environments have a slightly +higher average planet per star (8835/541 ≈ 16.3) than high- +metallicity environments (6722/455 ≈ 14.8). +Similar systems: Systems in the underlying Bern model that +are characterised by a similar architecture tend to have many wet +planets (∼ 10 per star) and few dry or moist planets (∼ 3 − 4 +per star). However, synthetically observed catalogues seem to +have a bias against the discovery of many wet planets. For the +similar class of compact multi-planetary systems, dry planets +are more common around a low-metallicity star. However, for +a high-metallicity star, the frequency of dry and wet planets is +roughly the same. For transiting systems, in the Bern KOBE +Multis, low-metallicity environments favour more dry planets +and equal proportions of wet and moist planets. Conversely, +in high-metallicity environments, wet planets occur more fre- +quently than dry or moist planets. For RV systems, the fre- +quency of each planet class is approximately the same in a low- +metallicity environment. High-metallicity environments almost +double the frequency of wet planets. The average planet per star +is similar around both low metallic (≈ 16.8) and high metallic +environments (≈ 17.3). +Mixed systems: Mixed class systems generally have many +wet planets. It is only for compact systems around high- +metallicity stars, the frequency of dry planets is higher than wet +planets. In all other cases, the frequency of wet planets is greater +than the frequency for dry or moist planets. The average planet +per star is similar around both low-metallicity (≈ 15.2) and high +metallicity environments (≈ 15.3). +Anti-Ordered systems: Systems with anti-ordered architec- +ture have a distinct core water mass fraction architecture. These +systems are rich in wet planets. In fact, about 80% of these sys- +tems follow the Aryabhata formation scenario described in Pa- +per II. Compact anti-ordered systems may have some dry plan- +ets. For transit and RV surveys, the frequency of dry planets is +zero in our simulations. The total number of planets per star +in anti-ordered systems is slightly higher around low metallic- +ity stars (159/19 ≈ 8.4), as compared to high metallicity stars +(504/65 ≈ 7.8). In the future, if an anti-ordered architecture +planetary system is to be discovered, it would be interesting to +study its core water mass fraction architecture as well. The cur- +rent work suggests that the Aryabhata’s number for these sys- +tems should be close to 0 and, irrespective of the detection tech- +nique, the system should would be expected to have many wet +planets (see Paper II); this is one of the main predictions arising +from this work. +Ordered systems: Juxtaposed directly to the anti-ordered sys- +tems, ordered systems in synthetically observed catalogues tend +to be rich in dry planets. These systems are distinct not only +because of their frequent dry planets, but also due to a low fre- +quency of wet planets. For all synthetic catalogues, moist plan- +ets occur more frequency than wet planets, which is a unique +distinguishing feature for these systems. For the Bern model, +these systems have low average planets per star: 5 around low- +metallicity stars and 3.1 around high-metallicity stars. +In summary, we note some salient features of these sys- +tem architectures. Generally, wet planets survive more fre- +quently around high-metallicity stars. One detection technique +that favours the discovery of close-in planets also favours the de- +tection of dry planets. The comparative frequency of planet (dry, +wet, or moist) per star seems to be intimately connected with +the mass architecture of the system. Similar and mixed systems +can host lots of dry or wet planets, depending on the metallicity +of the systems and detection technique. Anti-ordered systems, +forming prominently via the Aryabhata formation scenario, are +rich in wet planets. Ordered systems, in simulated observations, +are rich in dry planets and have more moist planets than wet +planets. The physical connection between the average planet per +star and the star’s metallicity is sensitive to the formation path- +ways that a system undergoes. +6. Habitability as a function of system architecture +In this paper thus far, we have described a new framework for +studying the architecture of planetary systems (Sect. 3), the char- +acteristics of the four classes of system architecture (Sect. 4), +and the relation between the mass architecture of a system and +its internal structure and composition architecture (Sect. 5). In +Article number, page 18 of 28 + +L. Mishra et al.: Architecture Framework I – Four classes of planetary system architecture +1 +2 +3 +0 +5 +10 +Overall +N*=541 Npl=8835 +[Fe/H] +0 +1 +2 +3 +Similar +N*=499 Npl=8346 +1 +2 +3 +Mixed +N*=21 +Npl=320 +1 +2 +3 +Anti-Ordered +N*=19 +Npl=159 +1 +2 +3 +Bern Model +Ordered +N*=2 +Npl=10 +1 +2 +3 +0 +2 +4 +Planets per star +N*=221 Npl=1312 +1 +2 +3 +N*=186 Npl=1152 +1 +2 +3 +N*=5 +Npl=27 +1 +2 +3 +N*=7 +Npl=29 +1 +2 +3 +Bern Compact Multis +N*=23 +Npl=104 +1 +2 +3 +0 +1 +2 +3 +N*=662 Npl=3569 +1 +2 +3 +N*=542 Npl=3034 +1 +2 +3 +N*=6 +Npl=27 +1 +2 +3 +N*=24 +Npl=102 +1 +2 +3 +Bern KOBE Multis +N*=90 +Npl=406 +Dry +Wet +Moist +0 +1 +2 +3 +N*=273 Npl=1796 +Dry +Wet +Moist +N*=228 Npl=1577 +Dry +Wet +Moist +N*=11 +Npl=63 +Dry +Wet +Moist +N*=2 +Npl=10 +Dry +Wet +Moist +Bern RV Multis +N*=32 +Npl=146 +1 +2 +3 +0 +5 +10 +Overall +N*=455 Npl=6722 +[Fe/H] > 0 +1 +2 +3 +Similar +N*=303 Npl=5230 +1 +2 +3 +Mixed +N*=61 +Npl=935 +1 +2 +3 +Anti-Ordered +N*=65 +Npl=504 +1 +2 +3 +Bern Model +Ordered +N*=13 +Npl=40 +1 +2 +3 +0 +2 +4 +Planets per star +N*=179 Npl=1100 +1 +2 +3 +N*=151 +Npl=967 +1 +2 +3 +N*=4 +Npl=21 +1 +2 +3 +N*=6 +Npl=30 +1 +2 +3 +Bern Compact Multis +N*=18 +Npl=82 +1 +2 +3 +0 +1 +2 +3 +N*=621 Npl=3146 +1 +2 +3 +N*=518 Npl=2682 +1 +2 +3 +N*=9 +Npl=36 +1 +2 +3 +N*=27 +Npl=111 +1 +2 +3 +Bern KOBE Multis +N*=67 +Npl=317 +Dry +Wet +Moist +0 +1 +2 +3 +N*=292 Npl=2032 +Dry +Wet +Moist +N*=232 Npl=1720 +Dry +Wet +Moist +N*=26 +Npl=163 +Dry +Wet +Moist +N*=5 +Npl=21 +Dry +Wet +Moist +Bern RV Multis +N*=29 +Npl=128 +Fig. 13. Frequency of planets. This diagram shows the average planet per star for dry, wet, and moist planets in several catalogues (rows), across +several architecture classes (columns), and around low (left) and high (right) metallicity stars. The planet per star is simply the total number of +planets divided by the total number of stars, after appropriate filters for metallicity, catalogue, or architecture. +this section, we speculate on the idea of studying habitability as +a function of system-level architecture. +Mankind has pondered the existence of other biotic life- +forms beyond Earth, as well as outside our own Solar System. +Our current understanding of habitability stems from and is fo- +cused at an individual planetary level. We consider whether hab- +itability could be correlated with other properties of a plane- +tary system, namely, whether habitability could be a system- +level phenomenon. In this section, we speculate on the role of +planetary-system level information on the existence of habit- +able worlds in such systems. The framework we present here +for studying the system-level architecture of a planetary sys- +tem brings to light several novel questions, probing the depen- +dence of habitability and occurrence of habitable worlds (and +related concepts) on the architecture of a said system. For ex- +ample, we wonder how the occurrence rate of habitable planets +in the galaxy depends on the occurrence of the four architecture +classes. +In this section, we address this question on three levels: sys- +tem, planet, and planet ratio. We use the concept of empirical +Habitable zone (EHZ) planets from Quanz et al. (2022); Koppa- +rapu et al. (2014). Planets with masses between [0.1, 5]M⊕ and +stellar insolation within [1.776, 0.32]S ⊕ are considered to be in- +side the EHZ. The stellar flux limits correspond to ‘recent Venus’ +and ‘early Mars’ scenarios and include the luminosity evolution +for a 1M⊙ Solar-twin. At the system level, we note the frequency +of systems of a particular architecture to host at least one planet +in the EHZ. At the planet level, we count the frequency of planets +in the EHZ across each system architecture class. At the planet +ratio level, we show the fraction of all EHZ planets across their +architecture class. Figure 14 shows the frequency of EHZ plan- +ets, at all three levels, as a function of their system architecture +for both synthetic and observed exoplanetary systems. +Out of all synthetic systems with a similar class architec- +ture, ≈ 77% host at least one EHZ planet. This is remarkably +higher than any other architecture class. ≈ 10% of systems with +mixed architecture host at least one EHZ planet. The frequency +drops to ≈ 1% for anti-ordered architecture systems and ≈ 0% +for ordered systems. One way to interpret these numbers could +be to look at the multiplicity distribution across each architecture +class in Fig. 7. The frequency of at least one EHZ planets across +architecture class seems to follow the multiplicity trends. Sim- +ilar and mixed architectures have comparably high number of +planets. The distribution of the Aryabhata’s number shows that +similar systems usually have higher Aryabhata’s number than +mixed systems, implying that similar systems tend to host more +planets which started from inside the ice line (see Paper II for +Aryabhata’s number). This may account for the large frequency +of similar systems which host at least one EHZ planet. The mul- +tiplicity distribution shows that anti-ordered systems often host +less planets than similar and mixed class systems, while ordered +systems have the lowest multiplicities. We see in Sect. 4.2 that +the similar class architecture is perhaps the most common archi- +tecture for planetary systems in our galaxy. These results from +the Bern model simulations suggest that observation campaigns +to detect habitable planets will find more EHZ planets in similar +class architectures. +For the observed multi-planetary systems in our catalogue, +about ≈ 13% of similar class systems have at least one EHZ +planet. About 7% of ordered class exoplanetary systems in our +catalogue host at least one EHZ planet. In the mixed class ob- +served systems in our catalogue, none of them have EHZ planets +and there are no known anti-ordered class systems in our cata- +logue. These frequencies are quite different from their theoretical +counterparts. While the lack of a complete and reliable obser- +vations catalogue may explain the discrepancy for similar class +systems – it does not completely explain the discrepancy for or- +dered systems. Our own planet resides in the ordered class sys- +tem of the Solar System, which is not supposed to be influenced +by issues such as completeness or detection biases. This reflects +the inability of Bern models to simulate a Solar System analogue +– pointing to a gap in our understanding of the physics that goes +into planetary formation and evolution. In addition, many ob- +served ordered class systems may have a different architecture +when more planets in these systems are detected. +Article number, page 19 of 28 + +A&A proofs: manuscript no. 43751corr +Similar +Anti-Ordered +Ordered +Mixed +0 +20 +40 +60 +80 +100 +Frequency [%] +Planets in EHZ +Bern Model: Systems +Observations: Systems +76.7% +1.2% +0.0% +9.8% +12.5% +0.0% +6.7% +0.0% +Similar +Anti-Ordered +Ordered +Mixed +0 +20 +40 +60 +80 +100 +Frequency [%] +Planets in EHZ +Bern Model: Planets +Observations: Planets +10.2% +0.2% +0.0% +1.0% +5.2% +0.0% +2.9% +0.0% +Similar +Anti-Ordered +Ordered +Mixed +0 +20 +40 +60 +80 +100 +Frequency [%] +Planets in EHZ +Bern Model: EHZ Planet Ratio +Observations: EHZ Planet Ratio +0.1% +0.0% +0.9% +75.0% +0.0% +25.0% +0.0% +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +fice +0 +5 +10 +15 +20 +25 +Density +"Dry" +"Moist" +"Wet" +Bern Model +EHZ Planets +Similar +Mixed +Fig. 14. Planets inside the empirical habitable zone (EHZ). The left-most plot shows the frequency of planetary systems, of a given architecture +class, which host at least one planet inside the EHZ. The central-left plot shows the fraction of planets inside a given architecture class which are +in the EHZ. The central-right plot shows the fraction of all EHZ planets within a given architecture class. The rightmost plot shows the distribution +of fice for EHZ planets across the architecture classes. The cartoon sketch of Earth emphasises that the only known life-harbouring planet resides +in an ordered system. The length of error bars visualises the total number of systems or planets in respective bin as: 100/ +√ +bin counts. The lengths +of the error bars represents the number of planetary systems (left-most panel) and the number of planets (two middle panels) which are inside +the bin. Large error bars in the leftmost panel, for example for anti-ordered architecture emerges from their low count (see Fig. 5). The Gaussian +kernel is estimated using Scott’s rule (Scott 2015) +At the planet level in our simulations, out of all synthetic +planets that exist in similar class systems, about 10% are inside +the EHZ. This frequency is, again, remarkably higher for any +other architecture class. About 1% of all simulated planets in a +mixed system are inside the EHZ. Close to 0% of all planets in +anti-ordered and ordered class architectures are inside the EHZ. +From our observational catalogue, while 5% of observed exo- +planets in similar class systems are inside the EHZ. About 3% +of observed exoplanets in ordered class systems are inside the +EHZ. +The planet ratio level shows the fraction of all EHZ planet +that belong to a particular architecture class. In the Bern model, +we see that out of all EHZ planets, about 99% are in the simi- +lar class. The share of EHZ planets by other architecture classes +is negligible. Amongst the observations, three-quarters of EHZ +planets are in similar class and the remaining are in ordered +class. The observations and theory are quite misaligned in this +scenario. We attribute this discrepancy to the absence of a com- +plete and reliable catalogue of observations. +Our observations catalogue has only 41 multi-planetary sys- +tems, of which only four host planets inside the EHZ. These +systems are Trappist-1 (three planets in EHZ), GJ 667 C (two +planets in EHZ), Solar System (two planets in EHZ), and Tau +Ceti (one planet in EHZ). The occurrence of architecture classes +and the frequency with which they host EHZ planets might be +better constrained with future observations. This may allow us +to have a better estimate of the occurrence rate of EHZ planets +as a function of architecture class. +Simulations suggest that ordered architecture is a rare out- +come of planet formation (about 1.5% of systems out of 1000 +were deemed to be ordered) and yet, we live in an ordered sys- +tem. These two statements can shed new light on the rarity of life +in the galaxy. We foresee that the famous Drake equation may be +suitably modified to take into account the occurrence rate of dif- +ferent architectures and thereby set more optimal constraints on +η⊕ (Sarkar 2022). +Since water plays a fundamental role for life forms on Earth, +it is interesting to probe the core water-ice fraction for the EHZ +planets. Figure 14 also shows the fice distribution for EHZ plan- +ets in the Bern model. As we see before, most of the EHZ plan- +ets are in the similar class and ≈ 1% of EHZ planets are in the +mixed class. EHZ planets in similar systems are ‘dry’, ’‘moist’, +and ‘wet’. In stark contrast, EHZ planets in mixed class are only +‘wet’ planets. We hope these results may be useful in guiding +future missions in finding EHZ planets that have the potential to +harbour life. +7. Summary, conclusions, and future work +In this paper, we introduce and explore a new framework for +studying the architecture of planetary systems. Our new frame- +work allows us to study, quantify, classify, the global architecture +of an entire planetary system at the system-level; and compare +the architecture of one planetary system with another. In Sect. 3, +we detailed the new architecture framework and presented an in- +depth discussion comparing our framework with other works in +the literature. We present the coefficient of similarity and the co- +efficient of variation as two quantities that quantify our concep- +tual ideas. Our framework gives rise to a new parameter space +(the CS versus CV plane) in which individual planetary systems +can be compared with one another. Throughout this paper, we +applied this framework to study the distribution and arrange- +ment of several planetary quantities within a planetary system, +thereby understanding the system architecture for that quantity. +In this manner, we studied the mass architecture, the radius ar- +chitecture, the core mass architecture, the core water mass frac- +tion architecture, and the density architecture of synthetic and +observed planetary systems. +To study some consequences of this framework, we applied +our method to several catalogues of planetary systems (intro- +duced in Sect. 2). We curated, especially for the purposes of this +study, a catalogue of observed multi-planetary systems that have +four or more planets and include mass measurements for at least +four planets. For engendering further studies, additional stellar +and planetary properties were collected and presented in Table 1. +We also used synthetic planetary systems simulated via the Bern +model. To facilitate a comparison of theory with observations, +we prepared three synthetic observed catalogues by applying the +detection biases on the simulated planetary systems. This led to +the Bern RV Multis, Bern KOBE Multis, and the Bern Compact +Multis catalogue. We note that there are caveats present in the +datasets we used. The model-dependent results we present here +may be improved upon in future studies using better theoretical +Article number, page 20 of 28 + +L. Mishra et al.: Architecture Framework I – Four classes of planetary system architecture +models and a more complete observational catalogues (e.g. from +PLATO). +Summary of architecture framework: +1. The architecture framework is model-independent and there- +fore does not suffer from any caveats emerging from planet +formation theory or observations. +2. The same architecture framework can be used to study the +multi-faceted aspects of planetary system architecture. When +the framework is applied to study planetary masses, the +framework informs us of the mass architecture of the sys- +tem, namely, the arrangement and distribution of masses in +the planetary system. In this way, we can use this framework +to study the mass architecture, radii architecture, eccentricity +architecture, and so on for the same system. In this series of +work, we identified the architecture of a system with its bulk +mass architecture. +3. Planetary system architecture can be one of four classes that +are derived from our framework: similar, mixed, ordered, and +anti-ordered. +4. A planetary system’s architecture is of similar class when the +masses of all the planets within such a system are similar to +each other. This architecture class corresponds to the ‘peas +in a pod’ architecture trend reported in the literature. +5. The architecture class of a planetary system is ordered (or +anti-ordered) when the planetary masses in such systems +tend to increase or decrease from inside-out. +6. Planetary systems of mixed class architecture host planets +whose masses show broad increasing and decreasing varia- +tions. +Our key model-dependent findings are as follows: +1. Frequency of architecture class: Systems with similar bulk +mass architecture are the most common outcome of simula- +tions, followed by the other three architecture classes. Our +model suggests that similar architecture should be the most +common exoplanetary system architecture in our Galaxy and +beyond. This explains why radius similarity in exoplanets +was already detected from the first four months of Kepler +data (Lissauer et al. 2011). +2. Distance bi-modality: We found hints of a bi-modality in +the exoplanetary distance distribution arising from the two +different modes of orbital migrations. This bi-modality is +readily visible (see Fig. 7) for similar and mixed mass ar- +chitecture exoplanetary systems observed via RV. +3. Core mass architectures: We found that for most systems, +the bulk mass architecture is inherited from the core mass ar- +chitecture. In addition, the accretion of gases tends to high- +light the underlying core mass architecture by amplifying it. +In this way, the observed mass architecture of a system could +serve as a gateway for studying the distribution and arrange- +ment of the planetary core masses, which tends to be simpler +for theoretical modelling. +4. In situ formation: We found that most planets belonging to +the similar bulk mass architecture class form in situ inside +the ice line. In contrast, planets inside the ice line belong- +ing to mixed, anti-ordered, and ordered show large inward +migrations. +5. Core water-ice mass fraction architectures: Synthetic +planetary systems were found to have two scenarios for their +core water mass fraction architecture. The default scenario +consists of relatively more dry planets in the inner parts of +a system and more wet planets in the outer parts of the sys- +tem. This is probably a direct consequence of the starting +location of planets: planets starting inside (or outside) the +ice line tend to be dry (or wet). About one-fifths of simulates +systems do not follow the default scenario described above. +We propose the ‘Aryabhata formation scenario’ to explain +their core-water mass fraction architecture (see Paper II). +6. Linking architecture and internal composition: We found +that wet planets are more likely to survive around high- +metallicity stars. Among other predictions, we showed that +anti-ordered observed systems should be rich in wet worlds, +while ordered observed systems are expected to have many +dry planets (based on the core-accretion planet formation +paradigm). +7. Density Architectures: Synthetic systems that are ordered +(or anti-ordered) in mass tend to also be ordered (or anti- +ordered) in their bulk densities. Some mass similar systems +may also have low dispersion in their planetary bulk densi- +ties. The density architecture is sensitive to the Aryabhata’s +number (i.e. the starting location of various surviving plan- +ets; see Paper II). The density architecture of observed sys- +tems is in good agreement with the density architecture of +synthetically observed simulated systems. Detection biases +seem to favour the discovery of planetary systems where the +densities show anti-ordering, mixing, or similarity. +8. Radius architectures: The radius architecture of most plan- +etary systems closely follows their mass architecture. There- +fore, most mass similar systems also show similarity in ra- +dius (also for mass mixed, ordered, or anti-ordered systems). +However, this is not always true. Future studies can calibrate +a classification scheme based on planetary radii. +9. Habitability as a system-level phenomenon: We reflected +on the prospect of studying habitability as a function of +system-level properties such as system architecture. Similar +architecture systems represent an excellent observation tar- +get for finding life outside the solar systems because these +systems tend to host many more planets inside the empirical +habitable zone that other architecture classes. +10. The current version of the Bern model seems to have dif- +ficulty in producing planets inside the EHZ of an ordered +architecture system. Nevertheless, more data is required to +conclude whether the existence of Earth, an inhabited planet +in an ordered system, is an exception or whether there are +additional gaps in our understanding of planet formation. +This paper is the first in a series. The current work presents a +new testing ground, the architecture space, for theoretical mod- +els and for comparing observations with theory. We can now +constrain our understanding of planet formation not only on the +level of an individual planet – but at the global systemic level. +This is a multi-faceted approach, since the system architecture +of several quantities can now be uniformly assessed and com- +pared with observations. In our next paper (Paper II), we show +another important aspect emerging from this architecture frame- +work which asserts that systems with comparable architecture +often have the same formation pathways. We present ideas to +further the nature versus nurture debate around planet forma- +tion. While similar architectures are usually a product of their +starting conditions, stochastic multi-body effects are responsible +for shaping the other three architecture classes. This work leads +to several future studies which will be presented in other papers +in this series. Davoult et al. (in prep.) explore how the present ar- +chitecture framework can be employed for an efficient usage of +telescope time to hunt for habitable worlds. Other possible ex- +plorations that emerge from this work include: (a) a data-driven +approach to classifying planetary architecture based on radii and +Article number, page 21 of 28 + +A&A proofs: manuscript no. 43751corr +(b) a suitable modification to Drake’s equation that accounts for +the empirical occurrence rate of system architectures. +Acknowledgements. The authors thank the anonymous referee for their care- +ful reading, constructive suggestions, and insightful questions, which has al- +lowed the quality of this manuscript to be improved. This work has been car- +ried out within the framework of the National Centre of Competence in Re- +search PlanetS supported by the Swiss National Science Foundation under grants +51NF40_182901 and 51NF40_205606. The authors acknowledge the finan- +cial support of the SNSF. Data: The synthetic planetary populations (NGPPS) +used in this work are available online at http://dace.unige.ch under section +“Formation & evolution”. This research has made use of the NASA Exoplanet +Archive, which is operated by the California Institute of Technology, under con- +tract with the National Aeronautics and Space Administration under the Exo- +planet Exploration Program: https://exoplanetarchive.ipac.caltech. +edu (DOI: 10.26133/NEA6). The artwork used to depict Earth in Fig. 14 is taken +from flaticon.com. 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The model in- +cludes stellar evolution for a solar-mass star, using evolution +tracks from Baraffe et al. (2015). The star interacts with the +protoplanetary disk and influences its thermodynamical proper- +ties. The protoplanetary disk has two phases: gas and solid. We +model this disk using the approaches of viscous angular momen- +tum transport (Lynden-Bell & Pringle 1974; Veras & Armitage +2004; Hueso & Guillot 2005). Turbulence is characterised by the +Shakura & Sunyaev (1973) approach, with α = 2 × 10−3. Gas +from the disk is accreted by planets, host star, and lost via photo- +evaporation. The 1D geometrically thin disk evolution is studied +up to 1000 au. The initial mass of this gas disk and its lifetime +are randomly drawn for each run of the simulation. The solid +phase of the disk is composed of a swarm of planetesimals. The +solid disk is modelled as a fluid which evolves via (a) accretion +by growing planets; (b) interaction with the gaseous disk; (c) dy- +namical stirring from planets and other planetesimals; and so on +(Fortier et al. 2013). The initial mass of the solid disk depends +on the metallicity of the star and also on the condensation state +of the molecules in the disk (Thiabaud et al. 2014). The host star +metallicity is randomly drawn for each run of the simulation. +We added 100 protoplanetary embryos to the protoplanetary +disk. The initial location of each embryo was varied from one +simulation to another. It was also ensured that no two embryos +start within 10 hill radii of each other (Kokubo & Ida 1998, +2002). Embryos accrete from their feeding zones and any over- +lap may lead to competition (Alibert et al. 2013). The accretion +rate depends on the collision probability between a protoplanet +and a planetesimal, which in turn is influenced by the dynamical +state of the solid disk. +Gas accretion occurs in several phases (Mordasini et al. +2012b). Initially, the gas disk smoothly transitions as a gaseous +envelope around all planets – the attached phase. For planets that +are massive enough to undergo runaway gas accretion, the rate +of gas supply from the disk may not be enough. In these scenar- +ios, the planet detaches from the gas disk and rapidly contracts +to RJ. After the gas disk dissipates, all planets are in the isolated +phase. Gas accretion from the disk is no longer possible and in +this phase, the planets contract and cool. For all the planets, their +internal structure is modelled at each time step. We assume plan- +ets are spherically symmetric and composed of accreted materi- +als that arranges itself in layers: iron code, silicate mantle, water +ice, and H/He gaseous envelope (if accreted). +Next, we use these recipes to simulate several thousands of +planetary systems in an approach called population synthesis +(Emsenhuber et al. 2021b). We start 1000 star-disk-embryo sys- +tems with some fixed as well as some randomly drawn proper- +ties. The initial properties are inspired by observations of disks +Tychoniec et al. (2018). The then numerically modelled these +systems, endowing them with additional physics at the same +time. Numerically, we incorporated multi-body dynamical in- +teractions via N-body simulations. Planet-disk interactions lead- +ing to orbital migration and eccentricity and inclination damping +were also incorporated in the N-body Coleman & Nelson (2014); +Paardekooper et al. (2011); Dittkrist et al. (2014). We followed +these numerically intensive steps for 20 Myrs and then stopped +the N-body calculations. The model then continued to evaluate +the internal structure of all planets in the system for 10 Gyrs. +The recent version of these simulations has been published +in the New Generation Planetary Population Synthesis (NGPPS) +series of papers (Emsenhuber et al. 2021a,b; Schlecker et al. +2021a,b; Burn et al. 2021; Mishra et al. 2021). The output of +these models have been compared with observations in several +works. Drazkowska et al. (2022) compares the occurrence rates +of synthetic systems with observations. Schlecker et al. (2021a) +studies the warm Super Earth and cold Jupiter correlation in the +synthetic systems. Mishra et al. (2021) analyse the ’peas in a +pod’ architecture and compare synthetic systems with observa- +tions from Weiss et al. (2018). Mulders et al. (2018) present a +detailed comparison of the synthetic models with Kepler obser- +vations. +Appendix B: Stellar and planetary data references +1. Sun: Archinal et al. (2018); Standish (1992); Wang et al. +(2018); Helffrich (2017); Jacobson et al. (2006); Jacobson +(2014, 2009) +2. Trappist-1: Agol et al. (2021); Gillon et al. (2017); Burgasser +& Mamajek (2017); Grimm et al. (2018) +3. TOI-178: Leleu et al. (2021) +4. HD 10180: Lovis et al. (2011); Kane & Gelino (2014) +5. HD 219134: Seager et al. (2021); Bonfanti & Gillon (2020); +Vogt et al. (2015) +6. HD 34445: Vogt et al. (2017) +7. Kepler-11: Berger et al. (2020); Lissauer et al. (2013) +8. Kepler-20: Fressin et al. (2011); Buchhave et al. (2016) +9. Kepler-80: MacDonald et al. (2016); Shallue & Vanderburg +(2018) +10. K2-138: Lopez et al. (2019) +11. 55 Cnc: Bourrier et al. (2018) +12. GJ 667 C: Anglada-Escudé et al. (2013) +13. HD 158259: Hara et al. (2020); Gáspár et al. (2016) +14. HD 40307: Díaz et al. (2016); Stassun et al. (2019) +15. Kepler-102: Berger et al. (2020); Marcy et al. (2014) +16. Kepler-33: Berger et al. (2020); Lissauer et al. (2012); Had- +den & Lithwick (2017) +17. Kepler-62: Berger et al. (2020); Borucki et al. (2013) +18. HD 20781: Udry et al. (2019) +19. TOI-561: Lacedelli et al. (2021); Weiss et al. (2021) +20. DMPP-1: Staab et al. (2020) +21. GJ 3293: Astudillo-Defru et al. (2017) +22. GJ 676 A: Sahlmann et al. (2016); Stassun et al. (2017) +23. GJ 876: Trifonov et al. (2018); Rojas-Ayala et al. (2012) +24. HD 141399: Hébrard et al. (2016) +25. HD 160691: Go´zdziewski et al. (2007); Pepe et al. (2007) +26. HD 20794: Go´zdziewski et al. (2007); Pepe et al. (2007) +27. HD 215152: Go´zdziewski et al. (2007); Pepe et al. (2007) +28. HR 8799: Marois et al. (2008); Gravity Collaboration et al. +(2019); Swastik et al. (2021) +29. K2-266: Rodriguez et al. (2018) +30. K2-285: Rodriguez et al. (2018) +31. Kepler-89: Berger et al. (2020); Weiss et al. (2013) +32. Kepler-106: Berger et al. (2020); Marcy et al. (2014) +33. Kepler-107: Berger et al. (2020); Bonomo et al. (2019) +34. Kepler-223: Berger et al. (2020); Mills et al. (2016) +35. Kepler-411: Berger et al. (2020); Sun et al. (2019) +36. Kepler-48: Berger et al. (2020); Marcy et al. (2014) +37. Kepler-65: Berger et al. (2020); Mills et al. (2019) +38. Kepler-79: Berger et al. (2020); Yoffe et al. (2021) +39. WASP-47: Vanderburg et al. (2017) +40. tau Cet: Vanderburg et al. (2017) +41. HD 164922: Benatti et al. (2020); Rosenthal et al. (2021a) +Article number, page 24 of 28 + +L. Mishra et al.: Architecture Framework I – Four classes of planetary system architecture +0.0 +0.2 +0.4 +0.6 +0.8 +Maximum tolerance (t) +0.0 +0.5 +1.0 +1.5 +2.0 +Value +max. +sim +th. max. +var +y = x +Fig. C.1. Maximum value of the coefficient of similarity (blue) and the +theoretical maximum value of the coefficient of variation (orange) is +plotted against the maximum tolerance, t. +Appendix C: Derivation of limits +We consider a set Q of quantities q, namely, Q = {qi} where +qi could be the mass, radius or other parameter of a planet, and +the index, i ∈ [1, n], identifies a planet (with 1 being the inner- +most planet). We assume that all qi ≥ 0. The quantities qi are +expressed as: +qi = q′ (1 ± ti). +(C.1) +The quantities, qi, are decomposed around some value q′ such +that all ti are minimised; ti is a measurement of the fractional +difference (or tolerance) between q′ and qi. Since all individual +tolerances are a positive quantity, they will satisfy the following +relation: +0 ≤ ti ≤ t. +(C.2) +Appendix C.1: Mean +Let us consider the mean of the quantities, ¯qi: +¯qi = Q +n = +� +i qi +n += q′ +n +�n ± t1 ± t2 ± · · · ± tn +�. +(C.3) +The mean takes its maximum value only when all individual ti +values take their maximum and are added up. This gives: +max ¯qi = q′ �1 + t�. +(C.4) +Similarly, the minimum value of the mean is: +min ¯qi = q′ �1 − t�. +(C.5) +The extreme value of the mean occurs when all the individual +quantities are extremised. However, in this scenario, since all +quantities are equal, the coefficient of variation is identically 0. +Appendix C.2: Coefficient of similarity +We start with the definition of the coefficient of similarity, +CS (q) = +1 +n − 1 +i=n−1 +� +i=1 +� +log qi+1 +qi +� +. +(C.6) +Inserting Eq. C.1, in the definition, we get: +CS (q) = +1 +n − 1 +i=n−1 +� +i=1 +� +log 1 ± ti+1 +1 ± ti +� +. +(C.7) +This formulation shows that the coefficient of similarity depends +only on the fractional differences (tolerances) between qi values +– and not on their actual values. This is a desirable property. +Next, we evaluate the maxCS as, +maxCS (q) = max +� +1 +n − 1 +i=n−1 +� +i=1 +� +log 1 ± ti+1 +1 ± ti +�� +, += +1 +n − 1 max +� i=n−1 +� +i=1 +� +log 1 ± ti+1 +1 ± ti +�� +, += +1 +n − 1 +i=n−1 +� +i=1 +log max +�1 ± ti+1 +1 ± ti +� +. +(C.8) +In the first step, we commuted the max operator with the frac- +tion (n − 1)−1 because we are interested in the maximum for a +constant n. Next, knowing that the maximum of a sum occurs at +the sum of maximum summands and that log is a monotonically +increasing function, we further commute the max operator. +We observe the following: +max +�1 ± ti+1 +1 ± ti +� +when +� +±ti+1 +→ +t +±ti +→ −t +� +. +(C.9) +This implies that +max CS (q) = log 1 + t +1 − t, +min CS (q) = − max CS = log 1 − t +1 + t, +(C.10) +where the second equality can be similarly derived. Fig. C.1 +shows the variation of max CS as a function of tolerance t. We +note that the limits of the coefficient of similarity do not depend +on n, and we verified our results with numerical simulations. ■ +Appendix C.3: Coefficient of Variation +We start with the definition of the coefficient of variation, +CV(q) = σ(q) +¯q , +(C.11) +and we note that the minimum value of the coefficient of vari- +ation is zero and it occurs when all qi values are equal, thereby +giving no variance. +In the literature, we can find some derivations for the max- +imum value of the coefficient of variation (Katsnelson & Kotz +1957; Sharma et al. 2010). Katsnelson & Kotz (1957) show that +the upper limit of the coefficient of variation is +√ +n − 1 when all +but one qi is zero. However, this limit is only a particular case +of our formulation (specifically, q1 = q′ and qi�1 = 0). Here, we +derive the limits for a more general scenario. +Article number, page 25 of 28 + +A&A proofs: manuscript no. 43751corr +We consider that: +C2 +V = 1 +n +i=n +� +i=1 +� +1 − qi +¯q +������ +=A +�2 +. +(C.12) +Here, we have squared the definition of CV and used the def- +inition of the standard deviation σ(q). As an aside, we note +that the equation above shows that the coefficient of variation +is zero when all qi = ¯q, as noted before. We note that the max- +imum value of C2 +V occurs when the term A (in parenthesis) is +maximised. Denoting � +i=1 qi by Q, we consider the term in the +parenthesis, +A = 1 − nqi +Q = Q − nqi +Q +. +(C.13) +The condition for the general maxima of the coefficient of +variation, in our formulation, is when one of the quantity (say q1 +takes the largest possible value, while all others take the smallest +possible value): +q1 = q′ (1 + t) +qi�1 = q′ (1 − t). +(C.14) +The mean in this scenario becomes (marked with ′′): +¯q′′ = q′(1 + t) + (n − 1) × q(1 − t) +n += q′ +n +� +n(1 − t) + 2t +� +. +(C.15) +The variance in this scenario becomes (marked with ′′): +σ′′2(q) = 1 +n +�� +q′(1 + t) − ¯q′′ +�������������������������� += 2q′t� +n−1 +n +� +�2 ++ (n − 1) +� +q′(1 − t) − ¯q′′ +�������������������������� += −2q′t +n +�2� +. +(C.16) +This gives: +σ′′(q) = +�2q′t +n +� √ +n − 1. +(C.17) +Finally, the general expression for the maximum value of the +coefficient of variation becomes: +maxCV(q) = σ′′(q) +¯q′′ += +2t +√ +n − 1 +n(1 − t) + 2t. +(C.18) +This expression recovers the particular case derived in lit- +erature when we set t = 1. From this expression, we note that +the upper limit of the coefficient of variation does not depend on +the actual values of the quantities, but it depends on the number +of quantities in the set, Q, and the maximum tolerance, t. This +new formulation allows us to extract the upper limit of the co- +efficient of variation for any set whose maximum tolerance, t, is +known. Interestingly, the above expression gives appropriate re- +sult when absurd inputs are considered. For example, when there +are no planets in a system, maxCV +���n=0 = +√ +−1, and when there +is only one planet in a system, maxCV +���n=1 = 0. For a system of +two planets, the upper limit is exactly the fractional difference +(or tolerance), that is, maxCV +���n=2 = t. +Furthermore, varying over n, and assuming t ∈ [0, 1), allows +us to derive the theoretical maximum possible value for the co- +efficient of variation. This occurs at n = +2 +1−t and gives: +max CV(q) +�����n= 2 +1−t +(q) = +t +√ +1 − t2 . +(C.19) +Figure C.1 shows the variation of the theoretical max, CV, as a +function of tolerance t.■ +Appendix D: Classification boundaries for +architectures classes +In this section, we present some considerations that motivate +the boundaries between the four architecture classes for plan- +etary masses. In the current formulation (Eq. 3), there are two +boundaries that need to be identified. We deal with the distinc- +tion between similar and mixed class first, and then distinguish +ordered/anti-ordered architecture classes. +Appendix D.1: Similar versus mixed +We saw in Sect. 3.2, it is difficult to distinguish between mixed +and similar architecture classes using the coefficient of similar- +ity alone. Mixed systems are characterised by large increasing +or decreasing variations, which may cancel each other out and +lead to small values of CS (M). Nevertheless, the coefficient of +variation can distinguish between large variations in values. Fig- +ure D.1 shows the CV(M) as a function of the number of planets +in a planetary system. The left panel shows all synthetic sys- +tems from the Bern model, while we only show systems with +|CS (M)| ≤ 0.2 in the right panel. +Clearly, there are two clusters of planetary systems. The clus- +ter on the lower right-hand side corresponds to similar class sys- +tems. Mixed systems, having large values of CV(M), are spread +over the top left region. It is clear that the boundary between +similar and mixed classes depends on the number of planets. The +black line (corresponding to y = +√ +n−1 +2 +) neatly separates the two +clusters. We have chosen this equation to disentangle similar ar- +chitectures from the mixed class. This equation has, incidentally, +two key properties: 1) it ensures that no two planet system can +be of mixed architecture and 2) it happens to be exactly half of +the maximum possible value of the coefficient of variation. +Appendix D.2: Ordered and anti-ordered +Having motivated the boundary between similar and mixed +class, we are now left with three groupings of architecture +classes. These three groupings correspond to CS (M) << 0 (anti- +ordered), CS (M) ∼ 0 (similar/mixed), and CS (M) >> 0 (or- +dered). This suggests that we require two boundaries to distin- +guish these three groups. However, we posit that the boundaries +between ordered and anti-ordered should be symmetric around +0. Thus, we are left with only one boundary. +ordered (or anti-ordered) systems differ in their architec- +ture from similar/mixed classes in that the quantity (mass here) +continues to show an increasing (or decreasing) trend with dis- +tance. For all planetary systems in the Bern model, we measure +the Spearman correlation coefficient, R, between the planetary +masses and their distance from the host star. The Spearman R, +measuring the monotonicity between two datasets, varies from +-1 to +1, with 0 indicating no correlation. A positive correlation +implies that as x increases, so does y. Negative correlations im- +ply that as x increases, y decreases. +Figure D.1 shows the CS (M) of synthetic systems as func- +tion of their Spearman correlation R (mass and distance). We +note that there is a large cluster of points towards CS (M) ∼ 0. +This group corresponds to the similar and mixed architecture +classes. There are some points to the top right (including those +with R = +1 – corresponding to planetary systems in which plan- +etary masses are monotonically increasing with distance). There +is a scatter of points towards the bottom-left (including some +systems with R = −1). +Article number, page 26 of 28 + +L. Mishra et al.: Architecture Framework I – Four classes of planetary system architecture +0 +10 +20 +30 +40 +Multiplicity +0 +1 +2 +3 +4 +5 +Coefficient of Variation (Mass) [unitless] +|CS(M)| +0.2 +y = +n +1 +2 +1.0 +0.5 +0.0 +0.5 +1.0 +Spearman Correlation: R(mass, distance) +1.0 +0.5 +0.0 +0.5 +1.0 +Coefficient of Similarity (Mass) [unitless] +0.1 +-0.1 +0.2 +-0.2 +0.3 +-0.3 +Bern Model +Fig. D.1. C +lassification boundaries for architecture classes. Left: Boundary between similar and mixed class. The panel show the coefficient of +variation for synthetic planetary systems as a function of the number of planets in a system for systems with |CS (M)| ≤ 0.2. Two +clusters are clearly distinguishable, allowing us to fix the boundary between the similar and mixed architecture classes. Right: +Boundary between ordered and anti-ordered. This plot shows the coefficient of similarity of synthetic planetary systems as a +function of the Spearman correlation coefficient between the planetary masses and distances of that system. Thick horizontal lines +correspond to potential boundaries. +First, we note that the comparison of the coefficient of sim- +ilarity with Spearman R fulfils some expectation. For example, +there are no points in bottom-right or top-left sections of this +plot. Second, our objective is to isolate the central cluster of +points from all other scattered points. We note that |CS (M)| = 0.1 +fails as a boundary, since it does not include the full central clus- +ter. Both |CS (M)| = 0.2 and 0.3 could succeed. Going beyond, +a value of 0.3 would add many unnecessary points to the central +cluster. +To further motivate our choice of boundary, namely, +|CS (M)| = 0.2, we show the mass-distance diagram of 12 ran- +domly selected systems with −0.3 < CS (M) < −0.2 (out of 19) +in Fig. D.2. We note that all systems show the qualitative fea- +tures of an anti-ordered system, namely, massive planets in the +inner region and small planets in the outer region. Since all of +these planets have their CS (M) < −0.2, we use |CS (M)| = 0.2 +as a boundary between ordered, anti-ordered, and similar+mixed +architecture classes. Future works may explore improvements to +our selected boundaries using additional ideas from K-means or +hierarchical clusterings. +Appendix E: A gallery of architecture types: +Mass-distance diagrams +100 +102 +104 +Mass [M +] +Anti-Ordered +CS(M) = -0.30 +CV(M) = 2.45 +Anti-Ordered +CS(M) = -0.26 +CV(M) = 2.27 +Anti-Ordered +CS(M) = -0.26 +CV(M) = 2.97 +0 +10 +20 +30 +40 +50 +60 +70 +Ice mass fraction in Core [%] +100 +102 +104 +Mass [M +] +Anti-Ordered +CS(M) = -0.24 +CV(M) = 2.61 +Anti-Ordered +CS(M) = -0.24 +CV(M) = 2.85 +Anti-Ordered +CS(M) = -0.23 +CV(M) = 2.17 +100 +102 +104 +Mass [M +] +Anti-Ordered +CS(M) = -0.23 +CV(M) = 2.18 +Anti-Ordered +CS(M) = -0.22 +CV(M) = 2.35 +Anti-Ordered +CS(M) = -0.22 +CV(M) = 2.35 +10 +2 +100 +102 +SMA [AU] +100 +102 +104 +Mass [M +] +Anti-Ordered +CS(M) = -0.22 +CV(M) = 2.35 +10 +2 +100 +102 +SMA [AU] +Anti-Ordered +CS(M) = -0.22 +CV(M) = 1.98 +10 +2 +100 +102 +SMA [AU] +Anti-Ordered +CS(M) = -0.21 +CV(M) = 2.63 +Fig. D.2. Mass-distance diagram. This plot shows the planetary masses +as a function of distance for some planetary systems with −0.3 < +CS (M) < −0.2. The dashed line connects that planets in the system +and serves to highlight the arrangement and distribution of masses. The +size of each circle corresponds to the planet’s radius and the colour of +each planet also shows its core water mass fraction. +Article number, page 27 of 28 + +A&A proofs: manuscript no. 43751corr +100 +102 +104 +Mass [M +] +Similar +CS(M) = -0.13 +CV(M) = 0.73 +Similar +CS(M) = -0.10 +CV(M) = 1.06 +Similar +CS(M) = -0.10 +CV(M) = 1.13 +Similar +CS(M) = -0.10 +CV(M) = 0.88 +0 +10 +20 +30 +40 +50 +60 +70 +Ice mass fraction in Core [%] +100 +102 +104 +Mass [M +] +Similar +CS(M) = -0.09 +CV(M) = 0.74 +Similar +CS(M) = -0.08 +CV(M) = 0.80 +Similar +CS(M) = -0.08 +CV(M) = 0.68 +Similar +CS(M) = -0.07 +CV(M) = 1.21 +100 +102 +104 +Mass [M +] +Similar +CS(M) = -0.06 +CV(M) = 0.91 +Similar +CS(M) = -0.06 +CV(M) = 0.57 +Similar +CS(M) = -0.04 +CV(M) = 0.87 +Similar +CS(M) = -0.04 +CV(M) = 0.67 +10 +2 +100 +102 +SMA [AU] +100 +102 +104 +Mass [M +] +Similar +CS(M) = -0.04 +CV(M) = 1.25 +10 +2 +100 +102 +SMA [AU] +Similar +CS(M) = -0.01 +CV(M) = 0.99 +10 +2 +100 +102 +SMA [AU] +Similar +CS(M) = -0.01 +CV(M) = 0.62 +10 +2 +100 +102 +SMA [AU] +Similar +CS(M) = -0.00 +CV(M) = 0.89 +100 +102 +104 +Mass [M +] +Mixed +CS(M) = -0.17 +CV(M) = 3.49 +Mixed +CS(M) = -0.17 +CV(M) = 2.03 +Mixed +CS(M) = -0.16 +CV(M) = 1.73 +Mixed +CS(M) = -0.16 +CV(M) = 2.73 +0 +10 +20 +30 +40 +50 +60 +70 +Ice mass fraction in Core [%] +100 +102 +104 +Mass [M +] +Mixed +CS(M) = -0.16 +CV(M) = 2.50 +Mixed +CS(M) = -0.15 +CV(M) = 2.16 +Mixed +CS(M) = -0.14 +CV(M) = 3.25 +Mixed +CS(M) = -0.13 +CV(M) = 2.12 +100 +102 +104 +Mass [M +] +Mixed +CS(M) = -0.12 +CV(M) = 2.19 +Mixed +CS(M) = -0.12 +CV(M) = 3.15 +Mixed +CS(M) = -0.11 +CV(M) = 3.27 +Mixed +CS(M) = -0.06 +CV(M) = 3.96 +10 +2 +100 +102 +SMA [AU] +100 +102 +104 +Mass [M +] +Mixed +CS(M) = -0.05 +CV(M) = 3.74 +10 +2 +100 +102 +SMA [AU] +Mixed +CS(M) = -0.03 +CV(M) = 3.40 +10 +2 +100 +102 +SMA [AU] +Mixed +CS(M) = -0.01 +CV(M) = 1.17 +10 +2 +100 +102 +SMA [AU] +Mixed +CS(M) = 0.13 +CV(M) = 2.88 +Fig. E.1. A gallery of planetary system architectures. These plots show the mass-distance diagram for similar (left) and mixed (right) planetary +systems from the Bern Model. Each circle represents a planet, its size corresponds to the planetary radius, and its colour represents the fraction of +ice in the planetary core. Each panel shows the CS (M) as well as the CV(M) of the system. +100 +102 +104 +Mass [M +] +Anti +Ordered +CS(M) = -2.33 +CV(M) = 1.41 +Anti +Ordered +CS(M) = -1.88 +CV(M) = 0.95 +Anti +Ordered +CS(M) = -1.34 +CV(M) = 1.41 +Anti +Ordered +CS(M) = -0.80 +CV(M) = 1.61 +0 +10 +20 +30 +40 +50 +60 +70 +Ice mass fraction in Core [%] +100 +102 +104 +Mass [M +] +Anti +Ordered +CS(M) = -0.63 +CV(M) = 2.27 +Anti +Ordered +CS(M) = -0.57 +CV(M) = 2.16 +Anti +Ordered +CS(M) = -0.55 +CV(M) = 2.83 +Anti +Ordered +CS(M) = -0.49 +CV(M) = 2.05 +100 +102 +104 +Mass [M +] +Anti +Ordered +CS(M) = -0.41 +CV(M) = 3.31 +Anti +Ordered +CS(M) = -0.36 +CV(M) = 2.53 +Anti +Ordered +CS(M) = -0.29 +CV(M) = 3.71 +Anti +Ordered +CS(M) = -0.28 +CV(M) = 2.98 +10 +2 +100 +102 +SMA [AU] +100 +102 +104 +Mass [M +] +Anti +Ordered +CS(M) = -0.26 +CV(M) = 2.27 +10 +2 +100 +102 +SMA [AU] +Anti +Ordered +CS(M) = -0.26 +CV(M) = 2.97 +10 +2 +100 +102 +SMA [AU] +Anti +Ordered +CS(M) = -0.24 +CV(M) = 2.85 +10 +2 +100 +102 +SMA [AU] +Anti +Ordered +CS(M) = -0.22 +CV(M) = 2.75 +100 +102 +104 +Mass [M +] +Ordered +CS(M) = 0.23 +CV(M) = 1.21 +Ordered +CS(M) = 0.23 +CV(M) = 2.12 +Ordered +CS(M) = 0.33 +CV(M) = 0.80 +Ordered +CS(M) = 0.52 +CV(M) = 0.53 +0 +10 +20 +30 +40 +50 +60 +70 +Ice mass fraction in Core [%] +100 +102 +104 +Mass [M +] +Ordered +CS(M) = 0.56 +CV(M) = 0.86 +Ordered +CS(M) = 0.78 +CV(M) = 1.49 +Ordered +CS(M) = 0.86 +CV(M) = 0.76 +Ordered +CS(M) = 0.99 +CV(M) = 0.81 +100 +102 +104 +Mass [M +] +Ordered +CS(M) = 1.02 +CV(M) = 1.38 +Ordered +CS(M) = 1.03 +CV(M) = 1.09 +Ordered +CS(M) = 1.09 +CV(M) = 1.04 +Ordered +CS(M) = 1.23 +CV(M) = 1.38 +10 +2 +100 +102 +SMA [AU] +100 +102 +104 +Mass [M +] +Ordered +CS(M) = 1.28 +CV(M) = 0.90 +10 +2 +100 +102 +SMA [AU] +Ordered +CS(M) = 1.64 +CV(M) = 1.13 +10 +2 +100 +102 +SMA [AU] +Ordered +CS(M) = 2.16 +CV(M) = 0.99 +Fig. E.2. A gallery of planetary system architectures. These plots show the mass-distance diagram for anti-ordered (left) and ordered (right) +planetary systems from the Bern Model. Each circle represents a planet, its size corresponds to the planetary radius, and its colour represents the +fraction of ice in the planetary core. Each panel shows the CS (M) as well as the CV(M) of the system. +Article number, page 28 of 28 + diff --git a/0tE0T4oBgHgl3EQfdQB7/content/tmp_files/load_file.txt b/0tE0T4oBgHgl3EQfdQB7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..83a72cbe7887dd43916bc6269b8edaed08ba0feb --- /dev/null +++ b/0tE0T4oBgHgl3EQfdQB7/content/tmp_files/load_file.txt @@ -0,0 +1,3768 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf,len=3767 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 43751corr ©ESO 2023 January 9, 2023 Framework for the architecture of exoplanetary systems I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Four classes of planetary system architecture⋆ Lokesh Mishra1, 2 , Yann Alibert1 , Stéphane Udry2 , and Christoph Mordasini1 1 Institute of Physics, University of Bern, Gesellschaftsstrasse 6, 3012 Bern, Switzerland e-mail: exomishra@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='com 2 Geneva Observatory, University of Geneva, Chemin Pegasi 51b, 1290 Versoix, Switzerland Received 10 04 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' accepted 05 12 2022 ABSTRACT We present a novel, model-independent framework for studying the architecture of an exoplanetary system at the system level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This framework allows us to characterise, quantify, and classify the architecture of an individual planetary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Our aim in this en- deavour is to generate a uniform systematic method to study the arrangement and distribution of various planetary quantities within a single planetary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We propose that the space of planetary system architectures be partitioned into four classes: similar, mixed, anti-ordered, and ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' A central aim of this paper is to introduce these four architecture classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We applied our framework to observed and synthetic multi-planetary systems, thereby studying their architectures of mass, radius, density, core mass, and the core water mass fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We explored the relationships between a system’s (mass) architecture and other properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Our work suggests that: (a) similar architectures are the most common outcome of planet formation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (b) internal structure and composition of planets shows a strong link with their system architecture;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (c) most systems inherit their mass architecture from their core mass architecture;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (d) most planets that started inside the ice line and formed in-situ are found in systems with a similar architecture;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' and (e) most anti-ordered systems are expected to be rich in wet planets, while most observed mass ordered systems are expected to have many dry planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We find, in good agreement with theory, that observations are generally biased towards the discovery of systems whose den- sity architectures are similar, mixed, or anti-ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This study probes novel questions and new parameter spaces for understanding theory and observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Future studies may utilise our framework to not only constrain the knowledge of individual planets, but also the multi-faceted architecture of an entire planetary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We also speculate on the role of system architectures in hosting habitable worlds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Planetary systems – Planets and satellites: detection – Planets and satellites: formation – Planets and satellites: physical evolution 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Introduction Over the last 25 years, our knowledge of exoplanetary astro- physics has improved dramatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' While the first decade was marked by sensational discoveries of individual exoplanets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Vidal-Madjar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Bouchy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Udry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Kalas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Charbonneau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Snellen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2010), we are now in an age of population-level ex- oplanetary statistics (for a recent review, see Zhu & Dong 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We now know that (statistically) almost every star hosts a planet and one in two Solar-like stars host a rocky planet in their habit- able zone (Hsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Bryson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Moreover, many exoplanet-hosting stars have multiple planets orbiting them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The arrangement of multiple planets and the collective dis- tribution of their physical properties around host star(s) char- acterises the architecture of a planetary system (Mishra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Exoplanets in some multi-planetary systems are thought to behave like ‘peas in a pod’ (Lissauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Ciardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Millholland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Weiss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The peas in a pod trend consists of the following correlations: size, whereby adjacent exoplanets are either similar or ordered in size (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' the outer planet is larger);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' mass, whereby adjacent exoplanets are ei- ⋆ Catalogue of observed planetary systems used in this work is avail- able online at https://cdsarc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='unistra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='fr/cgi-bin/qcat?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' J/A+A/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' ther similar or ordered in mass;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' spacing, whereby for a system with three or more planets, the spacing between an adjacent pair of exoplanets is similar to the spacing between the next consec- utive pair;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' packing, whereby smaller planets tend to be packed together closely and larger planets are in wider orbital configu- rations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' While the statistical method used by Weiss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2018) has been debated (Zhu 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Murchikova & Tremaine 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Weiss & Petigura 2020), support for the astrophysical nature of the peas in a pod correlations (as opposed to emerging from detec- tion biases) has emerged from theoretical studies and numerical simulations (Adams 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Adams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Mulders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In particular, Mishra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2021) reproduced the observations from Weiss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2018) us- ing a model of planet formation and evolution (the Bern Model Emsenhuber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2021a,b)) and a model for the detection bi- ases of a Kepler-like transit survey (using KOBE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We showed that when nature’s underlying exoplanetary population (consist- ing of detected and undetected exoplanets) resembles peas in a pod, then a population of transiting exoplanets will have correla- tions that are consistent with those found by Weiss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In addition, Mishra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2021) suggested that the four trends are not independent of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The size correlations seem to emerge from the mass correlations, while the mass and packing Article number, page 1 of 28 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='02374v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='EP] 6 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 43751corr trends could combine to give rise to the spacing trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The peas in a pod trends are amenable to a unification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Most of the current studies on this topic utilise statistical correlation coefficients at the population level, that is, the cor- relation is measured for adjacent planetary pairs from several planetary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' While useful in terms of testing the existence (or otherwise) of architecture trends, these coefficients may have limited utility for analysing the architecture of a single planetary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Being statistical in nature, a reliable estimate of these coefficients requires large datasets - which seems difficult for a single system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Although there are some planetary system-level studies (Kipping 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Alibert 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Mishra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Gilbert & Fabrycky 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Bashi & Zucker 2021, discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1), the current literature lacks a prescription for uniformly assessing the multi-faceted architectures of several quantities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' mass architecture, radius architecture, or eccentricity architecture) for a single planetary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We seek a framework that allows us to characterise the ar- chitecture of an individual planetary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Our motivations for developing such a framework arise from questions related to: formation, such as the extent to which a system’s architec- ture is shaped by initial conditions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' the environment in and around the star and protoplanetary disk formation regions) (Jin & Li 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Safsten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' evolution, the role of physical processes such as orbital migration or giant impacts in shaping the final architecture of planetary systems (Mulders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' identification, which particular stars host planets that resemble peas in a pod, and, in particular, whether the planets in systems like TOI-178 (Leleu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2021), Trappist-1 (Agol et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2021), or 55 Cancri (Bourrier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2018) show mass/size similarities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' other architectures, we know that there are many planetary sys- tems that do not follow the peas in a pod architecture (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' the Solar System).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Overall, it is not obvious how the architecture of any individual planetary system should be uniformly assessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In this series of papers, we propose a framework for examin- ing the architecture of planetary systems at the system level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The philosophy behind system level analysis is to consider the en- tire planetary system as a single unit of a physical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This framework allows us to not only quantify, compare, and inves- tigate a system’s architecture, but also offers some unexpected benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' As it turns out, the framework allows for a conceptu- ally intuitive partitioning of the space of possible architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We label the four classes of planetary system architectures as: similar, ordered, anti-ordered, and mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In this way, our work extends the trends initiated by the notion of peas in a pod ar- chitecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Furthermore, we verify the unification of the peas in a pod correlations proposed in Mishra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We find that, Similar architectures are the most common type of plane- tary system architectures and their high occurrence explains why the intra-system radius uniformity was already observable from the first four months of Kepler data (Lissauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Our framework engenders novel questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' For instance, if nature produces distinct classes of architecture in multi- planetary systems, then what is the frequency or occurrence rates of these architecture classes?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' How does the occurrence of an ar- chitecture class depend on stellar and protoplanetary disk en- vironment?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' How does the architecture of a system evolve over time?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' What is the role of stellar evolution, protoplanetary disk interactions, and planet formation in shaping the final architec- ture?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' How is a planet’s internal composition related to the sys- tem’s architecture?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Or does the ability of a planet to host life depends on the architecture of the planetary system?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In this se- ries of papers, we explore these questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Although the num- ber of multi-planetary systems is low today, this may change in the next few decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Thanks to large survey missions such as PLATO (Rauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2014), GAIA (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2016), TESS (Ricker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2015), LIFE (Quanz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2022), and others, the growing number of known multi-planetary systems will allow for a better understanding to emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We hope our work encourages observers to dedicate more observation time to detecting planets within a known planetary system, that is, in finding multi-planetary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The architecture classification scheme proposed in this pa- per is a model-independent framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' To demonstrate our clas- sification framework and explore its consequences, we applied our framework to simulated planetary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' To illustrate our framework on real systems, we also applied our framework to observed exoplanetary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We emphasise that while the re- sults emerging from the application of our framework on these datasets may suffer from some limitations (arising from theoret- ical modelling or detection biases for observed systems);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' how- ever, the concept of our architecture classification scheme, being model-independent, does not share these limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In this pa- per, we present the catalogues of planetary systems we apply our framework to in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2, along with a newly curated catalogue of observed exoplanetary systems and simulated planetary systems, using the Bern Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We introduce our framework in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 4, the characteristics of the architecture classes are dis- cussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We explore the link between the internal composition of planets and the system architecture class in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Then, in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 6, we speculate on how habitability could depend on the architecture of planetary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Our conclusions are given in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In a companion paper, we investigate the formation path- ways, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' the role of initial conditions and physical processes in shaping the final architecture (Mishra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2023) referred to as Paper II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Our work demonstrates that the processes of planet formation and evolution are imprinted on the entire system-level architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We find that protoplanetary disks with low solid- mass give rise to planetary systems endowed with a mass similar- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' On the other hand, massive disks and high metallicity often lead to mass Ordered, Anti-Ordered, or Mixed system architec- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Planet-planet and planet-disk interactions play a decisive role in shaping these three architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Catalogues 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Theoretical dataset: Bern Model In this series of works, we demonstrate our architecture frame- work by analysing the architecture of synthetic planetary sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' These systems were numerically computed using the Gen- eration III Bern Model of planet formation and evolution (Em- senhuber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2021a,b) that is based on the core-accretion paradigm of planet formation (Pollack et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Alibert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2004, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The model follows the growth of protoplanetary embryos embedded in a protoplanetary disk of gas and solids around a solar-type star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' A diverse range of physical processes are simultaneously occurring and coherently computed in this 1D star-disk-embryo system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' These include: stellar and disk physics (evolution of and interaction between star and viscous disk, condensation of volatile and refractory species, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' ), plane- tary formation physics (accretion of planetesimals and gases, in- ternal structure calculations, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' ), and additional physics (orbital and tidal migration, planet-planet N-body interactions, planet- disk interactions, atmospheric escape, deuterium fusion, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We describe these physical processes in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' A and a descriptive summary of these processes is provided in Mishra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2021, Article number, page 2 of 28 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Mishra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' : Architecture Framework I – Four classes of planetary system architecture Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Mass-distance diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This figure shows the masses and the distances of planets in all catalogues used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Shaded regions show the parameter space spanned by synthetic planets observed via radial velocity surveys (Bern RV Multis), transit surveys (Bern KOBE Multis), and ongoing missions (Bern Compact Multis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The parameter space for Bern KOBE Multis has been mapped from its original radius- period plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' in particular, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 1 and Sections 2, 3, and Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' More details can also be found in (Emsenhuber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2021a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We synthesised 1000 planetary systems, each starting with 100 lunar mass protoplanetary embryos, wherein the following initial conditions were varied: mass of protoplanetary gas disk, photo-evaporation rate, dust-to-gas ratio, disk inner edge, and the starting location of embryos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 1, we show all synthetic planets on the mass-distance diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' For each synthetic plane- tary system failed embryos, objects with mass less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1M⊕, were removed from further analysis1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Three observationally motivated catalogues were prepared from the synthetic dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This allowed us to facilitate a compar- ison of the architecture from observed planetary systems with the synthetic planetary systems and to make predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The param- eter space spanned by the planets in these catalogues is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' These catalogues are as follows: Bern RV Multis: We assume a radial velocity (RV) survey which can find planets with periods ≤ 15 yr and semi-amplitude KRV ≥ 20 cm/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' These numbers are motivated by (a) long- running RV surveys such as the HARPS survey (Mayor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2003, 2011) and the California Legacy Survey (Rosenthal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Fulton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (b) current precision achieved by ESPRESSO (Lillo-Box et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Netto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' and (c) making predictions for future RV surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Such RV detectable synthetic planetary systems with four or more planets form the 1 As long as the mass threshold for failed embryos is kept under 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1M⊕, the results presented in this paper are not sensitive to the threshold limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We removed these small objects since they (a) failed to grow as massive planets, (b) are insignificant to the dynamical evolution of the system, and (c) are currently unobservable in exoplanetary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' All results arising from the Bern RV Multis, Bern KOBE Multis, and Bern Com- pact Multis are insensitive to these failed embryos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Bern RV Multis catalogue, which includes 3 828 planets around 565 stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Bern KOBE Multis: We assume a Kepler-like transit survey which continuously observes 2 × 105 stars for 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5 yr (Thompson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' A planet which transits three or more times and pro- duces a transit S/N of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1 or more is considered detectable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The reliability and completeness of such a survey is replicated and those synthetic planets which would have been vetted as ‘plan- etary candidates’ by the Kepler Robovetter (Thompson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2018), are kept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Such transiting synthetic planetary systems with four or more planets form the Bern KOBE Multis catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' KOBE was developed and introduced in Mishra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' There are 6 715 planets around 1283 stars in this catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Bern Compact Multis: Ongoing transit missions such as CHEOPS and TESS have been successful in characterising com- pact multi-planetary systems, such as TOI-178 (Leleu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2021) and TOI-561 (Lacedelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Inspired by these dis- coveries, we investigated the architecture of compact planetary systems simulated by the Bern Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Our aim is to understand the architecture and make predictions for such systems based on the core-accretion paradigm (Pollack et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Alibert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2004, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' All planets with periods of ≤ 100 d and masses of ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1 M⊕ were retained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Synthetic planetary systems, in this pa- rameter space, with four or more planets form the Bern Compact Multis catalogue, with 2 412 planets around 400 stars included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Observational dataset: A new catalogue To demonstrate our framework on observed exoplanetary sys- tems, we have curated a new catalogue of known multi-planetary systems2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' A salient feature of this catalogue (and the philosophy behind this work) is its focus on considering planetary systems as a single unit of a physical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Unlike focussing on in- dividual exoplanets or a single detection technique, our aim is to study the planetary system as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' There are two serious challenges to this endeavour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Firstly, the biases present in de- tection methods tend to prevent a complete, reliable picture of an exoplanetary system from emerging (either via undetected or mischaracterised planets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Secondly, detecting planets on long orbital periods requires long-term, repeated observations, which is considerably challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We hope that upcoming missions and future surveys can mitigate these difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We included a planetary system in our catalogue if: (a) it has at least four known planets and (b) masses are available for at least four planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' For example, Kepler-33, a five planet system, is included because mass measurements are available for four of its planets3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The criterion of requiring minimum four plan- ets emerges due to (a) the requirement for enough planets for adequately characterising the architecture and (b) because for systems with lower number of planets, it is perhaps difficult to uniformly assess whether the low multiplicity is an outcome of natural processes or detection biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' To keep the comparison be- tween observations and theory uniform, all catalogues in this se- ries of works only consider planetary systems with four or more planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The architecture framework can, however, handle two- or three-planet systems as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' To make this catalogue useful to the wider community and enable future studies, we gathered several key stellar and exoplanetary properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' For host stars, we report the mass, radius, luminosity, effective temperature, metal- licity, age, and distance, along with their identification numbers 2 The catalogue was last updated in April 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 3 For this study, the distinction between mass and minimum mass is ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Article number, page 3 of 28 105 Bern Bern Bern Bern Model Compact KOBE RV Observations Multis Multis Multis Solar System 104 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Mass [M] 102 = 20 cm/s KRV 101 100 10-1 0 100 2 5 10-2 7 10-2 10-1 100 101 102 103 SMA[AU]A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 43751corr Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Observed multi-planetary systems: There are 41 planetary systems with 194 planets in this catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Only the first five rows are shown here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The entire table is available online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Online version includes additional identification columns: KIC ID, TIC ID, and GAIA ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Missing information is indicated by ‘–’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' References for individual systems are given in appendix Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Stellar parameters Hostname Multiplicity M⋆[M⊙] R⋆[R⊙] L⋆[L⊙] Teff[K] [Fe/H] Age [Gyr] Distance [pc] Sun 8 1 1 1 5, 772 0 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1 0 Trappist-1 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='001 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='53e − 04 2, 566 ± 026 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='08 07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='6 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2 012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 TOI-178 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1 ± 01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='08 4, 316 ± 070 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='05 07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1 062.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='7 HD 10180 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5 ± 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='02 5, 911 ± 019 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='01 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5 039.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 HD 219134 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='3 ± 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='01 4, 700 ± 020 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='04 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2 006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5 Planetary parameters Hostname Planet Mp[M⊕] Rp[R⊕] ap[AU] e i [◦] min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Mp Sun � j,s,u,n m,v,e,m, � ��������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='815 ± 00.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='383 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='000,��������� ��������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='723 ± −, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='387 ± −,��������� ��������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='01 ± −, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='21 ± −,��������� ��������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='39 ± −, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='00 ± 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='374 ± 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='069,��������� ��������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='097 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='014, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='116 ± 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='12, 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='73 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='17,��������� � F,F,F,F,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' � TOI-178 � 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+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' � HD 10180 � f,g,h c,d,e, � ��������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='014 ± 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='699, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='222 ± 00.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='039 ± −,��������� ��������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='04, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='00 ± −, ��������� ��������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='10, 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='13,��������� � F,F,T,T,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' � (when available) in the Kepler Input Catalogue (KIC), TESS In- put Catalogue (TIC), and GAIA ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' For planets, we report mass or minimum mass, radius, semi-major axis, eccentricity, and in- clination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In a conservative approach, errors (reported when pos- sible) are the maximum of the upper and lower error bounds available in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' When multiple publications reported planetary parameters, a more recent publication was preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' When a single publication reported parameters for all planets in a system, then such a consistent set of solution was given pref- erence (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' GJ 676 A or Kepler-11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' For stellar parameters, if a star was included in KIC, then the values from Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2020) are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Most other stellar parameters come from the TIC (Stassun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2019) or from individual publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' There are 41 planetary systems that meet our criteria and de- fine our multi-planetary system catalogue (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' With a total of 194 planets in our catalogue, the number of planetary systems with four, five, six, seven, and eight planets is 24, 7, 8, 1, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In this paper, we present the observed planetary systems as they are known today and we do not correct the observations for any detection biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Instead, to assist in making comparisons with the theory, detection biases will be placed on simulated plane- tary systems (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Figure 1 shows the mass of observed exoplanets as a function of their semi-major axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' While our observed multi-planetary systems catalogue en- genders system-level studies, its current form poses several tech- nical difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Foremost, the number of observations is only forty-one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Secondly, multiple detection methods, such as radial velocity or transits (etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=') were employed to observe these plan- etary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Each observation technique suffers from certain limitations and detection biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This implies that the observed systems in our catalogue do not constitute a homogeneous and complete set of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' These two limitations of the ob- servations catalogue prohibit us from deducing any statistically strong result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Nevertheless, we used the observed systems for (a) exemplifying system-level approach to real planetary systems and (b) using our framework on observations to explore trends in the architecture of observed systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Our results from the observed catalogue may be affected by another source of difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' There are two systems in our cata- logue that host some planets without known mass measurements (Kepler-33 b and Kepler-80 f and g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Since these two systems have at least four planets with known masses, they have been included in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' However, this does not impact the results of the present study in a drastic way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' All three planets in these systems without mass measurements are either the innermost and/or the outermost planets in their respective systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' There- fore, the missing measurements do not have a strong influence on the characterisable mass architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The missing measure- ment may have a strong effect if any planet with unknown mass was in between two planets with known masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Characterizing architecture: A new framework 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Literature review We review some approaches from other studies that have tried to capture planetary system-level properties in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Kipping (2018) investigated similarity and ordering (of planetary sizes) at the level of an individual system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Using an entropy based frame- work on Kepler systems, he concludes that initial conditions are inferable from the present-day architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' As we go on to show in this series, our work not only supports this conclusion, but additionally demonstrates the possible links between initial conditions and final architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Although the above-mentioned study considers a similar problem to the one we deal with here, our frameworks differ considerably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Built on step-functions and combinatorics, the aforementioned framework does not take into account the magnitude of variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Alibert (2019) proposed a concept of distance between two planetary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The Alibert distance captures inter-system differences, whereas our framework quantifies intra-system sim- ilarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The Alibert distance is useful to quantify the similarity (or dissimilarity) between two planetary systems and in unsuper- vised machine-learning algorithms to find clusters in the space of planetary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Bashi & Zucker (2021) recently proposed Article number, page 4 of 28 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Mishra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' : Architecture Framework I – Four classes of planetary system architecture another concept for distance based on a statistical distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The ‘weighted’ energy distance is the distance between two plane- tary systems, with each planet represented on the log-period and log-radius plane, utilising planetary masses (from a mass-radius relationship) as weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' As with the Alibert distance, the Bashi- Zucker distance requires two planetary systems and thus it is not suitable for characterising the global architecture for a single planetary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Gilbert & Fabrycky (2020) proposed seven parameters for quantifying the global structure of planetary systems: dynam- ical mass (ratio of mass in planets to stellar mass), mass par- titioning (normalised mass disequilibrium), mass monotonicity (weighted Spearman correlation coefficient), characteristic spac- ing (average mutual Hill radii), gap complexity, flatness, and multiplicity (n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Of these measurements, mass partitioning and mass monotonicity have close parallels with our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The input information required to compute mass partitioning, and monotonicity is exactly the same as the input information for our architecture framework, namely, a set of planetary masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' However, we find that the output displays a curious mix of con- cepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Mass partitioning is zero for a system in which all planets have the same mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' When one planet has some mass and all other planets have negligible mass, the mass partitioning for this system is unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' While this parameter captures the two extreme cases, it is difficult to interpret and employ this measure in cases other than these two extremes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Behaving similarly to a correla- tion coefficient, mass monotonicity has a range of [-1,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' It is de- fined as the Spearman correlation coefficient (between mass and distance) multiplied by the mass partitioning (which is weighted by n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Although the work of Gilbert & Fabrycky (2020) stud- ies the architecture of planetary systems at the system-level, we seek a framework which can also be used with planetary prop- erties other than mass, such as radius, bulk density, water mass fraction, eccentricities, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Millholland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2017) and Wang (2017) showed that the peas in a pod pattern reported by Ciardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Weiss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2018) also extends to planetary masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Millholland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2017), using planetary masses derived from transit-timing vari- ations, studied the clustering of planets in the mass-radius plane and found that the sum of distances (in the log mass-size space) between adjacent planets of real systems is much smaller than a bootstrapped randomised population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Based on a set of 29 RV observed systems, Wang (2017) infer two types of planetary sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Planetary systems with masses of ≲ 30M⊕ show intra- system mass uniformity, while systems with masses ≳ 100M⊕ do not follow the peas in a pod pattern – indicating that there are only two possibilities for the architecture structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' As we show in this series of works, their hypothesis of only two architecture types is too simple and cannot capture the richness of physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Concept With our framework, we initially aimed to capture the key aspect about the peas in a pod architecture trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' These trends are cor- relations between adjacent planets or between consecutive pairs of adjacent planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We want to capture these ideas at the level of a single planetary system through a unified framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We do this by studying how a quantity, qi, (such as mass, size, or period ratio) varies for all planets within a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Here, i in- dexes the planets within a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' For all quantities, we adopt an ‘inside-out’ convention, namely, we start with the innermost planet (qi=1) and go to the next adjacent planet (qi=2), and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' By comparing how qi varies for each planet inside-out, we Distance from star Quantity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Mass) Similar Anti-Ordered Ordered Mixed Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Classes of architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This schematic diagram shows the four architecture classes: similar, anti-ordered, mixed, and ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Depend- ing on how a quantity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' mass or size) varies from one planet to an- other, the architecture of a system can be identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' are actually estimating how qi varies with distance from the host star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In comparing a quantity, qi, with distance, four kind of vari- ations emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In one scenario, a quantity could show little to no variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In another case, the value of a quantity may increase with increasing distance or, conversely, the quantity could de- crease from one planet to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Finally, it is also possible for a quantity to not have any clear variations from one planet to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We identify these four scenarios as the four classes of architectures that can exist at the level of a single planetary sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This idea is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Mishra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2021) suggested that the mass correlations could originate from planet-formation physics and the correla- tions of size and spacing could be derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Therefore, we first apply our framework using planetary masses (except in Sects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 5 and 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' As depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2, when the masses of all planets within a system are similar to each other, we label the architec- ture of such systems as ‘similar’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This architecture class corre- sponds to the peas in a pod architecture reported in observations (Weiss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Millholland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' When the masses of planets tend to increase inside-out, the architecture of such sys- tems is labelled ‘ordered’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' If the planetary mass tends to decrease from the inner planet to the outer, we label the architecture of these systems as ‘anti-ordered’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Finally, if a large increasing and decreasing variation in the planetary masses is present, we label the architecture of such systems as ‘mixed’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The mixed architec- ture class is also useful in capturing all other architecture pat- terns which do not fall under the other three architecture classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Kipping (2018), for example, has analysed some interesting re- peating patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' By introducing these architecture classes, our framework organises the possibilities for system architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Article number, page 5 of 28 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 43751corr One might wonder, at this point, why introduce such a con- cept and the ensuing mathematical machinery?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' While part of this work began as an inspired exploration to categorise our under- standing of system architecture, it turns out that there are good physical reasons to pursue this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' As is shown in this and a companion paper, planetary systems that have the same archi- tecture tend to have a host of other properties in common, such as internal structures (core-mass, ice-mass) distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Most importantly, systems with a common architecture tend to have same formation pathways, initial conditions, and evolutionary histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Practically, this means that a quick glance at a system’s architecture may reveal a lot more about its formation scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Our architecture classification framework utilises two quan- tities – the coefficient of similarity and the coefficient of vari- ation, introduced in Sects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' These two coefficients allow us to quantify the conceptual ideas we have presented above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Together, these coefficients define a new space of possibilities for system architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5, we iden- tify the regions of this architecture space that correspond to the four architecture classes introduced above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' As this framework deals with the architecture of multi-planetary system, systems with only one planet are not studied within this framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Coefficient of similarity The term ‘coefficient of similarity’ is commonly used in the fields studying statistics of ecology and genetics (Gower 1971;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Dalirsefat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We borrow the term but develop our own concept and definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='Let q be a planetary quantity such as mass, size, period ratios of adjacent planets, bulk density, ec- centricity, and so on4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The value of this quantity for the ith planet in a system is denoted by qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The coefficient of similarity, CS , measures how q changes from one planet to another, inside-out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' For a system with n planets, it is defined as: CS (q) = 1 n − 1 i=n−1 � i=1 � log qi+1 qi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (1) There is a clear physical interpretation for CS (q): the coefficient of similarity measures the average order of magnitude variation in the quantity q from one planet to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The definition of the coefficient of similarity allows us to map the architecture of a planetary system on a one dimensional axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' When CS (q) ≈ 0, then the system’s architecture could imply a similarity in q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' When CS (q) is positive, then planets within a system are ordered in q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Conversely, CS (q) being negative, implies that the planets are anti-ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We have developed a mathematical formalism to study the sensitivity of the coefficient of similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In Appendix C, we de- rive the limiting values of the coefficient of similarity and present the results here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' For example, when the qi values for all planets in a system are within 10% of each other, then the maximum possible value of CS (q) is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='09 (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' For maximum tolerances of 20%, 40%, 60%, and 80%, the maximum possi- ble value of CS (q) are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='18, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='37, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='60, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='95 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1, we show the dependence of the maxCS (q) on t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The coefficient of similarity cannot distinguish between two classes of architecture: similar and mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Systems which show similarity will have CS (q) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' However, system with mixed ar- chitecture have large increasing and decreasing variations, such 4 For quantities which admit zero as a possible value, the coefficient of similarity may become ill-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This is a coordinate singularity and can be dealt with an appropriate treatment (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 4 Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' that the log of ratios qi+1 qi cancels itself out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Such systems will also have CS (q) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We propose the coefficient of variation to distin- guish these two architecture classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The coefficient of similarity depends on the actual order in which planets exist (inside-out) in a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' As we go on to show, the coefficient of variation does not depend on the ordering of planets in a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Coefficient of variation The coefficient of variation, CV, is a standard descriptive statistic used to measure the magnitude of variation in a set of numbers (Katsnelson & Kotz 1957;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Sharma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Abdi 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' It is defined as the ratio of the standard deviation with the mean: CV(q) = σ(q) ¯q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2) The coefficient of variation is a positive quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' When all qi have the same value then CV(q) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Planetary systems con- sisting of planets that have a small (or large) variability in their qi values will have a small (or large) value of the coefficient of variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Now, the distinction between systems showing simi- larity and mixed architecture is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' While similar systems will have a low value of the coefficient of variation, mixed systems will have a high value of coefficient of variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Since this coefficient is a well known statistical measure, there are some derivations for its limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' A classical result from Katsnelson & Kotz (1957) shows that, for a set of n numbers, the maximum value of the coefficient of variation is √ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' However, this result is only a particular case in our setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In Ap- pendix C, we develop a mathematical formalism to understand the limits of the coefficient of variation and present the results here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' When the qi values for all planets in a system are within 10%, 30%, 50%, 70%, and 90% of each other, the absolute the- oretical upper limit of CV(q) is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='10, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='31, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='58, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='98, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='06 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Figure C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1 shows how this upper limit varies with the maximum tolerance, t, for a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Classifying the architectures of planetary systems We are interested in obtaining a mapping from the scale- invariant coefficients to an architecture class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In Appendix D, we present some considerations that motivate the selection of boundaries between the four classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The selected boundaries were additionally tested on thousands of mock planetary systems to check their ability to correctly classify the four architecture classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We propose the following boundaries for identifying the architecture class based on planetary masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Architecture class Condition Anti-ordered CS (M) < −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2 Ordered CS (M) > +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2 Similar |CS (M)| ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2 and CV(M) ≤ √ n − 1 2 Mixed |CS (M)| ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2 and CV(M) > √ n − 1 2 (3) A natural (and welcome) outcome of these criteria is that a two-planet system can never have a mixed class architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The boundary between similar and mixed class is half the maximum possible value of the coefficient of variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' For the solar sys- tem, CS (M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='36 and CV(M) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This framework robustly Article number, page 6 of 28 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Mishra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' : Architecture Framework I – Four classes of planetary system architecture Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' New parameter space: Architectures of planetary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Both panels shows the coefficient of similarity (mass) as a function of the coefficient of variation (mass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The shaded regions show the allowed parameter space for planetary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The white gaps (between two shaded regions) mark the mathematically forbidden regions of this architecture space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Different parts of this parameter space are identified with four architecture classes, in accordance with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Each point corresponds to an individual planetary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' For visual clarity, the shaded and unshaded regions are drawn only for systems hosting up to fifteen planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Left: Planetary systems from the Bern model and observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Right: Synthetically observed systems depicting the detection biases of radial velocity and transit surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' identifies the architecture of the solar system as ordered5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This classification is in line with the historic understanding of the so- lar system architecture: small rocky planets on the inside and giant planets on the outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' If, however, Neptune were replaced with an Earth-like planet, the architecture of the solar system would be classified as mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Considering only the inner four planets of the solar system, CS (M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='10 and CV(M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='85, would make the architecture of the inner solar system belong to the similar class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The architecture of the outer four giants in the solar system is anti-ordered and we have CS (M) = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='42 and CV(M) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Figure 3 shows the CS (M) versus CV(M) space for plane- tary systems from several catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The Bern model planetary systems occupy all four regions of this architecture space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Ob- served planetary systems, however, span only a limited region of this parameter space, given the low multiplicity of observed planetary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The architecture space spanned by the ob- served planetary systems (shaded contour) is in agreement with the synthetically observed planetary systems from Bern Com- pact Multis, Bern KOBE Multis, and Bern RV Multis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The architecture for the systems in the synthetically observed catalogue was calculated based only on the planets that were de- tected (for RV/KOBE) or included (for Bern Compact Multis) in the above-mentioned catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' It is theoretically possible for a single Bern model system to exhibit different architectures de- pending on the planets which are detected or included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The re- verse is also true – the architecture of an observed planetary sys- tem may change if new planets are discovered or old controver- sial candidates are rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' While the ground truth architecture for observations seems elusive, a comparison with synthetic ob- 5 Even if the masses of each solar system planet were randomly varied within 85% of their original values, the emerging architecture is still ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' With 1M trials, varying the masses randomly within 90% of their original values lead to ordered (for ≈ 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='45% trials), mixed (for ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='55% trials), and similar (for ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='001% trials) architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' servations can bring forth patterns which are unexpected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' With this in mind, we consider the following example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Detection biases, in both radial velocities and transits, gen- erally disfavour the discovery of less-massive and small planets at larger distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This implies that anti-ordered architectures are difficult to detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In fact, we have no known example of a planetary system showing anti-ordered architecture in our obser- vations catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This is surprising for two major reasons: (a) theory suggests their existence: there are several synthetic plan- etary systems from the Bern Model whose architecture is anti- ordered;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (b) theory suggests their discovery: all three syntheti- cally observed catalogues contain some (albeit few) anti-ordered planetary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Since the number of systems in our catalogue is too low, we refrain from making any conclusions and, instead, we await the discovery of anti-ordered architectures in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' However, if such architectures are not found despite considerable efforts, this result will become a strong indicator for shaping our understanding of planet formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Another aspect of this new architecture space is the underly- ing mathematical structure6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 3, the shaded areas shown regions where a planetary system, with n ∈ [2, 15] planets, is al- lowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' A system with two planets, for example, can only occupy the shaded region labelled ‘n = 2’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' All non-shaded regions (in white – except the shaded regions for 16 or more planets which is not drawn), on this architecture space, is mathematically forbid- den.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' These are parts of the architecture parameter space that no planetary system, irrespective of its configuration, can occupy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 6 Visualizing this structure is easy (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (a) Construct mock planetary systems with masses, for each mock planet, randomly drawn from a uniform distribution with suitable limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (b) It is suggested to vary the number of planets in these mock systems randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (c) Calcu- late the CS (M) and the CV(M) using equations 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (d) Plot CS (M) versus CV(M) for this mock population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' For large number of systems the plot should be symmetric about CS (M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Article number, page 7 of 28 Coefficient of Similarity (Mass)[unitless] Bern Model Observations Solar System 2 4 u 0 1 2 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 4 Coefficient of Variation (Mass) [unitless]Coefficient of Similarity (Mass)[unitless] Observations Contour BernRVMultis Bern KOBE Multis Bern Compact Multis 4 0 1 2 3 4 Coefficient of Variation (Mass) [unitless]A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 43751corr This strong result stems from the mathematical limits that were derived for this work (see Sects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='3, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='4, and appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' For clarity and future convenience, we introduced some ter- minology to the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' When the architecture framework (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' CS and CV) is applied on planetary bulk masses, the resulting in- formation tells us the mass architecture of a system, namely, the arrangement and distribution of masses in said system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Similarly, when this framework is applied on radii, it gives us the radius ar- chitecture (arrangement and distribution of radii) for the system (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Similarly, we can obtain the bulk-density architecture (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2), core-mass architecture (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='3), water mass frac- tion architecture (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='4), period-ratio or spacing architecture, eccentricity-architecture, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In this series of papers, we identify a system’s architecture based on its bulk mass ar- chitecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Thus, when a system is said to be similar, we are referring to the similarity in terms of the mass architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Characteristics of architecture classes 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' General comments In earlier studies on the peas in a pod architecture, the strength of population-level (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' across many planetary systems) trends was quantified using Pearson correlations coefficient (Weiss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Zhu 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Chevance et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Millholland & Winn 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Mishra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The correlation coefficients were cal- culated using planetary quantities in the log space (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' by first taking the log10 of all quantities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This resulted in higher values of the correlation coefficient since quantities have limited range to perambulate in the log space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Consider planetary masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We calculated the correlation coefficient between the mass of adja- cent inner and outer planets in the Bern model population (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 7 in Mishra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The value of the coefficient is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='66 in the log space and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='16 in the linear space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This highlights that the planetary masses are more closely clustered in log than in linear space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We tested the same correlation for all systems in each ar- chitecture class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We expect planetary masses in mixed, ordered, and anti-ordered systems should (by definition) have low cor- relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' On the other hand, similar class architecture should exhibit a strong correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Surprisingly, in log space all archi- tecture classes show strong correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The coefficient value is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='67 for similar class, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='69 for mixed class, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='50 for ordered class, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='58 for anti-ordered class architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' However, in the linear space the coefficient values reflects our expectation: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='61 for the similar class, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='20 for the mixed class, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='16 for or- dered class, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='05 for anti-ordered class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This underscores that strong correlations in the log space may not be indicative of substantive architecture trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' It also shows that our framework is capable of identifying systems in which the ’peas in a pod’ architecture is discernible even in the linear space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' For all 41 observed planetary systems in our catalogue, we report their architecture classes in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Figure 6 shows the architecture of all observed multi-planetary systems in our cat- alogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The systems are sorted by their coefficient of similarity values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The figure also shows the four classes of architecture for a few randomly selected synthetic planetary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' To under- stand the characteristics of the different architectures, we study the distribution of planetary masses, radii, and semi-major axes as well as the multiplicity distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' For planetary systems across all catalogues, this is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We describe the characteristics of different architectures in the following subsec- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The discussion in the next subsection involves results de- rived from both observed and synthetic planetary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In ad- Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Architecture type of known multi-planetary systems (see Table 1 for catalogue and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 6 for architecture plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Hostname Multiplicity CS (M) CV(M) Architecture Class Solar System 8 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='36 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='85 Ordered Trappist-1 7 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='45 Similar TOI-178 6 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='46 Similar HD 10180 6 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='66 Similar HD 219134 6 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='27 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='49 Ordered HD 34445 6 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='84 Similar Kepler-11 6 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='03 Ordered Kepler-20 6 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='44 Similar Kepler-80 6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='19 Similar K2-138 6 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='61 Similar 55 Cnc 5 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='52 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='37 Ordered GJ 667 C 5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='29 Similar HD 158259 5 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='29 Similar HD 40307 5 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='33 Similar Kepler-102 5 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='41 Similar Kepler-33 5 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='67 Ordered Kepler-62 5 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} 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+page_content='20 Mixed HD 141399 4 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='40 Similar HD 160691 4 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='82 Ordered HD 20794 4 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='25 Similar HD 215152 4 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='23 Similar HR 8799 4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='17 Similar K2-266 4 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='60 Similar K2-285 4 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='31 Similar Kepler-89 4 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='91 Mixed Kepler-106 4 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='26 Similar Kepler-107 4 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='42 Similar Kepler-223 4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='22 Similar Kepler-411 4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='34 Similar Kepler-48 4 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='74 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='64 Ordered Kepler-65 4 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='68 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='63 Ordered Kepler-79 4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='24 Similar WASP-47 4 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='95 Ordered tau Cet 4 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='37 Similar HD 164922 4 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='49 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='29 Ordered dition, we present a gallery of mass-distance diagrams showing the four architecture classes in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 4 shows the coefficient of similarity of masses as a function of the total planetary mass in a system for all synthetic planetary systems from the Bern model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This figure shows sev- eral key aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Firstly, it illustrates the four architecture classes as separate clouds of scattered points strengthening the proposed four classes of planetary system architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Secondly, it shows that the architecture framework is scale-invariant, that is, the system architecture is sensitive only to the relative distribution of a quantity – and not its absolute value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' For example, while most similar system have ⪅ 100M⊕ mass in their planets (sug- gesting a lack of giant planets), there are some similar systems with mass values of ≈ 2000M⊕ for their planets and host giant planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Likewise, most ordered systems host giant planets and have ⪆ 2000M⊕ mass in their planets, there is an ordered sys- tems without any giant planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Also, it illustrates that the coeffi- cient of similarity partitions planetary systems into three groups: anti-ordered, similar and mixed in one group, and ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This demonstrates that the coefficient of variation is necessary to dis- tinguish between the similar and mixed systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Finally, the di- agram shows that the architecture class of a system has strong links with the total mass of planets in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This hints that there must be general patterns in the formation pathways of sys- Article number, page 8 of 28 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Mishra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' : Architecture Framework I – Four classes of planetary system architecture 10 0 10 1 10 2 10 3 10 4 Total Mass in Planets [M ] 3 2 1 0 1 2 3 Coefficient of Similarity (Mass) [unitless] Similar Anti-Ordered Mixed Ordered Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Four classes of system architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The diagram shows the coef- ficient of similarity for a system as a function of the sum of mass of each planet in a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Dashed horizontal lines correspond to CS = ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This diagram emphasises the four classes of planetary system architec- ture, namely: anti-ordered, similar, mixed, and ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' It also shows that the coefficient of similarity can not distinguish between similar and mixed architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' tems of the same architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This topic is discussed in Paper II, from this series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Frequency of architecture The frequency of each architecture class across all catalogues is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Similar systems are the most common archi- tecture classes emerging from simulations, with a frequency of ≈ 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' About ≈ 8% of synthetic systems show mixed and anti-ordered architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Ordered architecture is a rare out- come in simulations (≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In observations, similar class is the most common architecture (≈ 59%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Fifteen observed exo- planetary systems (out of forty-one) are part of the ordered ar- chitecture class (≈ 37%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' About ≈ 5% of observed planetary systems show mixed architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' There are no known examples of observed system with anti-ordered architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Comparing the frequency of architecture classes for ob- served systems with synthetically observed systems brings out some peculiar features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Firstly, theoretical catalogues seem to suggest that observations should find more similar systems and fewer ordered systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The frequency of similar (ordered) sys- tems in our observed catalogue is significantly lower (higher).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Secondly, while the frequency of mixed systems seems to be in agreement with synthetic observations, this agreement is not sta- tistically significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' These discrepancies probably arise from the incompleteness prevalent in our observations catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Transit surveys are con- ducted in a manner which allows the completeness and reliability of these survey to be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The completeness of RV surveys, on the other hand, is very difficult to estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Further, the obser- vation techniques used to find the exoplanets in our observations catalogue are heterogeneous, consisting of RV, transits, transit- Similar Anti-Ordered Ordered Mixed 0 20 40 60 80 100 Frequency [%] Bern Model Bern KOBE Multis Bern RV Multis Bern Compact Multis Observations Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Frequency diagram for the architecture classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Currently, there are no known examples of observed planetary systems with anti-ordered architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The length of error bars visualises the total number of sys- tems in each bin as: 100/ √ bin counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' timing variations, and direct imaging;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' this complicates the es- timation of completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The PLATO mission is an upcoming space mission that is equipped to allow for statistical estimates of cosmic occurrence rates of planetary system architecture in our galaxy (Rauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' If more exoplanetary systems are uniformly detected and characterised, then it would be possible to estimate the occurrence rate of the different classes of system architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' While such a result would constitute an important knowledge about our Universe, it could also become an excel- lent way of constraining our knowledge of initial conditions for planetary formation and the physical processes which shape the system architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The frequency of architecture class in sim- ulations is a direct consequence of the initial conditions and the physical processes modelled in the Bern model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Architecture class: similar Planetary systems have a similar architecture when all planets in the system have masses that are approximately similar to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' These planetary systems are the archetypical examples of the peas in a pod trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' There are several well-known planetary systems exhibiting similar architecture, such as Trappist-1 (Agol et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2021), TOI-178 (Leleu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2021), Kepler-20 (Buchhave et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2016), and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This architecture is the most common outcome of planetary formation and is also the most frequent architecture class in our observed catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Similar systems in the Bern model are composed of several low-mass planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' They tend to have limited diversity in plan- etary masses when compared with the observed systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The mass distribution, for similar systems in the Bern model, shows that there are many low-mass (< 1M⊕) planets in these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This peak is missing in observations as well as synthetic observa- tions as low mass exoplanets are difficult to observe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This could, however, be remedied in future as current radial velocity spectro- Article number, page 9 of 28 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 43751corr 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='06 Coefficient of Similarity (Mass) [unitless] 10 2 10 1 10 0 10 1 10 2 10 3 SMA [AU] 673 4 816 879 131 141 274 396 946 397 453 871 461 683 790 778 893 965 912 153 110 254 522 959 828 911 402 914 System ID 1 M 50 M 1 MJ 10 MJ Anti-Ordered Similar Ordered Mixed 10 2 10 3 Teq[K] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Architecture plot showing the architecture of observed (left) and randomly selected synthetic planetary systems (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Each row is for one planetary system and the circles in that row represent planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The area of the circle encodes planetary mass, and the colour shows the equilibrium temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The coefficient of similarity for each system is shown on the right y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The x-axis shows the semi-major axis, which is different for the two panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' graphs reach the ≈ 20 cm/s precision necessary for discovering exoplanets in the super-Earths and Earths mass range (Lillo-Box et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Netto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The radius distribution of similar systems implies that these systems are prominently composed of rocky planets, super-Earths and sub-Neptunes7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 7 Throughout this paper, we use planetary classes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' rocky, super- Earths, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=') from the radius based classification of Kopparapu et al.' metadata={'source': 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1 100 101 102 103 SMA [AU] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='8 Density Observations Similar Mixed Ordered 4 6 8 10 12 14 Multiplicity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 0.' metadata={'source': 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of the architecture classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' These plots show the distribution of various quantities (columns) as function of different cata- logues (rows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Left to right: Distributions of mass, radius, distance, and multiplicity in the following catalogues (top to bottom): Bern model, Bern RV Multis, Bern KOBE Multis, Bern Compact Multis, and observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' All catalogues are described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Some notable features from these plots are discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' All individual distributions are normalised such that the area under each curve sums to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The dotted vertical line in the radius distributions marks 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='75R⊕ – approximately, the location of the well-known gap in the radius distribution (Fulton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Since there are only two mixed systems with the same multiplicity (n = 4) in our observations catalogue, a vertical line replaces the density kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The Gaussian density kernels in all other cases were estimated using Scott’s rule (Scott 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Article number, page 11 of 28 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 43751corr The Bern RV Multis show a bimodal planetary distance dis- tribution for similar systems (as well as for mixed and ordered).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The approximate location of the gap is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='28 au or 55 d (for a solar mass star).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This bi-modality is not visible in our observed cata- logue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Planets in similar and mixed systems in the Bern Model also show a dip around this location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In the Bern Model, in- wardly migrating giant planets (≳ 100 M⊕) tend to stop around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='4 au or 100 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Inside this region, low-mass planets are popu- lous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We attribute this bi-modality to these two populations of planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This bi-modality probably arises because planets switch their orbital migration from type I to type II depending on their masses (Emsenhuber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This bi-modality cannot be seen in Bern Compact Multis because we only include planets with periods less than 100d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' For Bern KOBE Multis, the com- pleteness of the Kepler mission for large distant planets is poor (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2 in Mishra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' However, a dip at this loca- tion in Bern KOBE Multis is visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' It would be interesting to see if such a bi-modality is also present in the Kepler catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We tested the significance of this bi-modality with Hartigan’s dip test (Hartigan & Hartigan 1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The dip test is suggestive of the bi-modality for the Bern RV Multis and Bern KOBE Multis (p- value < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='05) and insignificant for the other catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' A system’s architecture is sensitive only to the relative distri- bution of a quantity (such as mass) amongst its planets and not the absolute distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' HR 8799 offers an example (Marois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2008) as a relatively young system with four directly im- aged giant planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Our framework identifies the architecture of this systems as similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Most observed similar systems are composed of low-mass planets (≲ 100M⊕), making HR 8799 a unique exception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This shows that the architecture framework is sensitive only to the relative variations in the mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Additionally, there are only two systems (out of 1000) in our simulated cata- logue where a similar architecture arises from only giant planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Even then, these two synthetic systems have only two giant plan- ets much closer to the star than the HR 8799 planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The Bern Model does not produce many HR 8799-like systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This sug- gests that a system with similar architecture made up of only giant planets is probably rare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' One possibility could be that sys- tems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' HR 8799) with such architecture are probably diffi- cult to form via core accretion pathway (Konopacky & Barman 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Such systems may require additional formation mecha- nisms such as protoplanetary disk instabilities (Schib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Boley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Kratter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Architecture class: mixed Planetary systems where the planetary masses (inside-out) show broad increasing and decreasing variations have mixed archi- tecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' GJ 876 and Kepler-89 host planetary systems with a mixed class architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' GJ 876 is an M dwarf low luminous (≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='01 L⊙) star hosting four planets with masses between 8 − 888M⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The outer three planets are in a Laplace mean- motion resonance (Millholland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Kepler-89, on the other hand, is an early F, highly luminous (≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5 L⊙) star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' It hosts a compact four planet system with masses between 10 − 100M⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Despite the starkly different stellar properties, the architecture of these two systems is analogous: CS (M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='12 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='17, CV(M) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='9, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' While the coefficient of simi- larity is low for both systems, the coefficient of variation is larger than √ 3/2, which helps us identify the architecture of these sys- tems as mixed class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Indeed, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 6 indicates that this identifica- tion is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The frequency of this architecture class in the Bern model is ≈ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The Bern model’s synthetic mixed architecture planetary systems (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 6 right) tend to have numerous Earth- mass planets outside 10 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This parameter space (mass-distance plane, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 1), however, remains inaccessible to most exoplanet detection techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' These systems are also composed of super- Earths, sub-Neptunes, Neptunes, and Jovian planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The bi- modality in distance distribution (discussed before) is prominent for these architectures in Bern RV Multis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We found a Harigan’s dip statistic of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='03 and p-value of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2 (Hartigan & Hartigan 1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Architecture class: anti-ordered Planetary systems where the planetary mass shows an overall decrease with distance have an anti-ordered architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' There are no observed examples of this architecture class in our cata- logue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The frequency of this architecture class in the Bern model is ≈ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' About ≈ 4% of systems in Bern KOBE Multis, ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2% of systems in Bern Compact Multis, and ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2% of systems in Bern RV Multis have this architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This shows that it is an observationally challenging system architecture to detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' However, even if 1% of observed exoplanetary systems are Anti-Ordered we should already have found about 30-40 such systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' More work is necessary to identify the handful of these systems from the already observed systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Many cur- rently known single hot Jupiter systems may host additional small, distant, and as yet undetected planets – revealing these potentially anti-ordered systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Anti-ordered systems in the Bern Model are mostly com- posed of low mass planets ≲ 5M⊕ and giants ≳ 100M⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In the Bern Model, the radius distribution of this architecture class peaks for Rocky and Super-Earths planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' It decreases for sub- Neptunes and Neptunes and then increases again for Jovian plan- ets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Many of the low-mass planets that make up this architecture class are outside 10au, making their detection very challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The multiplicity distribution shows that these systems tend to have fewer planets than similar or mixed architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This is an indication that the formation pathway of these architectures differs considerably from the other two types of architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Planets from anti-ordered architectures show a weak distance bi- modality feature (discussed earlier in this work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This is under- standable since these architectures consist of massive planets in the inner parts and less massive planets in the outer parts of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The distance bi-modality seems to arise from low mass planets (migrating via type I) inside 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='28au or 55 days and giant planets (migrating via type II) outside 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='28au or 55 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This adds further strength in attributing the distance bi-modality to planetary migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Architecture class: ordered Planetary systems where the planetary masses shows an overall increase with distance have an ordered architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The increas- ing mass may be monotonic (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' TOI-561, HD 20781, DMPP- 1,HD 160691, HD 164922) or non-monotonic (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' the Solar System, Kepler-11, 55 Cnc, Kepler-48, Kepler-65).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Ordered ar- chitecture is a rare outcome for the Bern model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Observations are generally biased against discovering small and less massive planets which are farther away from their host star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Such biases, however, make ordered systems the second most common archi- tecture class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Fifteen systems in our catalogue exhibit this archi- Article number, page 12 of 28 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Mishra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' : Architecture Framework I – Four classes of planetary system architecture tecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Unsurprisingly, the most notable known example of this architecture class is the Solar System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The mass and radius distributions of ordered architecture in the Bern Model shows considerable difference from other archi- tecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The mass distribution peaks around 1000M⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Most of the Bern model’s ordered systems tend to have at least one gi- ant planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' These systems are also composed of sub-Neptunes, Neptunes, and Jovian planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Internal composition across architecture classes So far we have seen the new architecture framework (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 3) and some characteristics of the four classes of architecture (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In this section, we study the connection between the bulk mass architecture classes and the internal composition of the planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This section demonstrates that the same architecture framework can be used to study the multi-faceted nature of planetary sys- tem architecture – from bulk mass architecture to density archi- tecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We study several different aspects of the planetary in- ternal composition: (a) radius architecture (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (b) bulk density architecture (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (c) Core/Envelope mass archi- tecture (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' and (d) fraction of volatiles and water ice in core architecture (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We explore these connections for planetary systems in the simulated (Bern model) and syntheti- cally observed catalogues (Bern RV Multis, Bern KOBE Mul- tis, Bern Compact Multis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' All results in this section are derived from synthetic planetary systems only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Radius architecture Weiss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2018) showed that the size of adjacent exoplanets were similar – coining the phrase ‘peas in a pod’ to describe this architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Millholland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Wang (2017) extended these ideas to planetary masses, showing that the masses of ad- jacent planets are also correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In Mishra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2021), we suggested that the peas in a pod trends in terms of size effec- tively emerge from the mass trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Here, we attempt to set our assumption on firmer ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Figure 8 (top) shows the coefficient of similarity for radii as a function of the coefficient of similarity of masses, for systems with two or more planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This allows us to compare the system- level radius architecture with the system-level mass architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We easily see that most systems seem to follow a linear rela- tionship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The Pearson correlation coefficient is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='89, indicating a strong positive correlation between the mass and radius archi- tecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The coefficient value increases to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='96, when systems with only three or more planets are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Since the mass- radius relation is not a bijective function (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' one-to-one corre- spondence), there are some systems that show a strong deviation from the linear relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Figure 8 (bottom) shows the radii architecture for the syn- thetic planetary systems8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This shows that most systems that are ordered (or anti-ordered) in mass are also ordered (or anti- ordered) in terms of radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The figure also shows that systems which are similar or mixed in mass architecture have CS (R) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Systems with mass similarity have lower CV(R) compared to sys- tems with mass mixture, suggesting that for most systems, the 8 A future study could investigate the boundaries for robust architec- ture identification, as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 3, but based on radius instead of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Such a classification is readily applicable since radius measurements tend to be uniformly available and are better agreed upon amongst sev- eral observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Data-driven approaches such as machine learning could be useful in such an endeavour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 3 2 1 0 1 2 3 Coefficient of Similarity (Mass) [unitless] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='8 Coefficient of Similarity (Radius) [unitless] RPearson = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='89 RSpearman = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='94 Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Fit: y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='23 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0006 Bern Model Observations - 41 systems Solar System Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Radii architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Top: The diagram shows the coefficient of similarity of radii as a function of the coefficient of similarity of masses, for synthetic and observed planetary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The dashed line shows the corresponding linear fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Bottom: Radius architecture of synthetic plane- tary systems contrasted with the mass architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In the bottom panel, the marker colour and shape indicates the bulk mass architecture of a system and its position on the diagram suggests its radii architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' radius architecture closely follows the mass architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' At the planetary level the radius of a planet is correlated with its mass via the planet’s chemical composition (Lopez & Fortney 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Our architecture framework shows that such relationships also exist at the system level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' A few mass-ordered systems show sim- ilarities in radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' These few systems have the following com- mon features: two mass-ordered giant planets with similar sizes (masses ∼ several MJ’s, and radius ≈ 1RJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This illustrates that Article number, page 13 of 28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='6 Based on Cs(M): [unitless] Similar Anti-Ordered 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='4 Mixed Ordered Coefficient of Similarity (Radius) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 Coefficient of Variation (Radius) [unitless]A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 43751corr while mass architecture and radius architecture are related, they are not always identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We conclude that the peas in a pod radius correlations gen- erally arise from the underlying mass architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We consider the mass architecture primal because planets, foremost, accrete mass from the protoplanetary disk and, consequently, are char- acterised by a size that is in accordance with their internal struc- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Density architecture Bulk density (or simply density) is a directly measurable quan- tity which is sensitive to the internal structure of a planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This makes density an important characteristic for understand- ing planetary structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The density of a planet depends on many parameters and many physical processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' For example, a planet’s mass may depend on its accretion history, starting loca- tion, amount of material in disk, competition with other planets, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Giant impacts may also affect a planet’s density, as explained in Bonomo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In this section, we study the arrangement and distribution of planetary density around their host star, namely, the density architecture of a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Figure 9 (left) shows the density of a planet, simulated via the Bern model, as a function of its mass and starting location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The figure also shows the density of solar system planets and few ob- served exoplanets (from our catalogue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The plot can be roughly divided into two halves: (a) planets with a mass of < 100 M⊕ and (b) planets with a mass of > 100 M⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In our simulations, most planets which started inside the ice line tend to have terrestrial Earth-like densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' These planets are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5 − 3R⊕ and ⪅ 10M⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Planets starting around or outside the ice line generally accrete more volatile rich material and H/He gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' These planets have lower densities due to their larger sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Planet which started outside the ice line (3-10 au) show a broad diversity in their den- sities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' As they accrete more gases, their density decreases fur- ther.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' These planets are roughly 2 − 10 R⊕ and are characterised by masses that vary by four orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Planets more massive than 100 M⊕ seem to lie on a single curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Since the size of these planets remains the same (≈ 1RJ or 11R⊕,), their densi- ties increases linearly with their masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Planets that started in the outer regions (30-40 au) cluster on the density-mass plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' These planets have low densities (< 2g/cm3) and low masses (⪅ 1 M⊕).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The density architecture for simulated systems in the Bern Model is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 9 (middle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' An important relation be- tween mass architecture and density architecture is seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Some systems which are ordered (or anti-ordered) in mass are also or- dered (or anti-ordered) in density, that is, these systems have large positive (or negative) CS (ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In other words, simulations suggest that planetary systems can also be ordered or anti- ordered in density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' A system is ordered in density when the in- ner planets have small densities and the outer planets have larger densities – and vice-versa for density anti-ordered systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Sys- tems with mass architectures of similar and mixed are strongly clustered around CS (ρ) ≈ 0 and CV(ρ) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The inset shows that similar mass systems tend to have small CV(ρ), while mixed mass systems have larger CV(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This implies that some systems that are similar (or mixed) in mass show some similarity (or mix- ture) in density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' A system with a similar density architecture will host planets that have approximately similar densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' However, the region CS (ρ) ≈ CV(ρ) ≈ 0 is empty, indicating the absence of planetary systems where the density of planets (inside out) is approximately the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' While there are exceptions, overall, for many systems, the density architecture seems to follow their mass architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This approximate link between the mass and density archi- tecture stems from massive planets (> 100 M⊕) whose densities increase with their mass (see 9 (left)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Systems which do not host any massive planet are mostly similar in their mass architecture and have CS (ρ) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The inset shows that the Aryabhata’s num- ber increases as a system approaches the CS (ρ) ≈ CV(ρ) ≈ 0 region (see Paper II for the definition of Aryabhata’s number).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' If a system has more surviving planets that started from inside the ice line, then the densities of these planets will be more similar to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This means that the density architecture of a system shows some dependence on the starting location of a planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We also investigated if the relation between the mass and density architectures is observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Figure 9 (right) shows the density architecture for systems from our synthetically observed catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Also shown is the density architecture of some ob- served exoplanetary systems for which the mass and radius mea- surements were available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The density architecture of syntheti- cally observed catalogues shows a trend which is quite unlike Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 9 (middle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' There is an unexpectedly good agreement be- tween the synthetically observed systems and the observed plan- etary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We attribute the peculiar shape of this plot to the difficulty of detecting distant planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Transit and RV observa- tions favour the detection of planets within ∼ 1 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Many close- in planets tend to have Earth-like densities, while planets far- ther out have lower densities (due to either their volatile rich or gaseous composition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Overall, this would lead to an observed density architecture where inner planets have higher densities and outer planets have lower densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' A situation such as this will be characterised by negative CS (ρ), which is readily seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 9 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In summary, many synthetic systems show a relationship be- tween their mass architectures and their density architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Bern model systems that are ordered or anti-ordered in their mass also tend to be ordered or anti-ordered in their densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The dispersion of planetary bulk densities in similar class sys- tems is lower than mixed class systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This relation seems to emerge from massive planets whose densities increases linearly with their masses (since they cannot grow their sizes any more).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' These relations can be considered as a prediction from this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' As future observations probe the outer parts of an exoplane- tary system, we may anticipate the discovery of several systems whose mass and density architectures are closely linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Core and envelope mass architecture In this section, we show that (a) most simulated planetary sys- tems inherit their architecture from the underlying core mass architecture;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (b) the accretion of gases tends to accentuate the underlying core mass architecture, and (c) the observed mass ar- chitecture of a planetary system is a gateway to studying the core mass architecture of the system, since the two are strongly cor- related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Exceptions to the first two statements tend to arise for those systems undergoing strong, multi-body dynamical effects such as planet-planet scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The fraction of mass which is partitioned into a planet’s core and its envelope is governed by planetary formation physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The end result is dictated by an interplay of several concurrent pro- cesses (see Emsenhuber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Mordasini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2012b, for discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In the core-accretion scenario, giant planets are formed when planetary cores can undergo run-away gas accre- tion (Pollack et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Alibert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2004, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Proto-planets that have failed to trigger runaway gas accretion comprise a di- Article number, page 14 of 28 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Mishra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' : Architecture Framework I – Four classes of planetary system architecture 100 102 104 Mass [M ] 10 2 10 1 100 101 102 Bulk Density [g/cm3] J S Similar Anti-Ordered Mixed Ordered Observations Solar System 10 1 100 101 Embryo Starting Distance [AU] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 CV ( ) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='00 CS ( ) 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1 0 Similar Anti-Ordered Mixed Ordered 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content="0 Aryabhata's Number 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 CV ( ) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='4 CS ( ) Bern RV Multis Bern KOBE Multis Bern Compact Multis Sun Trappist-1 TOI-178 Kepler-11 K2-138 TOI-561 K2-266 KOI-94 Kepler-107 Kepler-223 Other Observed Systems Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Density architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Left: Bulk density of simulated and few observed planets as a function of their mass and starting locations (for synthetic planets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The marker indicates the mass architecture of the system to which a synthetic planet belongs to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Middle: Density architecture, of synthetic planetary systems, as seen through the coefficient of similarity versus the coefficient of variation plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The marker shape and colour indicates their host system mass architecture and the system’s Aryabhata’s number (see Paper II), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Right: Density architecture of planetary systems from the simulated observed catalogue and few observed planetary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 CS (Core Mass) 2 1 0 1 2 CS (Mass) n i Menv [M ] Bern Model y = x Similar Anti-Ordered Mixed Ordered 100 101 102 103 104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 CV (Core Mass) 0 1 2 3 4 5 CV (Mass) n i Menv [M ] Bern Model y = x Similar Anti-Ordered Mixed Ordered 100 101 102 103 104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='00 CS (Mass) Bern Compact Multis 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5 CS (Core Mass) Bern KOBE Multis 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5 Bern RV Multis Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Mass architecture as a function of core-mass architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Panels compare the mass architecture with the core-mass architecture via the coefficient of similarity (left) and coefficient of variation (middle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In the left panel, the points corresponding to similar systems are very tightly clustered on the y = x line and are not visible due to over-plotting of points from other architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This signifies the core-mass architecture is very strongly correlated with the mass architecture for similar systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The sum of mass in the envelope of each planet in a system is indicated in colour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The right panel plots the coefficient of similarity for masses and core masses for systems in the synthetically observed catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' verse group of planets: Earths, Super-Earths, mini-Neptunes, and Neptunes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The bifurcation of a planet’s mass into its core and its enve- lope can probe the formation history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' For example, in our sim- ulations, most giant planets (⪆ 1 MJ) have about 1% of their mass in their cores and the rest is in their gassy envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' On the other hand, low mass planets (⪅ 10 M⊕) hardly accrete any gaseous envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' However, the mass in a planet’s core and en- velope is not an observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Even for the solar system planets, internal structure models guide our knowledge of core and enve- lope masses (see Helled et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2020, for a review on Uranus and Neptune).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' As giant planets dominated by their H/He envelopes are rare, we expect a strong correlation between the mass architecture (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' the arrangement and distribution of planetary masses) and the core-mass architecture (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' the arrangement and distribution of core-masses) to exist also at the system level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 10, we show the coefficient of similarity and the coefficient of variation of planetary mass as a function of the coefficient of similarity and the coefficient of variation of core mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The colour indicates the total mass of envelope accreted by all planets in a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Comparing the coefficient of similarity for planetary masses and core masses (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 10, left panel), we observe that a large frac- tion of systems (> 90%) follow the y=x line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This implies that for most planetary systems, the arrangement and distribution of core masses is imprinted on the mass architecture of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Sys- tems which show large deviations from the y=x line have gener- ally accreted a large amount of gaseous envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This suggests that the formation of one or more giant planet is partly responsi- ble for the deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We also observe another important feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Planetary systems that are ordered in mass are also often ordered in their core-masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Conversely, mass anti-ordered systems tend to be anti-ordered in their core masses as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In addition, or- dered systems are either on or above the y=x line, whereas anti- ordered systems are either on or below this line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This suggests that the accretion of gases generally accentuates the underlying core mass architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Considering the coefficient of variation for masses and core masses (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 10, middle), we see that most of the planetary sys- tems lie either on or above the y=x line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The CV value measures the amount of variation in a set of numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This suggests that the variation in total masses, for most systems, is either similar or larger than the variation in the core masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This is under- standable, since the amount of gas accreted by a planet shows a strong correlation with the mass of the planet’s core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' How- ever, there are a handful of systems where the variation in total Article number, page 15 of 28 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 43751corr mass is less than the variation in core masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Systems that are similar in the mass architecture are strongly clustered over the y=x line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This stems from the low amount of gas (0 − 20M⊕) accreted by planets in these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Figure 10 (middle) shows that mixed class systems, as opposed to similar systems, form a separate cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Physically, this difference is arising from the larger amount of gas (50−5 000M⊕) accreted by planets in these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Here, the question arises as to whether the strong correlation between mass architecture and core-mass architecture is observ- able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 10 (right), we show CS (M) as function of CS (Mcore) for the three synthetically observed catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' All three cata- logues probe the inner regions of a planetary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The figure shows that the correlation between mass architecture and core mass architecture is strong in all three catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This suggests that the observed mass architecture of a planetary system can be used to study the underlying core-mass architecture of the sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This is potentially useful to distinguish among competing models of planet formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Role of embryo starting location We have seen that the core mass architecture of a system strongly governs the overall architecture of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The arrangement of planets in a system also reflects the final distances of these planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' It is, therefore, instructive to understand some key as- pects which shape these two important properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The core mass and the final distance of a planet are strongly influenced by, among other effects, the distance at which an embryo starts in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Figure 11 shows the core mass (left) and the final distance (right) as a function of the starting distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In the Bern model, lunar mass (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='01M⊕) protoplanetary embryos are initialised with a random starting location between the inner edge of the disk and 40 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We also recall that failed embryos (objects with a total masses < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1M⊕) are removed from our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Emsenhuber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2021a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Burn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2021) analysed the nature of planetary migration using migration maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Both stud- ies show the existence of so-called convergence zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Within these zones, planets can migrate outwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' However, outside this zone inward migration is prevalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The existence of such con- vergence zones suggests that there ought to be regions of planet over-densities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' this are essentially regions where planets are ra- dially ‘stuck.’ These studies attribute the presence of these zones to dust opacity transitions and disc structures, finding that these zones evolve with the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' For a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='01M⊙ disc, around a solar mass star at 1Myr, these zones are: (a) for low-intermediate mass planets (⪅ 1M⊕) extending from disk inner edge to about 1au and (b) for intermediate mass planets (1 − 10M⊕) around 2-3 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Figure 11 (left) shows that even for embryos that start at the same initial distance, the mass accreted by a planetary core can differ by two to three orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' These differ- ences arise from (a) varying solid disc masses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (b) competition for accretion in the feeding zone (Alibert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (c) dy- namical state of solids in the disc resulting from planetesimal- planetesimal, planetesimal-protoplanet, planetesimal-gas disc interactions, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Nevertheless, the starting distance seems to play a significant role in this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The ice line seems to divide the parameter space into two regions: fewer planets inside the ice line have low mass cores (⪅ 1M⊕), while many planets outside the ice line have low-mass cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Inside the ice line, most planets have cores of 1−10M⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Plan- ets that start very close to the star (⪅ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1au) are unable to accrete a lot of material owing to their small Hill spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This explains their small cores masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Inside the ice line, planets belonging to systems of mixed, anti-ordered, and ordered architecture tend to have more massive cores than planets belonging to similar sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Around the ice line, planets show a large variety of core masses ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1M⊕ to 100M⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Outside the ice line we see the same trend as before: planets that are in similar systems, for the same starting location, usually have less massive cores than planets which belong to systems of other architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The final distance of a planet depends on several factors such as: (a) migration type (type I or type II), (b) planet’s mass, (c) local disc properties, and (d) multi-body effects such as N-body scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The joint distribution of a planet’s final and starting locations shows an intriguing trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Generally, for many plan- ets, the final distance strongly correlates with their starting loca- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Orbital migration allows planets to move (mostly) inwards – positioning many planets below the y=x line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' N-body effects (such as planet-planet scattering or outward migration) may scat- ter some planets further away from their host star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' These planets are located above the y=x line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Curiously, many planets which end up farther away than their starting location were initialised around the ice line and are mostly low massive (⪅ 20M⊕).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We attribute this over-density to the outward migration convergence zone around the ice line discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Another important finding is that planets inside the ice line in similar systems probably formed in situ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 11 (right) shows that most planets, inside the ice line, which did not migrate in- wards are part of similar architecture systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Conversely, most of the planets which have migrated inwards seem to belong to systems that have mixed, anti-ordered, and ordered architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Outside the ice line, many planets have migrated inwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Most planets starting around 20 au (or more) accrete little mass in their cores and show little radial displacement (Hansen & Murray 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Chiang & Laughlin 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The properties of these em- bryos may draw some influence from our modelling choice as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The N-body integrator in this model is used for 20 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Longer integration times may allow some embryos to have more massive cores via giant impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Core water-ice mass fraction architecture Our model calculates the internal structure of a planetary core (for details see Emsenhuber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Mordasini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2012a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We solved 1D differential equations demanding mass conser- vation and hydrostatic equilibrium, with a modified polytrope equation serving as the equation of state (Seager et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The chemical composition of each planetary core is also tracked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This is accomplished by tracking the chemical makeup of the accreted planetesimals and other colliding planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The underly- ing chemical models have thirty-two refractory and eight volatile species (Thiabaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Marboeuf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2014a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' These different chemical species are grouped into three different ma- terials which make the planet’s core, in our model: (a) iron, (b) silicates, and (c) ice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' All refractory species (except iron) make up the silicate mantle and all volatile species contribute to ice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Since H2O constitutes 60% of all ice by mass, we label this latter component as water ice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The water mass fraction ( fice) of each planetary core is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We assume that inside the H2O ice line, only refractory ele- ments contribute to the solid phase of a planetesimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Outside this evolving ice line, due to their condensation, volatile ele- ments also contribute to the solid phase of a planetesimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Fig- ure 12 (left) shows the water mass fraction of a planet’s core as a function of its initial location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Most planets which start inside the ice line have little to no volatiles in their cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' A jump in Article number, page 16 of 28 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Mishra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' : Architecture Framework I – Four classes of planetary system architecture Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Role of starting location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Plot shows the planetary core mass (left) and final distance (right) versus the starting distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The marker style indicates the architecture of the system to which the planet belongs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The vertical grey shaded region indicates the evolving locations of the ice line (Burn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The dotted line in the right panel shows the y=x line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 CV (fice) 1 0 1 2 3 4 5 CS (fice) Bern Model Similar Anti-Ordered Mixed Ordered 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content="0 Aryabhata's Number 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 fice 0 1 2 3 4 5 6 7 8 9 Density "Dry" "Moist" "Wet" Bern Model Similar Anti-Ordered Mixed Ordered Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Planetary core water-ice mass fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Left: Core water mass fraction of a planet as a function of its starting location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The architecture of the system to which a planet belongs to is shown by marker characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The vertical shaded regions shows the location of the ice line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Middle: Water mass fraction architecture seen through the coefficient of similarity versus the coefficient of variation plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The shape of the marker shows a system’s mass architecture, and the colour depicts its Aryabhata’s number (see Paper II for definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Right: Distribution of fice across architecture classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Depending on fice, planets are labelled as ‘dry’, ‘moist’, or ‘wet’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' fice is seen around the ice line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Outside the ice line, most plan- ets have at-least 40% fice in their cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This suggests that the history of formation and evolution of a planet is imprinted on its water mass fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We are interested in studying the ice mass fraction architec- ture of a planetary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' However, we cannot directly apply our framework (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 1, 2) because the water mass fraction is a quantity that admits 0 as a value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' While this can lead to ill- defined numbers, this issue has a simple remedy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' For quantities that can be 0, we propose the following modification to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 1: CS (q) = lim ϵ→0 1 n − 1 i=n−1 � i=1 � log qi+1 + ϵ qi + ϵ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (4) Numerically, we calculated the coefficient of similarity with ϵ = 10−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We verified this step by calculating the coefficient of similarity for quantities which do not admit zero (such as masses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In a bootstrapped numerical experiment of 10,000 tri- als, the coefficient of similarity for mass was calculated using both equations (1 and 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The relative difference between the two outcomes ranged between 10−12 to 10−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The ice mass fraction architecture of Bern Model systems is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 12 (middle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' A prominent feature from this fig- ure is that most systems have CS ( fice) either close to 0 or posi- tive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' A system with CS ( fice) ≈ 0 and low CV(fice) will be com- posed of planets whose core water mass fraction is similar to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' A system with positive CS ( fice) will be composed of planets whose core water mass fraction increases inside out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Article number, page 17 of 28 Bern Model 102 Similar Anti-Ordered Mixed Ordered 10 10 10-1 10-1 100 101 Embryo Starting Distance [AU]10 Bern Model Similar Anti-Ordered [AU] 102 Mixed Ordered Planet Final Distance [ 101 RV Multis KOBE 100 Multis Compact Multis 10- 10- 10-1 100 101 Embryo Starting Distance [AU]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='6 Bern Model Similar 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5 Anti-Ordered Mixed "Wet" Ordered 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2 "Moist\' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 10-1 100 101 Embryo Starting Distance [AU]A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 43751corr Figure 11 (right) tells us that many planets that started outside the ice line, and are water rich have not suffered any major ra- dial displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Thus, a positive CS ( fice) should be a default scenario for most planetary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' About 74% systems in the Bern model have CS ( fice) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Almost 97% of systems have CS ( fice) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We propose the ‘Aryabhata formation scenario’ to explain the ‘non-default’ systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This scenario and the related quantity ‘Aryabhata’s Number’ are described in Paper II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Frequency of dry, moist, and wet planets We are interested in exploring the link between the water mass fraction architecture and the mass architecture of a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' To this end, we divide planets into three categories based on their water mass fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' A planet is called ‘dry’ if fice ≤ 1%, ‘moist’ if fice ∈ (1, 40]%, and ‘wet’ if fice > 40%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' These labels serve to simplify our analysis and allows us to see general trends between system architecture and planetary composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The distribution of water mass fraction across systems of different architecture classes is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 12 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' While all three planet classes are present in all four architecture classes, there are some ob- servable trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Figure 12 (right) shows that similar architectures host many of the dry planets produced in the Bern model and anti-ordered architectures are mostly composed of wet planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This tells us that many of the planets that start inside the ice line become part of similar architecture systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Conversely, systems with anti- ordered architecture are mostly composed of planets that started outside the ice line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Mixed architecture systems are generally composed of more planets that started outside the ice line than inside, as compared to similar architecture systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Moist plan- ets are present in all architecture classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We quantify the fre- quency of dry, moist, and wet planets as a function of mass archi- tecture class (similar, mixed, ordered, or anti-ordered), metallic- ity (low or high), and source catalogues (Bern model, Bern Com- pact Multis, Bern KOBE Multis, and Bern RV Multis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Figure 13 shows the planets per star (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' the number of each planet type di- vided by the number of stars) across these forty sub-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Overall, compared to synthetically observed catalogues, Bern model simulations demonstrate more wet planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This is understandable since we are looking at the entire underly- ing population, which includes planets from the outer parts of these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Likewise, synthetically observed catalogues tend to have more dry planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Systems around low-metallicity stars (regardless of the catalogue) generally tend to have a higher frequency of dry planets as opposed to systems around high- metallicity stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The frequency of wet planets shows a notice- able increase for systems around high-metallicity stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Amongst the different catalogues, Bern Compact Multis have the highest frequency of dry planets, followed by Bern KOBE Multis, and Bern RV Multis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Low-metallicity environments have a slightly higher average planet per star (8835/541 ≈ 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='3) than high- metallicity environments (6722/455 ≈ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Similar systems: Systems in the underlying Bern model that are characterised by a similar architecture tend to have many wet planets (∼ 10 per star) and few dry or moist planets (∼ 3 − 4 per star).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' However, synthetically observed catalogues seem to have a bias against the discovery of many wet planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' For the similar class of compact multi-planetary systems, dry planets are more common around a low-metallicity star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' However, for a high-metallicity star, the frequency of dry and wet planets is roughly the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' For transiting systems, in the Bern KOBE Multis, low-metallicity environments favour more dry planets and equal proportions of wet and moist planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Conversely, in high-metallicity environments, wet planets occur more fre- quently than dry or moist planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' For RV systems, the fre- quency of each planet class is approximately the same in a low- metallicity environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' High-metallicity environments almost double the frequency of wet planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The average planet per star is similar around both low metallic (≈ 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='8) and high metallic environments (≈ 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Mixed systems: Mixed class systems generally have many wet planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' It is only for compact systems around high- metallicity stars, the frequency of dry planets is higher than wet planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In all other cases, the frequency of wet planets is greater than the frequency for dry or moist planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The average planet per star is similar around both low-metallicity (≈ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2) and high metallicity environments (≈ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Anti-Ordered systems: Systems with anti-ordered architec- ture have a distinct core water mass fraction architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' These systems are rich in wet planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In fact, about 80% of these sys- tems follow the Aryabhata formation scenario described in Pa- per II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Compact anti-ordered systems may have some dry plan- ets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' For transit and RV surveys, the frequency of dry planets is zero in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The total number of planets per star in anti-ordered systems is slightly higher around low metallic- ity stars (159/19 ≈ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='4), as compared to high metallicity stars (504/65 ≈ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In the future, if an anti-ordered architecture planetary system is to be discovered, it would be interesting to study its core water mass fraction architecture as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The cur- rent work suggests that the Aryabhata’s number for these sys- tems should be close to 0 and, irrespective of the detection tech- nique, the system should would be expected to have many wet planets (see Paper II);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' this is one of the main predictions arising from this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Ordered systems: Juxtaposed directly to the anti-ordered sys- tems, ordered systems in synthetically observed catalogues tend to be rich in dry planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' These systems are distinct not only because of their frequent dry planets, but also due to a low fre- quency of wet planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' For all synthetic catalogues, moist plan- ets occur more frequency than wet planets, which is a unique distinguishing feature for these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' For the Bern model, these systems have low average planets per star: 5 around low- metallicity stars and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1 around high-metallicity stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In summary, we note some salient features of these sys- tem architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Generally, wet planets survive more fre- quently around high-metallicity stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' One detection technique that favours the discovery of close-in planets also favours the de- tection of dry planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The comparative frequency of planet (dry, wet, or moist) per star seems to be intimately connected with the mass architecture of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Similar and mixed systems can host lots of dry or wet planets, depending on the metallicity of the systems and detection technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Anti-ordered systems, forming prominently via the Aryabhata formation scenario, are rich in wet planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Ordered systems, in simulated observations, are rich in dry planets and have more moist planets than wet planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The physical connection between the average planet per star and the star’s metallicity is sensitive to the formation path- ways that a system undergoes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Habitability as a function of system architecture In this paper thus far, we have described a new framework for studying the architecture of planetary systems (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 3), the char- acteristics of the four classes of system architecture (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 4), and the relation between the mass architecture of a system and its internal structure and composition architecture (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 5).' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='Moist ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='N*=5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='Npl=21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='Dry ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='Wet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='Moist ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='Bern RV Multis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='N*=29 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='Npl=128 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Frequency of planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This diagram shows the average planet per star for dry, wet, and moist planets in several catalogues (rows), across several architecture classes (columns), and around low (left) and high (right) metallicity stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The planet per star is simply the total number of planets divided by the total number of stars, after appropriate filters for metallicity, catalogue, or architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' this section, we speculate on the idea of studying habitability as a function of system-level architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Mankind has pondered the existence of other biotic life- forms beyond Earth, as well as outside our own Solar System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Our current understanding of habitability stems from and is fo- cused at an individual planetary level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We consider whether hab- itability could be correlated with other properties of a plane- tary system, namely, whether habitability could be a system- level phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In this section, we speculate on the role of planetary-system level information on the existence of habit- able worlds in such systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The framework we present here for studying the system-level architecture of a planetary sys- tem brings to light several novel questions, probing the depen- dence of habitability and occurrence of habitable worlds (and related concepts) on the architecture of a said system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' For ex- ample, we wonder how the occurrence rate of habitable planets in the galaxy depends on the occurrence of the four architecture classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In this section, we address this question on three levels: sys- tem, planet, and planet ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We use the concept of empirical Habitable zone (EHZ) planets from Quanz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Koppa- rapu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Planets with masses between [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1, 5]M⊕ and stellar insolation within [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='776, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='32]S ⊕ are considered to be in- side the EHZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The stellar flux limits correspond to ‘recent Venus’ and ‘early Mars’ scenarios and include the luminosity evolution for a 1M⊙ Solar-twin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' At the system level, we note the frequency of systems of a particular architecture to host at least one planet in the EHZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' At the planet level, we count the frequency of planets in the EHZ across each system architecture class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' At the planet ratio level, we show the fraction of all EHZ planets across their architecture class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Figure 14 shows the frequency of EHZ plan- ets, at all three levels, as a function of their system architecture for both synthetic and observed exoplanetary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Out of all synthetic systems with a similar class architec- ture, ≈ 77% host at least one EHZ planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This is remarkably higher than any other architecture class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' ≈ 10% of systems with mixed architecture host at least one EHZ planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The frequency drops to ≈ 1% for anti-ordered architecture systems and ≈ 0% for ordered systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' One way to interpret these numbers could be to look at the multiplicity distribution across each architecture class in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The frequency of at least one EHZ planets across architecture class seems to follow the multiplicity trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Sim- ilar and mixed architectures have comparably high number of planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The distribution of the Aryabhata’s number shows that similar systems usually have higher Aryabhata’s number than mixed systems, implying that similar systems tend to host more planets which started from inside the ice line (see Paper II for Aryabhata’s number).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This may account for the large frequency of similar systems which host at least one EHZ planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The mul- tiplicity distribution shows that anti-ordered systems often host less planets than similar and mixed class systems, while ordered systems have the lowest multiplicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We see in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2 that the similar class architecture is perhaps the most common archi- tecture for planetary systems in our galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' These results from the Bern model simulations suggest that observation campaigns to detect habitable planets will find more EHZ planets in similar class architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' For the observed multi-planetary systems in our catalogue, about ≈ 13% of similar class systems have at least one EHZ planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' About 7% of ordered class exoplanetary systems in our catalogue host at least one EHZ planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In the mixed class ob- served systems in our catalogue, none of them have EHZ planets and there are no known anti-ordered class systems in our cata- logue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' These frequencies are quite different from their theoretical counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' While the lack of a complete and reliable obser- vations catalogue may explain the discrepancy for similar class systems – it does not completely explain the discrepancy for or- dered systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Our own planet resides in the ordered class sys- tem of the Solar System, which is not supposed to be influenced by issues such as completeness or detection biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This reflects the inability of Bern models to simulate a Solar System analogue – pointing to a gap in our understanding of the physics that goes into planetary formation and evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In addition, many ob- served ordered class systems may have a different architecture when more planets in these systems are detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Article number, page 19 of 28 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 43751corr Similar Anti-Ordered Ordered Mixed 0 20 40 60 80 100 Frequency [%] Planets in EHZ Bern Model: Systems Observations: Systems 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='7% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0% 9.' metadata={'source': 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100 Frequency [%] Planets in EHZ Bern Model: EHZ Planet Ratio Observations: EHZ Planet Ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='9% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 fice 0 5 10 15 20 25 Density "Dry" "Moist" "Wet" Bern Model EHZ Planets Similar Mixed Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Planets inside the empirical habitable zone (EHZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The left-most plot shows the frequency of planetary systems, of a given architecture class, which host at least one planet inside the EHZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The central-left plot shows the fraction of planets inside a given architecture class which are in the EHZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The central-right plot shows the fraction of all EHZ planets within a given architecture class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The rightmost plot shows the distribution of fice for EHZ planets across the architecture classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The cartoon sketch of Earth emphasises that the only known life-harbouring planet resides in an ordered system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The length of error bars visualises the total number of systems or planets in respective bin as: 100/ √ bin counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The lengths of the error bars represents the number of planetary systems (left-most panel) and the number of planets (two middle panels) which are inside the bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Large error bars in the leftmost panel, for example for anti-ordered architecture emerges from their low count (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The Gaussian kernel is estimated using Scott’s rule (Scott 2015) At the planet level in our simulations, out of all synthetic planets that exist in similar class systems, about 10% are inside the EHZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This frequency is, again, remarkably higher for any other architecture class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' About 1% of all simulated planets in a mixed system are inside the EHZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Close to 0% of all planets in anti-ordered and ordered class architectures are inside the EHZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' From our observational catalogue, while 5% of observed exo- planets in similar class systems are inside the EHZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' About 3% of observed exoplanets in ordered class systems are inside the EHZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The planet ratio level shows the fraction of all EHZ planet that belong to a particular architecture class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In the Bern model, we see that out of all EHZ planets, about 99% are in the simi- lar class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The share of EHZ planets by other architecture classes is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Amongst the observations, three-quarters of EHZ planets are in similar class and the remaining are in ordered class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The observations and theory are quite misaligned in this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We attribute this discrepancy to the absence of a com- plete and reliable catalogue of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Our observations catalogue has only 41 multi-planetary sys- tems, of which only four host planets inside the EHZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' These systems are Trappist-1 (three planets in EHZ), GJ 667 C (two planets in EHZ), Solar System (two planets in EHZ), and Tau Ceti (one planet in EHZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The occurrence of architecture classes and the frequency with which they host EHZ planets might be better constrained with future observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This may allow us to have a better estimate of the occurrence rate of EHZ planets as a function of architecture class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Simulations suggest that ordered architecture is a rare out- come of planet formation (about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5% of systems out of 1000 were deemed to be ordered) and yet, we live in an ordered sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' These two statements can shed new light on the rarity of life in the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We foresee that the famous Drake equation may be suitably modified to take into account the occurrence rate of dif- ferent architectures and thereby set more optimal constraints on η⊕ (Sarkar 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Since water plays a fundamental role for life forms on Earth, it is interesting to probe the core water-ice fraction for the EHZ planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Figure 14 also shows the fice distribution for EHZ plan- ets in the Bern model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' As we see before, most of the EHZ plan- ets are in the similar class and ≈ 1% of EHZ planets are in the mixed class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' EHZ planets in similar systems are ‘dry’, ’‘moist’, and ‘wet’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In stark contrast, EHZ planets in mixed class are only ‘wet’ planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We hope these results may be useful in guiding future missions in finding EHZ planets that have the potential to harbour life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Summary, conclusions, and future work In this paper, we introduce and explore a new framework for studying the architecture of planetary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Our new frame- work allows us to study, quantify, classify, the global architecture of an entire planetary system at the system-level;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' and compare the architecture of one planetary system with another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 3, we detailed the new architecture framework and presented an in- depth discussion comparing our framework with other works in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We present the coefficient of similarity and the co- efficient of variation as two quantities that quantify our concep- tual ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Our framework gives rise to a new parameter space (the CS versus CV plane) in which individual planetary systems can be compared with one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Throughout this paper, we applied this framework to study the distribution and arrange- ment of several planetary quantities within a planetary system, thereby understanding the system architecture for that quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In this manner, we studied the mass architecture, the radius ar- chitecture, the core mass architecture, the core water mass frac- tion architecture, and the density architecture of synthetic and observed planetary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' To study some consequences of this framework, we applied our method to several catalogues of planetary systems (intro- duced in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We curated, especially for the purposes of this study, a catalogue of observed multi-planetary systems that have four or more planets and include mass measurements for at least four planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' For engendering further studies, additional stellar and planetary properties were collected and presented in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We also used synthetic planetary systems simulated via the Bern model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' To facilitate a comparison of theory with observations, we prepared three synthetic observed catalogues by applying the detection biases on the simulated planetary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This led to the Bern RV Multis, Bern KOBE Multis, and the Bern Compact Multis catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We note that there are caveats present in the datasets we used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The model-dependent results we present here may be improved upon in future studies using better theoretical Article number, page 20 of 28 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Mishra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' : Architecture Framework I – Four classes of planetary system architecture models and a more complete observational catalogues (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' from PLATO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Summary of architecture framework: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The architecture framework is model-independent and there- fore does not suffer from any caveats emerging from planet formation theory or observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The same architecture framework can be used to study the multi-faceted aspects of planetary system architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' When the framework is applied to study planetary masses, the framework informs us of the mass architecture of the sys- tem, namely, the arrangement and distribution of masses in the planetary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In this way, we can use this framework to study the mass architecture, radii architecture, eccentricity architecture, and so on for the same system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In this series of work, we identified the architecture of a system with its bulk mass architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Planetary system architecture can be one of four classes that are derived from our framework: similar, mixed, ordered, and anti-ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' A planetary system’s architecture is of similar class when the masses of all the planets within such a system are similar to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This architecture class corresponds to the ‘peas in a pod’ architecture trend reported in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The architecture class of a planetary system is ordered (or anti-ordered) when the planetary masses in such systems tend to increase or decrease from inside-out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Planetary systems of mixed class architecture host planets whose masses show broad increasing and decreasing varia- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Our key model-dependent findings are as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Frequency of architecture class: Systems with similar bulk mass architecture are the most common outcome of simula- tions, followed by the other three architecture classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Our model suggests that similar architecture should be the most common exoplanetary system architecture in our Galaxy and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This explains why radius similarity in exoplanets was already detected from the first four months of Kepler data (Lissauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Distance bi-modality: We found hints of a bi-modality in the exoplanetary distance distribution arising from the two different modes of orbital migrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This bi-modality is readily visible (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 7) for similar and mixed mass ar- chitecture exoplanetary systems observed via RV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Core mass architectures: We found that for most systems, the bulk mass architecture is inherited from the core mass ar- chitecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In addition, the accretion of gases tends to high- light the underlying core mass architecture by amplifying it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In this way, the observed mass architecture of a system could serve as a gateway for studying the distribution and arrange- ment of the planetary core masses, which tends to be simpler for theoretical modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In situ formation: We found that most planets belonging to the similar bulk mass architecture class form in situ inside the ice line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In contrast, planets inside the ice line belong- ing to mixed, anti-ordered, and ordered show large inward migrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Core water-ice mass fraction architectures: Synthetic planetary systems were found to have two scenarios for their core water mass fraction architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The default scenario consists of relatively more dry planets in the inner parts of a system and more wet planets in the outer parts of the sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This is probably a direct consequence of the starting location of planets: planets starting inside (or outside) the ice line tend to be dry (or wet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' About one-fifths of simulates systems do not follow the default scenario described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We propose the ‘Aryabhata formation scenario’ to explain their core-water mass fraction architecture (see Paper II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Linking architecture and internal composition: We found that wet planets are more likely to survive around high- metallicity stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Among other predictions, we showed that anti-ordered observed systems should be rich in wet worlds, while ordered observed systems are expected to have many dry planets (based on the core-accretion planet formation paradigm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Density Architectures: Synthetic systems that are ordered (or anti-ordered) in mass tend to also be ordered (or anti- ordered) in their bulk densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Some mass similar systems may also have low dispersion in their planetary bulk densi- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The density architecture is sensitive to the Aryabhata’s number (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' the starting location of various surviving plan- ets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' see Paper II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The density architecture of observed sys- tems is in good agreement with the density architecture of synthetically observed simulated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Detection biases seem to favour the discovery of planetary systems where the densities show anti-ordering, mixing, or similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Radius architectures: The radius architecture of most plan- etary systems closely follows their mass architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' There- fore, most mass similar systems also show similarity in ra- dius (also for mass mixed, ordered, or anti-ordered systems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' However, this is not always true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Future studies can calibrate a classification scheme based on planetary radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Habitability as a system-level phenomenon: We reflected on the prospect of studying habitability as a function of system-level properties such as system architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Similar architecture systems represent an excellent observation tar- get for finding life outside the solar systems because these systems tend to host many more planets inside the empirical habitable zone that other architecture classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The current version of the Bern model seems to have dif- ficulty in producing planets inside the EHZ of an ordered architecture system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Nevertheless, more data is required to conclude whether the existence of Earth, an inhabited planet in an ordered system, is an exception or whether there are additional gaps in our understanding of planet formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This paper is the first in a series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The current work presents a new testing ground, the architecture space, for theoretical mod- els and for comparing observations with theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We can now constrain our understanding of planet formation not only on the level of an individual planet – but at the global systemic level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This is a multi-faceted approach, since the system architecture of several quantities can now be uniformly assessed and com- pared with observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In our next paper (Paper II), we show another important aspect emerging from this architecture frame- work which asserts that systems with comparable architecture often have the same formation pathways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We present ideas to further the nature versus nurture debate around planet forma- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' While similar architectures are usually a product of their starting conditions, stochastic multi-body effects are responsible for shaping the other three architecture classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This work leads to several future studies which will be presented in other papers in this series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Davoult et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=') explore how the present ar- chitecture framework can be employed for an efficient usage of telescope time to hunt for habitable worlds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Other possible ex- plorations that emerge from this work include: (a) a data-driven approach to classifying planetary architecture based on radii and Article number, page 21 of 28 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 43751corr (b) a suitable modification to Drake’s equation that accounts for the empirical occurrence rate of system architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The authors thank the anonymous referee for their care- ful reading, constructive suggestions, and insightful questions, which has al- lowed the quality of this manuscript to be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This work has been car- ried out within the framework of the National Centre of Competence in Re- search PlanetS supported by the Swiss National Science Foundation under grants 51NF40_182901 and 51NF40_205606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The authors acknowledge the finan- cial support of the SNSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Data: The synthetic planetary populations (NGPPS) used in this work are available online at http://dace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='unige.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='ch under section “Formation & evolution”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under con- tract with the National Aeronautics and Space Administration under the Exo- planet Exploration Program: https://exoplanetarchive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='ipac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' edu (DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='26133/NEA6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The artwork used to depict Earth in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 14 is taken from flaticon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Software: KOBE (Mishra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Mishra 2021), Python (Van Rossum & Drake 2009), NumPy (Oliphant 2006), SciPy (Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2020), Seaborn (Waskom & the seaborn development team 2020), Pandas (pan- das development team 2020), Matplotlib (Hunter 2007).' metadata={'source': 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+page_content=' 2020, ApJ, 893, L1 Yoffe, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=', Ofir, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=', & Aharonson, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2021, ApJ, 908, 114 Zhu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2020, AJ, 159, 188 Zhu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' & Dong, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2021, ARA&A, 59, 291 Article number, page 23 of 28 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 43751corr Appendix A: Bern Model: Additional details In this section, we provide some additional details on the physics included in the Bern model and how it is utilised to simulate syn- thetic planetary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Finally, we give an overview on com- parisons between the output of the Bern Model and observed planetary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' For the historic development, we refer to Al- ibert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2004, 2005);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Mordasini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Alibert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Mordasini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2012a,b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Alibert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Fortier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Marboeuf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2014b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Thiabaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Dittkrist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2014) and reviews in Benz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Mordasini (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The Bern model is based on the core accretion paradigm of planetary formation (Pollack et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The model in- cludes stellar evolution for a solar-mass star, using evolution tracks from Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The star interacts with the protoplanetary disk and influences its thermodynamical proper- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The protoplanetary disk has two phases: gas and solid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We model this disk using the approaches of viscous angular momen- tum transport (Lynden-Bell & Pringle 1974;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Veras & Armitage 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Hueso & Guillot 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Turbulence is characterised by the Shakura & Sunyaev (1973) approach, with α = 2 × 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Gas from the disk is accreted by planets, host star, and lost via photo- evaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The 1D geometrically thin disk evolution is studied up to 1000 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The initial mass of this gas disk and its lifetime are randomly drawn for each run of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The solid phase of the disk is composed of a swarm of planetesimals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The solid disk is modelled as a fluid which evolves via (a) accretion by growing planets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (b) interaction with the gaseous disk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (c) dy- namical stirring from planets and other planetesimals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' and so on (Fortier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The initial mass of the solid disk depends on the metallicity of the star and also on the condensation state of the molecules in the disk (Thiabaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The host star metallicity is randomly drawn for each run of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We added 100 protoplanetary embryos to the protoplanetary disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The initial location of each embryo was varied from one simulation to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' It was also ensured that no two embryos start within 10 hill radii of each other (Kokubo & Ida 1998, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Embryos accrete from their feeding zones and any over- lap may lead to competition (Alibert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The accretion rate depends on the collision probability between a protoplanet and a planetesimal, which in turn is influenced by the dynamical state of the solid disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Gas accretion occurs in several phases (Mordasini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2012b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Initially, the gas disk smoothly transitions as a gaseous envelope around all planets – the attached phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' For planets that are massive enough to undergo runaway gas accretion, the rate of gas supply from the disk may not be enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In these scenar- ios, the planet detaches from the gas disk and rapidly contracts to RJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' After the gas disk dissipates, all planets are in the isolated phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Gas accretion from the disk is no longer possible and in this phase, the planets contract and cool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' For all the planets, their internal structure is modelled at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We assume plan- ets are spherically symmetric and composed of accreted materi- als that arranges itself in layers: iron code, silicate mantle, water ice, and H/He gaseous envelope (if accreted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Next, we use these recipes to simulate several thousands of planetary systems in an approach called population synthesis (Emsenhuber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We start 1000 star-disk-embryo sys- tems with some fixed as well as some randomly drawn proper- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The initial properties are inspired by observations of disks Tychoniec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The then numerically modelled these systems, endowing them with additional physics at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Numerically, we incorporated multi-body dynamical in- teractions via N-body simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Planet-disk interactions lead- ing to orbital migration and eccentricity and inclination damping were also incorporated in the N-body Coleman & Nelson (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Paardekooper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Dittkrist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We followed these numerically intensive steps for 20 Myrs and then stopped the N-body calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The model then continued to evaluate the internal structure of all planets in the system for 10 Gyrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The recent version of these simulations has been published in the New Generation Planetary Population Synthesis (NGPPS) series of papers (Emsenhuber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2021a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Schlecker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2021a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Burn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Mishra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The output of these models have been compared with observations in several works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Drazkowska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2022) compares the occurrence rates of synthetic systems with observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Schlecker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2021a) studies the warm Super Earth and cold Jupiter correlation in the synthetic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Mishra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2021) analyse the ’peas in a pod’ architecture and compare synthetic systems with observa- tions from Weiss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Mulders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2018) present a detailed comparison of the synthetic models with Kepler obser- vations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Appendix B: Stellar and planetary data references 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Sun: Archinal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Standish (1992);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Helffrich (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Jacobson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Jacobson (2014, 2009) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Trappist-1: Agol et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Gillon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Burgasser & Mamajek (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Grimm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2018) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' TOI-178: Leleu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2021) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' HD 10180: Lovis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Kane & Gelino (2014) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' HD 219134: Seager et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Bonfanti & Gillon (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Vogt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2015) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' HD 34445: Vogt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2017) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Kepler-11: Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Lissauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2013) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Kepler-20: Fressin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Buchhave et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2016) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Kepler-80: MacDonald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Shallue & Vanderburg (2018) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' K2-138: Lopez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2019) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 55 Cnc: Bourrier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2018) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' GJ 667 C: Anglada-Escudé et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2013) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' HD 158259: Hara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Gáspár et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2016) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' HD 40307: Díaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Stassun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2019) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Kepler-102: Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Marcy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2014) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Kepler-33: Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Lissauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Had- den & Lithwick (2017) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Kepler-62: Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Borucki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2013) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' HD 20781: Udry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2019) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' TOI-561: Lacedelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Weiss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2021) 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' DMPP-1: Staab et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2020) 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' GJ 3293: Astudillo-Defru et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2017) 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' GJ 676 A: Sahlmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Stassun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2017) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' GJ 876: Trifonov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Rojas-Ayala et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2012) 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' HD 141399: Hébrard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2016) 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' HD 160691: Go´zdziewski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Pepe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2007) 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' HD 20794: Go´zdziewski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Pepe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2007) 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' HD 215152: Go´zdziewski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Pepe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2007) 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' HR 8799: Marois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Gravity Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Swastik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2021) 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' K2-266: Rodriguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2018) 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' K2-285: Rodriguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2018) 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Kepler-89: Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Weiss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2013) 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Kepler-106: Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Marcy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2014) 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Kepler-107: Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Bonomo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2019) 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Kepler-223: Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Mills et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2016) 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Kepler-411: Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2019) 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Kepler-48: Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Marcy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2014) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Kepler-65: Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Mills et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2019) 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Kepler-79: Berger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Yoffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2021) 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' WASP-47: Vanderburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2017) 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' tau Cet: Vanderburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2017) 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' HD 164922: Benatti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Rosenthal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (2021a) Article number, page 24 of 28 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Mishra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' : Architecture Framework I – Four classes of planetary system architecture 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='8 Maximum tolerance (t) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 Value max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' sim th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' var y = x Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Maximum value of the coefficient of similarity (blue) and the theoretical maximum value of the coefficient of variation (orange) is plotted against the maximum tolerance, t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Appendix C: Derivation of limits We consider a set Q of quantities q, namely, Q = {qi} where qi could be the mass, radius or other parameter of a planet, and the index, i ∈ [1, n], identifies a planet (with 1 being the inner- most planet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We assume that all qi ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The quantities qi are expressed as: qi = q′ (1 ± ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1) The quantities, qi, are decomposed around some value q′ such that all ti are minimised;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' ti is a measurement of the fractional difference (or tolerance) between q′ and qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Since all individual tolerances are a positive quantity, they will satisfy the following relation: 0 ≤ ti ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2) Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1: Mean Let us consider the mean of the quantities, ¯qi: ¯qi = Q n = � i qi n = q′ n �n ± t1 ± t2 ± · · · ± tn �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='3) The mean takes its maximum value only when all individual ti values take their maximum and are added up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This gives: max ¯qi = q′ �1 + t�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='4) Similarly, the minimum value of the mean is: min ¯qi = q′ �1 − t�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5) The extreme value of the mean occurs when all the individual quantities are extremised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' However, in this scenario, since all quantities are equal, the coefficient of variation is identically 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2: Coefficient of similarity We start with the definition of the coefficient of similarity, CS (q) = 1 n − 1 i=n−1 � i=1 � log qi+1 qi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='6) Inserting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1, in the definition, we get: CS (q) = 1 n − 1 i=n−1 � i=1 � log 1 ± ti+1 1 ± ti � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='7) This formulation shows that the coefficient of similarity depends only on the fractional differences (tolerances) between qi values – and not on their actual values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This is a desirable property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Next, we evaluate the maxCS as, maxCS (q) = max � 1 n − 1 i=n−1 � i=1 � log 1 ± ti+1 1 ± ti �� , = 1 n − 1 max � i=n−1 � i=1 � log 1 ± ti+1 1 ± ti �� , = 1 n − 1 i=n−1 � i=1 log max �1 ± ti+1 1 ± ti � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='8) In the first step, we commuted the max operator with the frac- tion (n − 1)−1 because we are interested in the maximum for a constant n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Next, knowing that the maximum of a sum occurs at the sum of maximum summands and that log is a monotonically increasing function, we further commute the max operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We observe the following: max �1 ± ti+1 1 ± ti � when � ±ti+1 → +t ±ti → −t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='9) This implies that max CS (q) = log 1 + t 1 − t, min CS (q) = − max CS = log 1 − t 1 + t, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='10) where the second equality can be similarly derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1 shows the variation of max CS as a function of tolerance t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We note that the limits of the coefficient of similarity do not depend on n, and we verified our results with numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' ■ Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='3: Coefficient of Variation We start with the definition of the coefficient of variation, CV(q) = σ(q) ¯q , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='11) and we note that the minimum value of the coefficient of vari- ation is zero and it occurs when all qi values are equal, thereby giving no variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In the literature, we can find some derivations for the max- imum value of the coefficient of variation (Katsnelson & Kotz 1957;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Sharma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Katsnelson & Kotz (1957) show that the upper limit of the coefficient of variation is √ n − 1 when all but one qi is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' However, this limit is only a particular case of our formulation (specifically, q1 = q′ and qi�1 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Here, we derive the limits for a more general scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Article number, page 25 of 28 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 43751corr We consider that: C2 V = 1 n i=n � i=1 � 1 − qi ¯q ������ =A �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='12) Here, we have squared the definition of CV and used the def- inition of the standard deviation σ(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' As an aside, we note that the equation above shows that the coefficient of variation is zero when all qi = ¯q, as noted before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We note that the max- imum value of C2 V occurs when the term A (in parenthesis) is maximised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Denoting � i=1 qi by Q, we consider the term in the parenthesis, A = 1 − nqi Q = Q − nqi Q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='13) The condition for the general maxima of the coefficient of variation, in our formulation, is when one of the quantity (say q1 takes the largest possible value, while all others take the smallest possible value): q1 = q′ (1 + t) qi�1 = q′ (1 − t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='14) The mean in this scenario becomes (marked with ′′): ¯q′′ = q′(1 + t) + (n − 1) × q(1 − t) n = q′ n � n(1 − t) + 2t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='15) The variance in this scenario becomes (marked with ′′): σ′′2(q) = 1 n �� q′(1 + t) − ¯q′′ �������������������������� = 2q′t� n−1 n � �2 + (n − 1) � q′(1 − t) − ¯q′′ �������������������������� = −2q′t n �2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='16) This gives: σ′′(q) = �2q′t n � √ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='17) Finally, the general expression for the maximum value of the coefficient of variation becomes: maxCV(q) = σ′′(q) ¯q′′ = 2t √ n − 1 n(1 − t) + 2t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='18) This expression recovers the particular case derived in lit- erature when we set t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' From this expression, we note that the upper limit of the coefficient of variation does not depend on the actual values of the quantities, but it depends on the number of quantities in the set, Q, and the maximum tolerance, t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This new formulation allows us to extract the upper limit of the co- efficient of variation for any set whose maximum tolerance, t, is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Interestingly, the above expression gives appropriate re- sult when absurd inputs are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' For example, when there are no planets in a system, maxCV ���n=0 = √ −1, and when there is only one planet in a system, maxCV ���n=1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' For a system of two planets, the upper limit is exactly the fractional difference (or tolerance), that is, maxCV ���n=2 = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Furthermore, varying over n, and assuming t ∈ [0, 1), allows us to derive the theoretical maximum possible value for the co- efficient of variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This occurs at n = 2 1−t and gives: max CV(q) �����n= 2 1−t (q) = t √ 1 − t2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='19) Figure C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1 shows the variation of the theoretical max, CV, as a function of tolerance t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='■ Appendix D: Classification boundaries for architectures classes In this section, we present some considerations that motivate the boundaries between the four architecture classes for plan- etary masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' In the current formulation (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 3), there are two boundaries that need to be identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We deal with the distinc- tion between similar and mixed class first, and then distinguish ordered/anti-ordered architecture classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1: Similar versus mixed We saw in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2, it is difficult to distinguish between mixed and similar architecture classes using the coefficient of similar- ity alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Mixed systems are characterised by large increasing or decreasing variations, which may cancel each other out and lead to small values of CS (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Nevertheless, the coefficient of variation can distinguish between large variations in values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Fig- ure D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1 shows the CV(M) as a function of the number of planets in a planetary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The left panel shows all synthetic sys- tems from the Bern model, while we only show systems with |CS (M)| ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2 in the right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Clearly, there are two clusters of planetary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The clus- ter on the lower right-hand side corresponds to similar class sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Mixed systems, having large values of CV(M), are spread over the top left region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' It is clear that the boundary between similar and mixed classes depends on the number of planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The black line (corresponding to y = √ n−1 2 ) neatly separates the two clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We have chosen this equation to disentangle similar ar- chitectures from the mixed class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This equation has, incidentally, two key properties: 1) it ensures that no two planet system can be of mixed architecture and 2) it happens to be exactly half of the maximum possible value of the coefficient of variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2: Ordered and anti-ordered Having motivated the boundary between similar and mixed class, we are now left with three groupings of architecture classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' These three groupings correspond to CS (M) << 0 (anti- ordered), CS (M) ∼ 0 (similar/mixed), and CS (M) >> 0 (or- dered).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This suggests that we require two boundaries to distin- guish these three groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' However, we posit that the boundaries between ordered and anti-ordered should be symmetric around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Thus, we are left with only one boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' ordered (or anti-ordered) systems differ in their architec- ture from similar/mixed classes in that the quantity (mass here) continues to show an increasing (or decreasing) trend with dis- tance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' For all planetary systems in the Bern model, we measure the Spearman correlation coefficient, R, between the planetary masses and their distance from the host star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The Spearman R, measuring the monotonicity between two datasets, varies from 1 to +1, with 0 indicating no correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' A positive correlation implies that as x increases, so does y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Negative correlations im- ply that as x increases, y decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Figure D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1 shows the CS (M) of synthetic systems as func- tion of their Spearman correlation R (mass and distance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We note that there is a large cluster of points towards CS (M) ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This group corresponds to the similar and mixed architecture classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' There are some points to the top right (including those with R = +1 – corresponding to planetary systems in which plan- etary masses are monotonically increasing with distance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' There is a scatter of points towards the bottom-left (including some systems with R = −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Article number, page 26 of 28 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Mishra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' : Architecture Framework I – Four classes of planetary system architecture 0 10 20 30 40 Multiplicity 0 1 2 3 4 5 Coefficient of Variation (Mass) [unitless] |CS(M)| 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2 y = n 1 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 Spearman Correlation: R(mass, distance) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='0 Coefficient of Similarity (Mass) [unitless] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='3 Bern Model Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' C lassification boundaries for architecture classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Left: Boundary between similar and mixed class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The panel show the coefficient of variation for synthetic planetary systems as a function of the number of planets in a system for systems with |CS (M)| ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Two clusters are clearly distinguishable, allowing us to fix the boundary between the similar and mixed architecture classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Right: Boundary between ordered and anti-ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This plot shows the coefficient of similarity of synthetic planetary systems as a function of the Spearman correlation coefficient between the planetary masses and distances of that system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Thick horizontal lines correspond to potential boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' First, we note that the comparison of the coefficient of sim- ilarity with Spearman R fulfils some expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' For example, there are no points in bottom-right or top-left sections of this plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Second, our objective is to isolate the central cluster of points from all other scattered points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We note that |CS (M)| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1 fails as a boundary, since it does not include the full central clus- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Both |CS (M)| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='3 could succeed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Going beyond, a value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='3 would add many unnecessary points to the central cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' To further motivate our choice of boundary, namely, |CS (M)| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2, we show the mass-distance diagram of 12 ran- domly selected systems with −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='3 < CS (M) < −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2 (out of 19) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' We note that all systems show the qualitative fea- tures of an anti-ordered system, namely, massive planets in the inner region and small planets in the outer region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Since all of these planets have their CS (M) < −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2, we use |CS (M)| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2 as a boundary between ordered, anti-ordered, and similar+mixed architecture classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Future works may explore improvements to our selected boundaries using additional ideas from K-means or hierarchical clusterings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Appendix E: A gallery of architecture types: Mass-distance diagrams 100 102 104 Mass [M ] Anti-Ordered CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='30 CV(M) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='45 Anti-Ordered CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='26 CV(M) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='27 Anti-Ordered CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='26 CV(M) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='97 0 10 20 30 40 50 60 70 Ice mass fraction in Core [%] 100 102 104 Mass [M ] Anti-Ordered CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='24 CV(M) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='61 Anti-Ordered CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='24 CV(M) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='85 Anti-Ordered CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='23 CV(M) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='17 100 102 104 Mass [M ] Anti-Ordered CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='23 CV(M) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='18 Anti-Ordered CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='22 CV(M) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='35 Anti-Ordered CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='22 CV(M) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='35 10 2 100 102 SMA [AU] 100 102 104 Mass [M ] Anti-Ordered CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='22 CV(M) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='35 10 2 100 102 SMA [AU] Anti-Ordered CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='22 CV(M) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='98 10 2 100 102 SMA [AU] Anti-Ordered CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='21 CV(M) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='63 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Mass-distance diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' This plot shows the planetary masses as a function of distance for some planetary systems with −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='3 < CS (M) < −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The dashed line connects that planets in the system and serves to highlight the arrangement and distribution of masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' The size of each circle corresponds to the planet’s radius and the colour of each planet also shows its core water mass fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Article number, page 27 of 28 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 43751corr 100 102 104 Mass [M ] Similar CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='13 CV(M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='73 Similar CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='10 CV(M) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='06 Similar CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='10 CV(M) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='13 Similar CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='10 CV(M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='88 0 10 20 30 40 50 60 70 Ice mass fraction in Core [%] 100 102 104 Mass [M ] Similar CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='09 CV(M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='74 Similar CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='08 CV(M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='80 Similar CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='08 CV(M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='68 Similar CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='07 CV(M) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='21 100 102 104 Mass [M ] Similar CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='06 CV(M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='91 Similar CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='06 CV(M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='57 Similar CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='04 CV(M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='87 Similar CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='04 CV(M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='67 10 2 100 102 SMA [AU] 100 102 104 Mass [M ] Similar CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='04 CV(M) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='25 10 2 100 102 SMA [AU] Similar CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='01 CV(M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='99 10 2 100 102 SMA [AU] Similar CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='01 CV(M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='62 10 2 100 102 SMA [AU] Similar CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='00 CV(M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='89 100 102 104 Mass [M ] Mixed CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='17 CV(M) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='49 Mixed CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='17 CV(M) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='03 Mixed CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='16 CV(M) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='73 Mixed CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='16 CV(M) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='73 0 10 20 30 40 50 60 70 Ice mass fraction in Core [%] 100 102 104 Mass [M ] Mixed CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='16 CV(M) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='50 Mixed CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='15 CV(M) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='16 Mixed CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='14 CV(M) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='25 Mixed CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='13 CV(M) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='12 100 102 104 Mass [M ] Mixed CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='12 CV(M) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='19 Mixed CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='12 CV(M) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='15 Mixed CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='11 CV(M) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='27 Mixed CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='06 CV(M) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='96 10 2 100 102 SMA [AU] 100 102 104 Mass [M ] Mixed CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='05 CV(M) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='74 10 2 100 102 SMA [AU] Mixed CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='03 CV(M) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='40 10 2 100 102 SMA [AU] Mixed CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='01 CV(M) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='17 10 2 100 102 SMA [AU] Mixed CS(M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='13 CV(M) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='88 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' A gallery of planetary system architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' These plots show the mass-distance diagram for similar (left) and mixed (right) planetary systems from the Bern Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Each circle represents a planet, its size corresponds to the planetary radius, and its colour represents the fraction of ice in the planetary core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Each panel shows the CS (M) as well as the CV(M) of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' 100 102 104 Mass [M ] Anti Ordered CS(M) = -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='33 CV(M) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='41 Anti Ordered CS(M) = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='88 CV(M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='95 Anti Ordered CS(M) = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='34 CV(M) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='41 Anti Ordered CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='80 CV(M) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='61 0 10 20 30 40 50 60 70 Ice mass fraction in Core [%] 100 102 104 Mass [M ] Anti Ordered CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='63 CV(M) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='27 Anti Ordered CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='57 CV(M) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='16 Anti Ordered CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='55 CV(M) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='83 Anti Ordered CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='49 CV(M) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='05 100 102 104 Mass [M ] Anti Ordered CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='41 CV(M) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='31 Anti Ordered CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='36 CV(M) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='53 Anti Ordered CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='29 CV(M) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='71 Anti Ordered CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='28 CV(M) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='98 10 2 100 102 SMA [AU] 100 102 104 Mass [M ] Anti Ordered CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='26 CV(M) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='27 10 2 100 102 SMA [AU] Anti Ordered CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='26 CV(M) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='97 10 2 100 102 SMA [AU] Anti Ordered CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='24 CV(M) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='85 10 2 100 102 SMA [AU] Anti Ordered CS(M) = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='22 CV(M) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='75 100 102 104 Mass [M ] Ordered CS(M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='23 CV(M) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='21 Ordered CS(M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='23 CV(M) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='12 Ordered CS(M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='33 CV(M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='80 Ordered CS(M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='52 CV(M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='53 0 10 20 30 40 50 60 70 Ice mass fraction in Core [%] 100 102 104 Mass [M ] Ordered CS(M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='56 CV(M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='86 Ordered CS(M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='78 CV(M) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='49 Ordered CS(M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='86 CV(M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='76 Ordered CS(M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='99 CV(M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='81 100 102 104 Mass [M ] Ordered CS(M) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='02 CV(M) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='38 Ordered CS(M) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='03 CV(M) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='09 Ordered CS(M) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='09 CV(M) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='04 Ordered CS(M) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='23 CV(M) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='38 10 2 100 102 SMA [AU] 100 102 104 Mass [M ] Ordered CS(M) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='28 CV(M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='90 10 2 100 102 SMA [AU] Ordered CS(M) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='64 CV(M) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='13 10 2 100 102 SMA [AU] Ordered CS(M) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='16 CV(M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='99 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' A gallery of planetary system architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' These plots show the mass-distance diagram for anti-ordered (left) and ordered (right) planetary systems from the Bern Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Each circle represents a planet, its size corresponds to the planetary radius, and its colour represents the fraction of ice in the planetary core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Each panel shows the CS (M) as well as the CV(M) of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} +page_content=' Article number, page 28 of 28' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfdQB7/content/2301.02374v1.pdf'} diff --git a/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf b/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..fbc495745ed6e59908fc8b3b5b90debd4a5a0e17 --- /dev/null +++ b/1dAzT4oBgHgl3EQfDfo-/content/2301.00976v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:3b2e8e609bfcb013397873fe4a33840e0251b7c695ea5a42fd8aa1ba01d8167e +size 245499 diff --git a/1dAzT4oBgHgl3EQfDfo-/vector_store/index.pkl b/1dAzT4oBgHgl3EQfDfo-/vector_store/index.pkl new 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neural networks are vulnerable to adversarial attacks. +In this paper, we take the role of investigators who want to +trace the attack and identify the source, that is, the particular +model which the adversarial examples are generated from. +Techniques derived would aid forensic investigation of at- +tack incidents and serve as deterrence to potential attacks. We +consider the buyers-seller setting where a machine learning +model is to be distributed to various buyers and each buyer +receives a slightly different copy with same functionality. A +malicious buyer generates adversarial examples from a par- +ticular copy Mi and uses them to attack other copies. From +these adversarial examples, the investigator wants to iden- +tify the source Mi. To address this problem, we propose a +two-stage separate-and-trace framework. The model separa- +tion stage generates multiple copies of a model for a same +classification task. This process injects unique characteristics +into each copy so that adversarial examples generated have +distinct and traceable features. We give a parallel structure +which embeds a “tracer” in each copy, and a noise-sensitive +training loss to achieve this goal. The tracing stage takes in +adversarial examples and a few candidate models, and iden- +tifies the likely source. Based on the unique features induced +by the noise-sensitive loss function, we could effectively trace +the potential adversarial copy by considering the output logits +from each tracer. Empirical results show that it is possible to +trace the origin of the adversarial example and the mechanism +can be applied to a wide range of architectures and datasets. +1 +Introduction +Deep learning models are vulnerable to adversarial attacks. +By introducing specific perturbations on input samples, the +network model could be misled to give wrong predictions +even when the perturbed sample looks visually close to +the clean image (Szegedy et al. 2014; Goodfellow, Shlens, +and Szegedy 2014; Moosavi-Dezfooli, Fawzi, and Frossard +2016; Carlini and Wagner 2017). There are many existing +works on defending against such attacks (Kurakin, Good- +fellow, and Bengio 2016; Meng and Chen 2017; Gu and +Rigazio 2014; Hinton, Vinyals, and Dean 2015). Unfortu- +nately, although current defenses could mitigate the attack +to some extent, the threat is still far from being completely +eliminated. In this paper, we look into the forensic aspect: +from the adversarial examples, can we determine which +*Corresponding Authors. +Figure 1: Buyers-seller setting. The seller has multiple mod- +els Mi, i ∈ [1, m] that are to be distributed to different +buyers. A malicious buyer batt attempts to attack the vic- +tim buyer bvic by generating the adversarial examples with +his own model Matt. +model the adversarial examples were derived from? Tech- +niques derived could aid forensic investigation of attack in- +cidents and provide deterrence to future attacks. +We consider a buyers-seller setting (Zhang, Tann, and +Chang 2021), which is similar to the buyers-seller setting +in digital rights protection (Memon and Wong 2001). +Buyers-seller Setting. +Under this setting, the seller S dis- +tributes m classification models Mi, i ∈ [1, m] to different +buyers bi’s as shown in Fig. 1. These models are trained for +a same classification task using a same training dataset. The +models are made accessible to the buyer as black boxes, for +instance, the models could be embedded in hardware such as +FPGA and ASIC, or are provided in a Machine Learning as a +Service (MLaaS) platform. Hence, the buyer only has black- +box access, which means that he can only query the model +for the hard label. In addition, we assume that the buyers do +not know the training datasets. The seller has full knowledge +and thus has white-box access to all the distributed models. +Attack and Traceability. +A malicious buyer wants to at- +tack other victim buyers. The malicious buyer does not have +direct access to other models and thus generates the exam- +ples from its own model and then deploys the found exam- +ples. For example, the malicious buyer might generate an ad- +versarial example of a road sign using its self-driving vehi- +cle, and then physically defaces the road sign to trick passing +vehicles. Now, as forensic investigators who have obtained +the defaced road sign, we want to understand how the ad- +versarial example is generated and trace the models used in +generating the example. +arXiv:2301.01218v1 [cs.CR] 31 Dec 2022 + +M1 +M2 +M1 +M3 +Matt +Attacking +20 +Generating +Adversarial +ExamplesProposed Framework. +There are two stages in our solu- +tion: model separation and origin tracing. During the model +separation stage, given a classification task, we want to gen- +erate multiple models that have high accuracy on the clas- +sification task and yet are sufficiently different for tracing. +In other words, we want to proactively enhance differences +among the models in order to facilitate tracing. To achieve +that, we propose a parallel network structure that pairs a +unique tracer with the original classification model. The role +of the tracer is to modify the output, so as to induce the at- +tacker to adversarial examples with unique features. We give +a noise-sensitive training loss for the tracer. +During the tracing stage, given m different classification +models Mi, i ∈ [1, m] and the found adversarial example, +we want to determine which model is most likely used in +generating the adversarial examples. This is achieved by ex- +ploiting the different tracers that are earlier embedded into +the parallel models. Our proposed method compares the out- +put logits (the output of the network before softmax) of those +tracers to identify the source. +In a certain sense, traceability is similar to neural network +watermarking and can be viewed as a stronger form of water- +marking. Neural network watermarking schemes (Boenisch +2020) attempt to generate multiple models so that an investi- +gator can trace the source of a modified copy. In traceability, +the investigator can trace the source based on the generated +adversarial examples. +Contributions. +1. We point out a new aspect in defending against adver- +sarial attacks, that is, tracing the origin of adversarial +samples among multiple classifiers. Techniques derived +would aid forensic investigation of attack incidents and +provide deterrence to future attacks. +2. We propose a framework to achieve traceability in the +buyers-seller setting. The framework consists of two +stages: a model separation stage, and a tracing stage. +The model separation stage generates multiple “well- +separated” models and this is achieved by a parallel +network structure that pairs a tracer with the classifier. +The tracing mechanism exploits the characteristics of the +paired tracers to decide the origin of the given adversarial +examples. +3. We investigate the effectiveness of the separation and the +subsequent tracing. Experimental studies show that the +proposed mechanism can effectively trace to the source. +For example, the tracing accuracy achieves more than +97% when applying to “ResNet18-CIFAR10” task. We +also observe a clear separation of the source tracer’s log- +its distribution, from the non-source’s logits distribution +(e.g. Fig. 5a-5c). +2 +Related Work +In this paper, we adopt black-box settings where the adver- +sary can only query the model and get the hard label (final +decision) of the output. Many existing attacks assume white- +box settings. Attack such as FGSM (Goodfellow, Shlens, +and Szegedy 2014), PGD (Kurakin, Goodfellow, and Bengio +2016), JSMA (Papernot et al. 2016), DeepFool (Moosavi- +Dezfooli, Fawzi, and Frossard 2016), CW (Carlini and Wag- +ner 2017) and EAD (Chen et al. 2018) usually directly rely +on the gradient information provided by the victim model. +As the detailed information of the model is hidden in black- +box settings, black-box attacks are often considered more +difficult and there are fewer works. Chen et. al. introduced +a black-box attack called Zeroth Order Optimization (ZOO) +(Chen et al. 2017). ZOO can approximate the gradients of +the objective function with finite-difference numerical esti- +mates by only querying the network model. Thus the ap- +proximated gradient is utilized to generate the adversarial +examples. Guo et. al. proposed a simple black-box adver- +sarial attack called “SimBA” (Guo et al. 2019) to generate +adversarial examples with a set of orthogonal vectors. By +testing the output logits with the added chosen vector, the +optimization direction can be effectively found. Brendel et. +al. developed a decision-based adversarial attack which is +known as “Boundary attack” (Brendel, Rauber, and Bethge +2018), it worked by iteratively perturbing another initial im- +age that belongs to a different label toward the decision +boundaries between the original label and the adjacent la- +bel. By querying the model with enough perturbed images, +the boundary as well as the perturbation can be found thus +generating the adversarial examples. Chen et. al. proposed +another decision based attack named hop-skip-jump attack +(HSJA) (Chen, Jordan, and Wainwright 2020) recently. By +only utilizing the binary information at the decision bound- +ary and the Monte-Carlo estimation, the gradient direction of +the network can be found so as to realize the adversarial ex- +amples generation. Based on (Chen, Jordan, and Wainwright +2020), Li et. al. (Li et al. 2020) proposed a query-efficient +boundary-based black-box attack named QEBA which es- +timate the gradient of the boundary in several transformed +space and effectively reduce the query numbers in gener- +ating the adversarial examples. Maho et. al. (Maho, Furon, +and Le Merrer 2021) proposed a surrogate-free black-box +attack which do not estimate the gradient but searching the +boundary based on polar coordinates, compared with (Chen, +Jordan, and Wainwright 2020) and (Li et al. 2020), (Maho, +Furon, and Le Merrer 2021) achieves less distortion with +less query numbers. +3 +Proposed Framework +3.1 +Main Idea +We design a framework that contains two stages: model sep- +aration and origin tracing. +During the model separation stage, we want to generate +multiple models which are sufficiently different under ad- +versarial attack while remaining highly accurate on the clas- +sification task. Our main idea is a parallel network structure +which pairs a unique tracer with the original classifier. The +specific structure will be illustrated in Section 3.2. +As for origin tracing, we exploit unique characteristics of +different tracers in the parallel structure, which can be ob- +served in the tracers’ logits. Hence, our tracing process is +conducted by feeding the adversarial examples into the trac- +ers and analyzing their output. + +Figure 2: The framework of the proposed method. The left part of the framework indicates the separation process of the seller’s +distributed models Mi, i ∈ [1, m]. The right part of the framework illustrates the origin tracing process. +The whole framework of the proposed scheme is shown +in Fig. 2. As illustrated in Fig.2, each distributed model Mi +consists of a tracer Ti and the original classification model +C, and the tracer is trained with a proposed noise-sensitive +loss LNS. During the tracing stage, the adversarial examples +are fed into each Ti and the outputs are analyzed to identify +the origin. +3.2 +Model Separation +We design a parallel network structure to generate the dis- +tributed models Mi, i ∈ [1, m], which contains a tracer +model Ti and a main model C, as shown in Fig. 3a. Ti is +used for injecting unique features and setting traps for the +attacker. C is the network trained for the original task. The +final results are determined by both C and Ti with a weight +parameter α. In each distributed model, C is fixed and only +Ti is different. +The specific structure of Ti is shown in Fig. 3b, it is +linearly cascaded with one “SingleConv” block (Conv-BN- +ReLU), two “Res-block” (He et al. 2016), one “Conv” block, +one full connection block and one “Tanh” activation layer. +The training process of Ti can be described as: +1) Given the training dataset1 and tracer Ti, we first ini- +tialize Ti with random parameters. +2) For each training epoch, we add random noise No 2 on +the input image x to generate the noised image xNo. +3) Then we feed both x and xNo into Ti and get the out- +puts Ox and OxNo. We attempt to make Ti sensitive to noise, +so Ox and OxNo should be as different as possible. The loss +function of Ti can be written as: +LNS = +|Ox ◦ OxNo| +∥Ox∥2∥OxNo∥2 += +|Ti(θTi, x) ◦ Ti(θTi, xNo)| +∥Ti(θTi, x)∥2∥Ti(θTi, xNo)∥2 +(1) +1The training dataset for Ti only contains 1000 random sampled +images from the dataset of the original classification task +2No follows a uniform distribution over [0, 0.03) +where ◦ represents the Hadamard product. θTi indicates the +parameters of Ti. +Each distributed Ti for different buyers is generated by +randomly initializing and then training. We believed the ran- +domness in initialization is enough to guarantee the differ- +ence from different Ti. It should be noted that when pro- +ducing a new distributed copy, we only have to train one +new tracer without setting more constraints on former trac- +ers. So such a separation method can be applied to multiple +distributed models independently. +As for C, it is trained in a normal way which utilizes the +whole training dataset and cross-entropy loss. For the main +classification task, C only has to be trained once. Besides, the +training of C is independent of the training of Ti. After train- +ing C, we could get a high accuracy classification model. +The final distributed model Mi is parallel combined with C +and Ti. The specific workflow of Mi can be described as: +For input image x, Ti and C both receive the same x and +output two different vectors OTi and OC respectively. OTi +and OC have the same size and will be further added in a +weighted way to generate the final outputs OF , as shown in +Eq. 2. +OF = OC + α × OTi +(2) +where α is the weight parameter. It is worth noting that for +the output of C, we use the normalization form of it, which +can be formulated as: +OC = +C(x) − min(C(x)) +max(C(x)) − min(C(x)) +(3) +where x indicates the input image, max and min indicate +the maximum value and minimum value respectively. +By +utilizing +the +aforementioned +model +separation +method, two properties are well satisfied: (I) The attack +could be tricked to focus more on Ti than C. Since after the +training, Ti will be sensitive to random noise. Therefore, the +output of Ti is easy to be changed by adding noise. Com- +pared with C, the boundary of Ti is more likely to be esti- +mated and Ti is more likely to be attacked. Thus, the attacker + +Model Separation Stage +Origin Tracing Stage + Attacker's Model ! +Main Model +Initialized Tracer Model +Distributed Tracer +(source model) +Ms +c +T1 +T2 +Tini +Tini +Tini +2 +m +Adversarial +Trace's Outputs Obtaining +Model Combination +Tracer Model Training +Examples +Outputs OTi(x) +Outputs +Outputs +Outputs +Adversarial +Tracer +Image +Example +OT1 +OT2 +OTs +OTm +Ti +X +Tracer +Noise-sensitive +Loss L'is. +Image +Main +Ti + Noised +x + Image +c +Attacked Model Tracing +Identified Tracer : +xNo +Outputs OTi(xNo) +Mi +Ts +Outputs +Tracing mechanism +OTm +no +Generated Models +OTs +arg max 0 +ho +true +Identified Model : +att: attacked label +M1 +M2 +M3 +M4 +Ms +Mm +true: true label +Ms(a) Parallel network structure. +(b) The architecture of tracer. +(c) Differences in logits. +Figure 3: The specific network design in model separation. +will fall into the trap of Ti and the generated adversarial per- +turbations will bring the feature of the source Ti. (II) Based +on random initialization, each distributed Ti will correspond +to different adversarial perturbations. This property helps us +in tracing, since the source Ts which generates adversarial +examples will output unique responses compared with other +Ti, i ̸= s when feeding the generated adversarial examples, +as shown in Fig. 3c. +3.3 +Tracing the Origin +The tracing process is conducted by two related compo- +nents: +• The first component keeps white-box copies for each of +the m distributed copies 3. This component allows us to +obtain the output logits of each tracer on an input x. +• The second component is an output logits-based mech- +anism. It gives a decision on which copy i is the most +likely one to generate the adversarial example. +The specific tracing process can be described as follows: +1) Given an appeared adversarial examples denoted as +xatt, we feed the adversarial example into all Ti, i ∈ [1, m] +and obtain the output logits of them, noted as OTi, i ∈ +[1, m]. +2) Then we extract two values that are corresponding to +the attacked label and true label in each OTi, denoted as OTi +att +and OTi +true respectively. 4 +3) The source model can be determined by: +s = arg max +i,i∈[1,m] +(OTi +att − OTi +true) +(4) +To simplify the description, we denote the difference of out- +put logits (OTi +att−OTi +true) as DOL. The tracer corresponded to +the largest DOL is regarded as the source model. The reason +is as follows: +Since the perturbation are highly related to Ti, when feed- +ing the same adversarial example, the outputs of Ti and Tj +3This setting is reasonable because when an adversarial attack +appeared, the model seller who has all the details of the distributed +network takes responsible to trace the attacker. +4Attacked label can be easily determined by the output logits +and the true label can be tagged by the model owner. If this sample +cannot be accurately tagged by the owner, then this sample is not +regarded as an adversarial example. +(i ̸= j) will be certainly different. For source model Ts +where the adversarial examples are generated from, OTs is +likely to render a large value on the adversarial label and a +small value on the ground-truth label. Since the weight of +OTs in the final OFs is small, so in order to achieve ad- +versarial attack, OTs will be modified as much as possible. +Thus DOL of Ts should be large. But for victim model Tv, +the DOL will be small. Therefore, according to the value of +DOL, we can trace the origin of the adversarial example. +4 +Experimental Results +4.1 +Implementation Details +In order to show the effectiveness of the proposed frame- +work, we perform the experiments on two network architec- +ture (ResNet18 (He et al. 2016) and VGG16 (Simonyan and +Zisserman 2014)) with two small image datasets (CIFAR10 +(Krizhevsky, Hinton et al. 2009) of 10 classes and GTSRB +(Houben et al. 2013) of 43 classes) and two deeper network +architecture (ResNet50 and VGG19) with one big image +dataset (mini-ImageNet (Ravi and Larochelle 2016) of 100 +classes). The main classifier C in experiments is trained for +200 epochs. All the model training is implemented by Py- +Torch and executed on NVIDIA RTX 2080ti. For gradient +descent, Adam (Kingma and Ba 2015) with learning rate of +1e-4 is applied as the optimization method. +4.2 +The Classification Accuracy of The Proposed +Architecture +The most influenced parameter for the classification accu- +racy is the weight parameter α. α determines the partici- +pation ratio of Ti in final outputs. To investigate the influ- +ence of α, we change the value of α from 0 (baseline) to 0.2 +and record the corresponding classification accuracy of each +task, the results are shown in Table 1. +It can be seen from Table 1 that for CIFAR10 and GTSRB, +the growth of α will seldom decrease the accuracy of the +classification task. Compared with the baseline (α = 0), the +small value of α will keep the accuracy at the same level as +the baseline. But for mini-ImageNet, the accuracy decreases +more as α increases, we believe it is due to the complexity +of the classification task. But even though, the decrease rate +is still within 3% when α is not larger than 0.15. + +Tracer Model +Ti +Image +Main Model +x +cTracer Model T;Tv +Ti +Ti +c +c +c +cα +CIFAR10 +GTSRB +Mini-ImageNet +ResNet18 +VGG16 +ResNet18 +VGG16 +ResNet50 +VGG19 +0 +94.30% +93.68% +96.19% +97.59% +73.12% +75.79% +0.05 +94.24% +93.64% +96.14% +97.52% +72.32% +75.04% +0.1 +94.24% +93.63% +96.07% +97.36% +71.88% +74.96% +0.15 +94.07% +93.63% +95.72% +96.84% +70.50% +73.75% +0.2 +93.95% +93.57% +95.09% +95.52% +68.14% +71.75% +Table 1: The classification accuracy with different α. +4.3 +Traceability of different black-box attack +It should be noted that the change of α will not only influ- +ence the accuracy but also affect the process of black-box +adversarial attack. Therefore, in order to explore the influ- +ence of α, the following experiments will be conducted with +α = 0.05, 0.1 and 0.15. +Setup and Code. To verify the traceability of the pro- +posed mechanism, we conduct experiments on two dis- +tributed models. We set one model as the source model Ms +to perform the adversarial attack and set the other model +as the victim model Mv. The goal is to test whether the +proposed scheme can effectively trace the source model +from the generated adversarial examples. The black-box at- +tack we choose is Boundary (Brendel, Rauber, and Bethge +2018), HSJA (Chen, Jordan, and Wainwright 2020), QEBA +(Li et al. 2020) and SurFree (Maho, Furon, and Le Merrer +2021). For Boundary (Brendel, Rauber, and Bethge 2018) +and HSJA (Chen, Jordan, and Wainwright 2020), we use Ad- +versarial Robustness Toolbox (ART) (Nicolae et al. 2018) +platform to conduct the experiments. For QEBA (Li et al. +2020) and SurFree (Maho, Furon, and Le Merrer 2021), we +pull implementations from their respective GitHub reposito- +ries 5 6 with default parameters. For each α, each network +architecture, each dataset and each attack, we generate 1000 +successful attacked adversarial examples of Ms and con- +duct the tracing experiment. +Evaluation Metrics. Traceability is evaluated by tracing +accuracy, which is calculated by: +Acc = Ncorrect +NAll +(5) +where Ncorrect indicates the number of correct-tracing sam- +ples and NAll indicates the total number of samples, which +is set as 1000 in the experiments. +The tracing performance of different attacks with different +settings is shown in Table 2. It can be seen that when apply- +ing ResNet-based architecture as the backbone of C, the trac- +ing accuracy is higher than 90%. Especially for α = 0.15, +most of the tracing accuracy is higher than 96%, which indi- +cates the effectiveness of the proposed mechanism. Besides, +for a different level of classification task and different attack- +ing methods, the tracing accuracy can stay at a high level, +which shows the great adaptability of the proposed scheme. +The influence of α. We can see from Table 2 that the trac- +ing accuracy increases with the increase of α. We conclude +5QEBA:https://github.com/AI-secure/QEBA +6SurFree:https://github.com/t-maho/SurFree +the reason as: α determines the participation rate of tracer +Ti in final output logits, the larger α will make the final de- +cision boundary rely more on T . Therefore, when α gets +larger, making DOL of T larger would be a better choice to +realize the adversarial attack. The bigger DOL of T will cer- +tainly lead to better tracing performance. To verify the cor- +rectness of the explanation, we show the distribution of DOL +for task “ResNet18-CIFAR10” with different attacks in Fig. +4. We first generate 1000 adversarial examples of model Mi +for each α (0.05,0.1,0.15) with Boundary, HSJA, QEBA and +SurFree attack, then we record the DOLs of Ti. The distri- +bution of DOLs are shown in Fig. 4. +(a) The results of Boundary. +(b) The results of HSJA. +(c) The results of QEBA. +(d) The results of SurFree. +Figure 4: The distributions of output differences with differ- +ent black-box attacks. +It can be seen that compared with α = 0.05 and α = 0.1, +the DOL of α = 0.15 concentrate more on larger values, +which indicates that the larger α will result to larger DOL. +The influence of network architecture. The tracing re- +sults vary with different networks and different datasets. +With the same dataset, the tracing accuracy of ResNet18 will +be higher than that of VGG16. We attribute the reason to +the complexity of the model architecture. According to (Su +et al. 2018), compared with ResNet, the structure of VGG is +less robust, so VGG-based C might be easier to be adversar- +ial attacked. Therefore, once C is attacked, there is a certain +probability that Ti is not attacked as we expected, so DOL of +Ti will not produce the expected features for tracing. Fortu- +nately, the network architecture can be designed by us, so in +practice, choosing a robust architecture would be better for +tracing. +The influence of classification task. In our experiments, +we test the classification task with different classes. It can +be seen that with the increase of classification task complex- +ity, traceability performance decreases slightly. But in most +cases, when α = 0.15, the traceability ability can still reach +more than 90%. +The influence of black-box attack. The mechanism of +the black-box attack greatly influences the tracing perfor- +mance. For Boundary attack(Brendel, Rauber, and Bethge + +600 +α = 0.05 +500 +α = 0.1 +umbers of sampl +α = 0.15 +400 +300 +200 +100 +0.7 +1.6 +1.9 +Outout crferences800 +α = 0.05 +α = 0.1 +0 + sam +600 +α = 0.15 +of +400 +mbers +200 +0.5 +0.8 +Outout dirference800 +α = 0.05 +α = 0.1 +Jumbers of samp +600 +α= 0.15 +400 +200 +3 +1.9 +Outout dirferences800 +α = 0.05 +α = 0.1 +Q +Jumbers of samr +600 +α = 0.15 +400 +200 +0.5 +0.8 +2 +Output differencesAttack +Boundary +HSJA +QEBA +SurFree +alpha +0.05 +0.1 +0.15 +0.05 +0.1 +0.15 +0.05 +0.1 +0.15 +0.05 +0.1 +0.15 +CIFAR10 +ResNet18 +98.1% +98.9 % +99.2 % +98.2% +99.1% +99.3% +99.6% +99.7% +99.7% +94.5% +95.7% +97.9% +VGG16 +92.1% +95.6 % +98.2% +92.3 % +96.4 % +97.9 % +92.6 % +96.6% +99.2% +64.2 % +82.1 % +87.8 % +GTSRB +ResNet18 +97.6% +97.6 % +98.9 % +97.6 % +97.7 % +98.7 % +97.6% +97.7% +99.6 % +89.8 % +95.7 % +96.8% +VGG16 +94.1 % +96.8 % +97.6 % +95.5 % +97.3 % +98.3% +86.3% +92.6% +95.0 % +89.7% +95.7% +96.8% +mini ImageNet +ResNet50 +96.2% +96.4 % +98.7 % +94.5% +95.5 % +97.5 % +91.7% +93.8% +95.4 % +82.1 % +87.3 % +90.5% +VGG19 +89.4 % +94.7 % +98.2% +93.4 % +95.1 % +95.4% +89.5% +90.4% +90.8 % +75.7 % +88.7 % +88.8% +Table 2: The trace accuracy of different attacks. +2018), HSJA(Chen, Jordan, and Wainwright 2020) and +QEBA(Li et al. 2020), the tracing accuracy shows similar +results, but for SurFree (Maho, Furon, and Le Merrer 2021), +the tracing accuracy will be worse than that of the other +attacks. The reason is that Boundary attack, HSJA(Chen, +Jordan, and Wainwright 2020), QEBA(Li et al. 2020) are +gradient-estimation-based attacks, which tries to use random +noise to estimate the gradient of the network and further +attack along the gradient. Since the gradient is highly re- +lated to Ti, such attacks are more likely to be trapped by +Ti. But SurFree(Maho, Furon, and Le Merrer 2021) is at- +tacking based on geometric characteristics of the boundary, +which may ignore the trap of Ti especially when α is small. +So compared with Boundary attack(Brendel, Rauber, and +Bethge 2018), HSJA(Chen, Jordan, and Wainwright 2020) +and QEBA(Li et al. 2020), the proposed mechanism may +get worse performance when facing SurFree(Maho, Furon, +and Le Merrer 2021) attack. +4.4 +The influence of distributed copy numbers +In this section, we will discuss the traceability of the algo- +rithm in multiple distributed copies. When training tracer Ti, +the parameter is randomly initialized and each Ti is trained +independently. So the distribution of DOL corresponding to +any two branches should follow independent and identically +distribution. Therefore, the traceability results of multiple +copies could be calculated from the results of two copies. +In order to verify the correctness, we perform the following +experiments. +For experiment verification, we trained 10 different Ti +first, then we randomly choose one Ms as the source model +to generate the adversarial examples. We record the tracing +performance on the n, n ∈ [2, 10] models. +To estimate the tracing results for n, n ∈ [2, 10] models, +we utilize the Monte-Carlo sampling method in the distri- +bution of two models’ DOL. The specific procedure is de- +scribed as: +1). We randomly choose one source model Ms and one +other victim model Mv as the fundamental models, then we +perform the black-box attack on Ms with 1000 different im- +ages and record the DOL of Ts and Tv. +2). We draw the distribution of DOL corresponding to Ts +and Tv as the basic distribution, denoted as Ds and Dv, as +shown in Fig. 5a- 5c. +3). For the tracing results of n, n ∈ [2, 10] models, we +conduct the sampling process (take one sample Ss from Ds +and n − 1 sample Sn−1 +v +from Dv) 10000 times. +4) For each sampling, if Ss > max(Sn−1 +v +), we consider +it as a correct tracing sample. We record the total number +of correct tracing N n +C in 10000 samplings. The final tracing +accuracy of n models can be calculated with N n +C/10000. +The results are shown in Fig. 5d-5f. The attack we choose +is HSJA(Chen, Jordan, and Wainwright 2020), and α is fixed +as 0.15. It can be seen that with the increasing number of +distributed copies, the tracing accuracy gradually decreases. +But with 10 branches, it can still maintain more than 90% +accuracy for CIFAR10 and GTSRB. Besides, the estimated +tracing performance is almost the same as the actual experi- +ment results, which indicates the correctness of our analysis. +5 +Discussion +5.1 +The importance of noise-sensitive loss +In the proposed mechanism, making Ti easier to be attacked +is the key for tracing. We design the noise-sensitive loss to +meet the requirement. In this section, experiments will be +conducted to show the importance of noise-sensitive loss. +We use two randomly initialized tracers as the comparison +to conduct the tracing experiment on 1000 adversarial im- +ages. The adversarial attack is set as HSJA(Chen, Jordan, +and Wainwright 2020), α is fixed as 0.15. The experimental +results are shown in Table 3. +Attack +CIFAR10 +GTSRB +mini-ImageNet +ResNet18 +VGG16 +ResNet18 +VGG16 +ResNet50 +VGG19 +Random +57.9% +62.4% +53.9% +57.0% +56.2% +59.8% +Proposed +99.3% +97.9% +98.7% +98.3% +97.5% +95.4% +Table 3: The trace accuracy of HSJA attack with different T . +It can be seen that without noise-sensitive loss, the trac- +ing accuracy of the random initialized tracer only achieves +60%, which is much lower than the proposed noise-sensitive +tracer. This indicates that noise-sensitive loss is very impor- +tant in realizing accurate tracing, only setting different pa- +rameters of tracer is not enough to trap the attack to result in +specific features. +5.2 +Non-transferability and traceability +The concept of traceability is related but not equivalent +to non-transferability. A non-transferable adversarial exam- +ple works only on the victim model it is generated from. +Therefore, tracing such non-transferable example may be +a straightforward task. On the other hand, a transferable +sample may be generic enough to work on many copies/- +models. The task of tracing becomes more meaningful in + +(a) The distribution of CIFAR10. +(b) The distribution of GTSRB. +(c) The distribution of mini-ImageNet. +(d) The tracing results of CIFAR10. +(e) The tracing results of GTSRB. +(f) The tracing results of mini-ImageNet. +Figure 5: The distribution of DOL with HSJA and ResNet backbone and tracing performance of multiple branches. +this scenario. Our ability to trace a non-transferable exam- +ple demonstrates that the process of adversarial attack intro- +duces distinct traceable features which are unique to each +victim model. In this sense, traceability can serve as a fail- +safe property in defending adversarial attacks. There are +many defense methods can satisfy non-transferrability, but +once the defense fails, the model will not be effectively pro- +tected. But our experimental results show that for the pro- +posed method, even if the defense fails, we still have a cer- +tain probability to trace the attacked model, as shown in +Table 4. We use the data of “ResNet-CIFAR10” task with +HSJA (Chen, Jordan, and Wainwright 2020) and QEBA (Li +et al. 2020) as examples to show the specific tracing results. +Attack +α +NTr +NTr(+) +Tr +Tr(+) +Tr Rate +Total Rate +HSJA +0.05 +672 +672 +328 +313 +95.43% +98.50% +0.1 +973 +973 +27 +19 +70.37% +99.20% +0.15 +993 +993 +7 +0 +0% +99.30% +QEBA +0.05 +840 +840 +160 +156 +97.50% +99.60% +0.1 +879 +879 +121 +118 +97.52% +99.70% +0.15 +859 +859 +141 +138 +97.87% +99.7% +Table 4: The trace accuracy of different attacks. +In Table 4, NTr and Tr indicate the number of non- +transferrable samples and transferrable samples respectively. +NTr(+) and Tr(+) indicate the number of successful tracing +samples. We can see that for QEBA with α = 0.05, 0.1, and +0.15, the traceability to transferrable samples is all keep at +a high level which is greater than 97%. As for HSJA, when +α = 0.05, 328 samples can be transferred, and the trace- +ability of transferrable examples achieves 95.43%. When +α = 0.15, although the traceability of transferrable exam- +ples decreases to 0%, only 7 samples are transferrable. So +the total tracing rate is still at a high level. In general, the pro- +posed method either guarantees the high non-transferability +or the high tracing accuracy for transferred samples. +5.3 +Limitations and adaptive attacks +Although the proposed system maintains certain traceability +in the buyers-seller setting, there are still some limitations +that need to be addressed. For example, once the attacker +finds a way to attack C and bypass Ti, the tracing perfor- +mance may degrade. But we found that attacking such sys- +tem could be a challenging topic itself (in our setting) as +the attackers do not have access to all other copies and thus +are unable to avoid the differences that our tracer exploits. +Besides, it seems a more adaptive attack also comes with +“cost”. For instance, the approach of attacking C and by- +passing Ti would degrade the visual quality of the attack. +So future work may be paid on how to evade the attack by +utilizing such “cost”. +6 +Conclusion +This paper researches a new aspect of defending against ad- +versarial attacks that is traceability of adversarial attacks. +The techniques derived could aid forensic investigation of +known attacks, and provide deterrence to future attacks in +the buyers-seller setting. As for the mechanism, we de- +sign a framework which contains two related components +(model separation and origin tracing) to realize traceabil- +ity. For model separation, we propose a parallel network +structure which pairs a unique tracer with the original classi- +fier and a noise-sensitive training loss. Tracer model injects +the unique features and ensures the differences between dis- +tributed models. As for origin tracing, we design an output- +logits-based tracing mechanism. Based on this, the traceabil- +ity of the attacked models can be realized when obtaining + +400 +Source +350 +INon-Source +300 +250 +200 +150 +100 +50 +1.5450 +400 +Source +INon-Source +350 +300 +250 +200 +150 +100 +50500 +450 +Source +INon-Source +400 +350 +300 +250 +200 +150 +100 +50110 +105 +Tracing Accuracy ( +100 +95 +90 +ResNet18-R ++--ResNet18-S +85 +-VGG16-R ++-- VGG16-S +80 +Number of Distributed Models100 +Tracing Accuracy (%) +66 +98 +96 +95 +94 +ResNet18-R +--ResNet18-S +93 +VGG16-R ++--VGG16-S +92 +10 +Number of Distributed Models110.00 +100.00 +Tracing Accuracy +90.00 +80.00 +70.0 +ResNet50-R +ResNet50-S +60.0 +—VGG19-R ++-- VGG19-S +50.00 +2 +3 +4 +5 +D +10 +Number of Distributed Modelsthe adversarial examples. The experiment of multi-dataset +and multi-network model shows that it is possible to achieve +traceability through the adversarial examples. +References +Boenisch, F. 2020. 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Mitigat- +ing Adversarial Attacks by Distributing Different Copies to +Different Users. arXiv preprint arXiv:2111.15160. + diff --git a/1tAzT4oBgHgl3EQfRfvj/content/tmp_files/load_file.txt b/1tAzT4oBgHgl3EQfRfvj/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..820fad56ca334fec90bb68a3b180439ef9e4e77b --- /dev/null +++ b/1tAzT4oBgHgl3EQfRfvj/content/tmp_files/load_file.txt @@ -0,0 +1,838 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf,len=837 +page_content='Tracing the Origin of Adversarial Attack for Forensic Investigation and Deterrence Han Fang1, Jiyi Zhang 1, Yupeng Qiu 1, Ke Xu 2, Chengfang Fang 2, Ee-Chien Chang 1* 1 National University of Singapore 2 Huawei International fanghan@nus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='sg, jiyizhang@u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='nus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='edu, qiu yupeng@u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='nus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='edu, changec@comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='nus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='sg Abstract Deep neural networks are vulnerable to adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' In this paper, we take the role of investigators who want to trace the attack and identify the source, that is, the particular model which the adversarial examples are generated from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Techniques derived would aid forensic investigation of at- tack incidents and serve as deterrence to potential attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' We consider the buyers-seller setting where a machine learning model is to be distributed to various buyers and each buyer receives a slightly different copy with same functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' A malicious buyer generates adversarial examples from a par- ticular copy Mi and uses them to attack other copies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' From these adversarial examples, the investigator wants to iden- tify the source Mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' To address this problem, we propose a two-stage separate-and-trace framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The model separa- tion stage generates multiple copies of a model for a same classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' This process injects unique characteristics into each copy so that adversarial examples generated have distinct and traceable features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' We give a parallel structure which embeds a “tracer” in each copy, and a noise-sensitive training loss to achieve this goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The tracing stage takes in adversarial examples and a few candidate models, and iden- tifies the likely source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Based on the unique features induced by the noise-sensitive loss function, we could effectively trace the potential adversarial copy by considering the output logits from each tracer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Empirical results show that it is possible to trace the origin of the adversarial example and the mechanism can be applied to a wide range of architectures and datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 1 Introduction Deep learning models are vulnerable to adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' By introducing specific perturbations on input samples, the network model could be misled to give wrong predictions even when the perturbed sample looks visually close to the clean image (Szegedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Goodfellow, Shlens, and Szegedy 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Moosavi-Dezfooli, Fawzi, and Frossard 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Carlini and Wagner 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' There are many existing works on defending against such attacks (Kurakin, Good- fellow, and Bengio 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Meng and Chen 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Gu and Rigazio 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Hinton, Vinyals, and Dean 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Unfortu- nately, although current defenses could mitigate the attack to some extent, the threat is still far from being completely eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' In this paper, we look into the forensic aspect: from the adversarial examples, can we determine which Corresponding Authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Figure 1: Buyers-seller setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The seller has multiple mod- els Mi, i ∈ [1, m] that are to be distributed to different buyers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' A malicious buyer batt attempts to attack the vic- tim buyer bvic by generating the adversarial examples with his own model Matt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' model the adversarial examples were derived from?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Tech- niques derived could aid forensic investigation of attack in- cidents and provide deterrence to future attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' We consider a buyers-seller setting (Zhang, Tann, and Chang 2021), which is similar to the buyers-seller setting in digital rights protection (Memon and Wong 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Buyers-seller Setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Under this setting, the seller S dis- tributes m classification models Mi, i ∈ [1, m] to different buyers bi’s as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' These models are trained for a same classification task using a same training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The models are made accessible to the buyer as black boxes, for instance, the models could be embedded in hardware such as FPGA and ASIC, or are provided in a Machine Learning as a Service (MLaaS) platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Hence, the buyer only has black- box access, which means that he can only query the model for the hard label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' In addition, we assume that the buyers do not know the training datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The seller has full knowledge and thus has white-box access to all the distributed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Attack and Traceability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' A malicious buyer wants to at- tack other victim buyers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The malicious buyer does not have direct access to other models and thus generates the exam- ples from its own model and then deploys the found exam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' For example, the malicious buyer might generate an ad- versarial example of a road sign using its self-driving vehi- cle, and then physically defaces the road sign to trick passing vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Now, as forensic investigators who have obtained the defaced road sign, we want to understand how the ad- versarial example is generated and trace the models used in generating the example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='01218v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='CR] 31 Dec 2022 M1 M2 M1 M3 Matt Attacking 20 Generating Adversarial ExamplesProposed Framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' There are two stages in our solu- tion: model separation and origin tracing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' During the model separation stage, given a classification task, we want to gen- erate multiple models that have high accuracy on the clas- sification task and yet are sufficiently different for tracing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' In other words, we want to proactively enhance differences among the models in order to facilitate tracing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' To achieve that, we propose a parallel network structure that pairs a unique tracer with the original classification model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The role of the tracer is to modify the output, so as to induce the at- tacker to adversarial examples with unique features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' We give a noise-sensitive training loss for the tracer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' During the tracing stage, given m different classification models Mi, i ∈ [1, m] and the found adversarial example, we want to determine which model is most likely used in generating the adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' This is achieved by ex- ploiting the different tracers that are earlier embedded into the parallel models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Our proposed method compares the out- put logits (the output of the network before softmax) of those tracers to identify the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' In a certain sense, traceability is similar to neural network watermarking and can be viewed as a stronger form of water- marking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Neural network watermarking schemes (Boenisch 2020) attempt to generate multiple models so that an investi- gator can trace the source of a modified copy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' In traceability, the investigator can trace the source based on the generated adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' We point out a new aspect in defending against adver- sarial attacks, that is, tracing the origin of adversarial samples among multiple classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Techniques derived would aid forensic investigation of attack incidents and provide deterrence to future attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' We propose a framework to achieve traceability in the buyers-seller setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The framework consists of two stages: a model separation stage, and a tracing stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The model separation stage generates multiple “well- separated” models and this is achieved by a parallel network structure that pairs a tracer with the classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The tracing mechanism exploits the characteristics of the paired tracers to decide the origin of the given adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' We investigate the effectiveness of the separation and the subsequent tracing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Experimental studies show that the proposed mechanism can effectively trace to the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' For example, the tracing accuracy achieves more than 97% when applying to “ResNet18-CIFAR10” task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' We also observe a clear separation of the source tracer’s log- its distribution, from the non-source’s logits distribution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 5a-5c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 2 Related Work In this paper, we adopt black-box settings where the adver- sary can only query the model and get the hard label (final decision) of the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Many existing attacks assume white- box settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Attack such as FGSM (Goodfellow, Shlens, and Szegedy 2014), PGD (Kurakin, Goodfellow, and Bengio 2016), JSMA (Papernot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 2016), DeepFool (Moosavi- Dezfooli, Fawzi, and Frossard 2016), CW (Carlini and Wag- ner 2017) and EAD (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 2018) usually directly rely on the gradient information provided by the victim model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' As the detailed information of the model is hidden in black- box settings, black-box attacks are often considered more difficult and there are fewer works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Chen et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' introduced a black-box attack called Zeroth Order Optimization (ZOO) (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' ZOO can approximate the gradients of the objective function with finite-difference numerical esti- mates by only querying the network model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Thus the ap- proximated gradient is utilized to generate the adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Guo et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' proposed a simple black-box adver- sarial attack called “SimBA” (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 2019) to generate adversarial examples with a set of orthogonal vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' By testing the output logits with the added chosen vector, the optimization direction can be effectively found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Brendel et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' developed a decision-based adversarial attack which is known as “Boundary attack” (Brendel, Rauber, and Bethge 2018), it worked by iteratively perturbing another initial im- age that belongs to a different label toward the decision boundaries between the original label and the adjacent la- bel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' By querying the model with enough perturbed images, the boundary as well as the perturbation can be found thus generating the adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Chen et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' proposed another decision based attack named hop-skip-jump attack (HSJA) (Chen, Jordan, and Wainwright 2020) recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' By only utilizing the binary information at the decision bound- ary and the Monte-Carlo estimation, the gradient direction of the network can be found so as to realize the adversarial ex- amples generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Based on (Chen, Jordan, and Wainwright 2020), Li et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 2020) proposed a query-efficient boundary-based black-box attack named QEBA which es- timate the gradient of the boundary in several transformed space and effectively reduce the query numbers in gener- ating the adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Maho et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' (Maho, Furon, and Le Merrer 2021) proposed a surrogate-free black-box attack which do not estimate the gradient but searching the boundary based on polar coordinates, compared with (Chen, Jordan, and Wainwright 2020) and (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 2020), (Maho, Furon, and Le Merrer 2021) achieves less distortion with less query numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 3 Proposed Framework 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='1 Main Idea We design a framework that contains two stages: model sep- aration and origin tracing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' During the model separation stage, we want to generate multiple models which are sufficiently different under ad- versarial attack while remaining highly accurate on the clas- sification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Our main idea is a parallel network structure which pairs a unique tracer with the original classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The specific structure will be illustrated in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' As for origin tracing, we exploit unique characteristics of different tracers in the parallel structure, which can be ob- served in the tracers’ logits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Hence, our tracing process is conducted by feeding the adversarial examples into the trac- ers and analyzing their output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Figure 2: The framework of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The left part of the framework indicates the separation process of the seller’s distributed models Mi, i ∈ [1, m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The right part of the framework illustrates the origin tracing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The whole framework of the proposed scheme is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='2, each distributed model Mi consists of a tracer Ti and the original classification model C, and the tracer is trained with a proposed noise-sensitive loss LNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' During the tracing stage, the adversarial examples are fed into each Ti and the outputs are analyzed to identify the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='2 Model Separation We design a parallel network structure to generate the dis- tributed models Mi, i ∈ [1, m], which contains a tracer model Ti and a main model C, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Ti is used for injecting unique features and setting traps for the attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' C is the network trained for the original task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The final results are determined by both C and Ti with a weight parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' In each distributed model, C is fixed and only Ti is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The specific structure of Ti is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 3b, it is linearly cascaded with one “SingleConv” block (Conv-BN- ReLU), two “Res-block” (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 2016), one “Conv” block, one full connection block and one “Tanh” activation layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The training process of Ti can be described as: 1) Given the training dataset1 and tracer Ti, we first ini- tialize Ti with random parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 2) For each training epoch, we add random noise No 2 on the input image x to generate the noised image xNo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 3) Then we feed both x and xNo into Ti and get the out- puts Ox and OxNo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' We attempt to make Ti sensitive to noise, so Ox and OxNo should be as different as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The loss function of Ti can be written as: LNS = |Ox ◦ OxNo| ∥Ox∥2∥OxNo∥2 = |Ti(θTi, x) ◦ Ti(θTi, xNo)| ∥Ti(θTi, x)∥2∥Ti(θTi, xNo)∥2 (1) 1The training dataset for Ti only contains 1000 random sampled images from the dataset of the original classification task 2No follows a uniform distribution over [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='03) where ◦ represents the Hadamard product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' θTi indicates the parameters of Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Each distributed Ti for different buyers is generated by randomly initializing and then training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' We believed the ran- domness in initialization is enough to guarantee the differ- ence from different Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' It should be noted that when pro- ducing a new distributed copy, we only have to train one new tracer without setting more constraints on former trac- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' So such a separation method can be applied to multiple distributed models independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' As for C, it is trained in a normal way which utilizes the whole training dataset and cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' For the main classification task, C only has to be trained once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Besides, the training of C is independent of the training of Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' After train- ing C, we could get a high accuracy classification model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The final distributed model Mi is parallel combined with C and Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The specific workflow of Mi can be described as: For input image x, Ti and C both receive the same x and output two different vectors OTi and OC respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' OTi and OC have the same size and will be further added in a weighted way to generate the final outputs OF , as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' OF = OC + α × OTi (2) where α is the weight parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' It is worth noting that for the output of C, we use the normalization form of it, which can be formulated as: OC = C(x) − min(C(x)) max(C(x)) − min(C(x)) (3) where x indicates the input image, max and min indicate the maximum value and minimum value respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' By utilizing the aforementioned model separation method, two properties are well satisfied: (I) The attack could be tricked to focus more on Ti than C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Since after the training, Ti will be sensitive to random noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Therefore, the output of Ti is easy to be changed by adding noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Com- pared with C, the boundary of Ti is more likely to be esti- mated and Ti is more likely to be attacked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=" Thus, the attacker Model Separation Stage Origin Tracing Stage Attacker's Model !" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=" Main Model Initialized Tracer Model Distributed Tracer (source model) Ms c T1 T2 Tini Tini Tini 2 m Adversarial Trace's Outputs Obtaining Model Combination Tracer Model Training Examples Outputs OTi(x) Outputs Outputs Outputs Adversarial Tracer Image Example OT1 OT2 OTs OTm Ti X Tracer Noise-sensitive Loss L'is." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Image Main Ti Noised x Image c Attacked Model Tracing Identified Tracer : xNo Outputs OTi(xNo) Mi Ts Outputs Tracing mechanism OTm no Generated Models OTs arg max 0 ho true Identified Model : att: attacked label M1 M2 M3 M4 Ms Mm true: true label Ms(a) Parallel network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' (b) The architecture of tracer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' (c) Differences in logits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Figure 3: The specific network design in model separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' will fall into the trap of Ti and the generated adversarial per- turbations will bring the feature of the source Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' (II) Based on random initialization, each distributed Ti will correspond to different adversarial perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' This property helps us in tracing, since the source Ts which generates adversarial examples will output unique responses compared with other Ti, i ̸= s when feeding the generated adversarial examples, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='3 Tracing the Origin The tracing process is conducted by two related compo- nents: The first component keeps white-box copies for each of the m distributed copies 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' This component allows us to obtain the output logits of each tracer on an input x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The second component is an output logits-based mech- anism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' It gives a decision on which copy i is the most likely one to generate the adversarial example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The specific tracing process can be described as follows: 1) Given an appeared adversarial examples denoted as xatt, we feed the adversarial example into all Ti, i ∈ [1, m] and obtain the output logits of them, noted as OTi, i ∈ [1, m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 2) Then we extract two values that are corresponding to the attacked label and true label in each OTi, denoted as OTi att and OTi true respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 4 3) The source model can be determined by: s = arg max i,i∈[1,m] (OTi att − OTi true) (4) To simplify the description, we denote the difference of out- put logits (OTi att−OTi true) as DOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The tracer corresponded to the largest DOL is regarded as the source model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The reason is as follows: Since the perturbation are highly related to Ti, when feed- ing the same adversarial example, the outputs of Ti and Tj 3This setting is reasonable because when an adversarial attack appeared, the model seller who has all the details of the distributed network takes responsible to trace the attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 4Attacked label can be easily determined by the output logits and the true label can be tagged by the model owner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' If this sample cannot be accurately tagged by the owner, then this sample is not regarded as an adversarial example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' (i ̸= j) will be certainly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' For source model Ts where the adversarial examples are generated from, OTs is likely to render a large value on the adversarial label and a small value on the ground-truth label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Since the weight of OTs in the final OFs is small, so in order to achieve ad- versarial attack, OTs will be modified as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Thus DOL of Ts should be large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' But for victim model Tv, the DOL will be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Therefore, according to the value of DOL, we can trace the origin of the adversarial example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 4 Experimental Results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='1 Implementation Details In order to show the effectiveness of the proposed frame- work, we perform the experiments on two network architec- ture (ResNet18 (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 2016) and VGG16 (Simonyan and Zisserman 2014)) with two small image datasets (CIFAR10 (Krizhevsky, Hinton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 2009) of 10 classes and GTSRB (Houben et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 2013) of 43 classes) and two deeper network architecture (ResNet50 and VGG19) with one big image dataset (mini-ImageNet (Ravi and Larochelle 2016) of 100 classes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The main classifier C in experiments is trained for 200 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' All the model training is implemented by Py- Torch and executed on NVIDIA RTX 2080ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' For gradient descent, Adam (Kingma and Ba 2015) with learning rate of 1e-4 is applied as the optimization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='2 The Classification Accuracy of The Proposed Architecture The most influenced parameter for the classification accu- racy is the weight parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' α determines the partici- pation ratio of Ti in final outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' To investigate the influ- ence of α, we change the value of α from 0 (baseline) to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='2 and record the corresponding classification accuracy of each task, the results are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' It can be seen from Table 1 that for CIFAR10 and GTSRB, the growth of α will seldom decrease the accuracy of the classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Compared with the baseline (α = 0), the small value of α will keep the accuracy at the same level as the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' But for mini-ImageNet, the accuracy decreases more as α increases, we believe it is due to the complexity of the classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' But even though, the decrease rate is still within 3% when α is not larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Tracer Model Ti Image Main Model x cTracer Model T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='Tv Ti Ti c c c cα CIFAR10 GTSRB Mini-ImageNet ResNet18 VGG16 ResNet18 VGG16 ResNet50 VGG19 0 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='30% 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='68% 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='19% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='59% 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='12% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='79% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='05 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='24% 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='64% 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='14% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='52% 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='32% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='04% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='1 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='24% 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='63% 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='07% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='36% 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='88% 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='96% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='15 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='07% 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='63% 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='72% 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='84% 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='50% 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='75% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='2 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='95% 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='57% 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='09% 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='52% 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='14% 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='75% Table 1: The classification accuracy with different α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='3 Traceability of different black-box attack It should be noted that the change of α will not only influ- ence the accuracy but also affect the process of black-box adversarial attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Therefore, in order to explore the influ- ence of α, the following experiments will be conducted with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Setup and Code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' To verify the traceability of the pro- posed mechanism, we conduct experiments on two dis- tributed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' We set one model as the source model Ms to perform the adversarial attack and set the other model as the victim model Mv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The goal is to test whether the proposed scheme can effectively trace the source model from the generated adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The black-box at- tack we choose is Boundary (Brendel, Rauber, and Bethge 2018), HSJA (Chen, Jordan, and Wainwright 2020), QEBA (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 2020) and SurFree (Maho, Furon, and Le Merrer 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' For Boundary (Brendel, Rauber, and Bethge 2018) and HSJA (Chen, Jordan, and Wainwright 2020), we use Ad- versarial Robustness Toolbox (ART) (Nicolae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 2018) platform to conduct the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' For QEBA (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 2020) and SurFree (Maho, Furon, and Le Merrer 2021), we pull implementations from their respective GitHub reposito- ries 5 6 with default parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' For each α, each network architecture, each dataset and each attack, we generate 1000 successful attacked adversarial examples of Ms and con- duct the tracing experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Evaluation Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Traceability is evaluated by tracing accuracy, which is calculated by: Acc = Ncorrect NAll (5) where Ncorrect indicates the number of correct-tracing sam- ples and NAll indicates the total number of samples, which is set as 1000 in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The tracing performance of different attacks with different settings is shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' It can be seen that when apply- ing ResNet-based architecture as the backbone of C, the trac- ing accuracy is higher than 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Especially for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='15, most of the tracing accuracy is higher than 96%, which indi- cates the effectiveness of the proposed mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Besides, for a different level of classification task and different attack- ing methods, the tracing accuracy can stay at a high level, which shows the great adaptability of the proposed scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The influence of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' We can see from Table 2 that the trac- ing accuracy increases with the increase of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' We conclude 5QEBA:https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='com/AI-secure/QEBA 6SurFree:https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='com/t-maho/SurFree the reason as: α determines the participation rate of tracer Ti in final output logits, the larger α will make the final de- cision boundary rely more on T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Therefore, when α gets larger, making DOL of T larger would be a better choice to realize the adversarial attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The bigger DOL of T will cer- tainly lead to better tracing performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' To verify the cor- rectness of the explanation, we show the distribution of DOL for task “ResNet18-CIFAR10” with different attacks in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' We first generate 1000 adversarial examples of model Mi for each α (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='05,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='15) with Boundary, HSJA, QEBA and SurFree attack, then we record the DOLs of Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The distri- bution of DOLs are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' (a) The results of Boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' (b) The results of HSJA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' (c) The results of QEBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' (d) The results of SurFree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Figure 4: The distributions of output differences with differ- ent black-box attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' It can be seen that compared with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='05 and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='1, the DOL of α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='15 concentrate more on larger values, which indicates that the larger α will result to larger DOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The influence of network architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The tracing re- sults vary with different networks and different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' With the same dataset, the tracing accuracy of ResNet18 will be higher than that of VGG16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' We attribute the reason to the complexity of the model architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' According to (Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 2018), compared with ResNet, the structure of VGG is less robust, so VGG-based C might be easier to be adversar- ial attacked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Therefore, once C is attacked, there is a certain probability that Ti is not attacked as we expected, so DOL of Ti will not produce the expected features for tracing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Fortu- nately, the network architecture can be designed by us, so in practice, choosing a robust architecture would be better for tracing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The influence of classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' In our experiments, we test the classification task with different classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' It can be seen that with the increase of classification task complex- ity, traceability performance decreases slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' But in most cases, when α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='15, the traceability ability can still reach more than 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The influence of black-box attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The mechanism of the black-box attack greatly influences the tracing perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' For Boundary attack(Brendel, Rauber, and Bethge 600 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='05 500 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='1 umbers of sampl α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='15 400 300 200 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='9 Outout crferences800 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='05 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='1 0 sam 600 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='15 of 400 mbers 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='8 Outout dirference800 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='05 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='1 Jumbers of samp 600 α= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='15 400 200 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='9 Outout dirferences800 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='05 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='1 Q Jumbers of samr 600 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='15 400 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='8 2 Output differencesAttack Boundary HSJA QEBA SurFree alpha 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='15 CIFAR10 ResNet18 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='1% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='9 % 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='2 % 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='2% 99.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='6 % 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='6% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='2% 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='2 % 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='1 % 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='8 % GTSRB ResNet18 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='6% 97.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='6 % 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='8 % 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='7 % 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='8% VGG16 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='1 % 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='8 % 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='6 % 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='5 % 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='3 % 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='3% 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='3% 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='6% 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='0 % 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='7% 95.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='5 % 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='7% 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='8% 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='4 % 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='1 % 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='3 % 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='5% VGG19 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='4 % 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='7 % 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='2% 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='4 % 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='1 % 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='4% 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='5% 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='4% 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='8 % 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='7 % 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='7 % 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='8% Table 2: The trace accuracy of different attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 2018), HSJA(Chen, Jordan, and Wainwright 2020) and QEBA(Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 2020), the tracing accuracy shows similar results, but for SurFree (Maho, Furon, and Le Merrer 2021), the tracing accuracy will be worse than that of the other attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The reason is that Boundary attack, HSJA(Chen, Jordan, and Wainwright 2020), QEBA(Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 2020) are gradient-estimation-based attacks, which tries to use random noise to estimate the gradient of the network and further attack along the gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Since the gradient is highly re- lated to Ti, such attacks are more likely to be trapped by Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' But SurFree(Maho, Furon, and Le Merrer 2021) is at- tacking based on geometric characteristics of the boundary, which may ignore the trap of Ti especially when α is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' So compared with Boundary attack(Brendel, Rauber, and Bethge 2018), HSJA(Chen, Jordan, and Wainwright 2020) and QEBA(Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 2020), the proposed mechanism may get worse performance when facing SurFree(Maho, Furon, and Le Merrer 2021) attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='4 The influence of distributed copy numbers In this section, we will discuss the traceability of the algo- rithm in multiple distributed copies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' When training tracer Ti, the parameter is randomly initialized and each Ti is trained independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' So the distribution of DOL corresponding to any two branches should follow independent and identically distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Therefore, the traceability results of multiple copies could be calculated from the results of two copies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' In order to verify the correctness, we perform the following experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' For experiment verification, we trained 10 different Ti first, then we randomly choose one Ms as the source model to generate the adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' We record the tracing performance on the n, n ∈ [2, 10] models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' To estimate the tracing results for n, n ∈ [2, 10] models, we utilize the Monte-Carlo sampling method in the distri- bution of two models’ DOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The specific procedure is de- scribed as: 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' We randomly choose one source model Ms and one other victim model Mv as the fundamental models, then we perform the black-box attack on Ms with 1000 different im- ages and record the DOL of Ts and Tv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' We draw the distribution of DOL corresponding to Ts and Tv as the basic distribution, denoted as Ds and Dv, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 5a- 5c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' For the tracing results of n, n ∈ [2, 10] models, we conduct the sampling process (take one sample Ss from Ds and n − 1 sample Sn−1 v from Dv) 10000 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 4) For each sampling, if Ss > max(Sn−1 v ), we consider it as a correct tracing sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' We record the total number of correct tracing N n C in 10000 samplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The final tracing accuracy of n models can be calculated with N n C/10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 5d-5f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The attack we choose is HSJA(Chen, Jordan, and Wainwright 2020), and α is fixed as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' It can be seen that with the increasing number of distributed copies, the tracing accuracy gradually decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' But with 10 branches, it can still maintain more than 90% accuracy for CIFAR10 and GTSRB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Besides, the estimated tracing performance is almost the same as the actual experi- ment results, which indicates the correctness of our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 5 Discussion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='1 The importance of noise-sensitive loss In the proposed mechanism, making Ti easier to be attacked is the key for tracing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' We design the noise-sensitive loss to meet the requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' In this section, experiments will be conducted to show the importance of noise-sensitive loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' We use two randomly initialized tracers as the comparison to conduct the tracing experiment on 1000 adversarial im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The adversarial attack is set as HSJA(Chen, Jordan, and Wainwright 2020), α is fixed as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The experimental results are shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Attack CIFAR10 GTSRB mini-ImageNet ResNet18 VGG16 ResNet18 VGG16 ResNet50 VGG19 Random 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='9% 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='4% 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='9% 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='0% 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='2% 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='8% Proposed 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='3% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='9% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='7% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='3% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='5% 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='4% Table 3: The trace accuracy of HSJA attack with different T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' It can be seen that without noise-sensitive loss, the trac- ing accuracy of the random initialized tracer only achieves 60%, which is much lower than the proposed noise-sensitive tracer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' This indicates that noise-sensitive loss is very impor- tant in realizing accurate tracing, only setting different pa- rameters of tracer is not enough to trap the attack to result in specific features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='2 Non-transferability and traceability The concept of traceability is related but not equivalent to non-transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' A non-transferable adversarial exam- ple works only on the victim model it is generated from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Therefore, tracing such non-transferable example may be a straightforward task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' On the other hand, a transferable sample may be generic enough to work on many copies/- models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The task of tracing becomes more meaningful in (a) The distribution of CIFAR10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' (b) The distribution of GTSRB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' (c) The distribution of mini-ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' (d) The tracing results of CIFAR10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' (e) The tracing results of GTSRB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' (f) The tracing results of mini-ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Figure 5: The distribution of DOL with HSJA and ResNet backbone and tracing performance of multiple branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Our ability to trace a non-transferable exam- ple demonstrates that the process of adversarial attack intro- duces distinct traceable features which are unique to each victim model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' In this sense, traceability can serve as a fail- safe property in defending adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' There are many defense methods can satisfy non-transferrability, but once the defense fails, the model will not be effectively pro- tected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' But our experimental results show that for the pro- posed method, even if the defense fails, we still have a cer- tain probability to trace the attacked model, as shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' We use the data of “ResNet-CIFAR10” task with HSJA (Chen, Jordan, and Wainwright 2020) and QEBA (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 2020) as examples to show the specific tracing results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Attack α NTr NTr(+) Tr Tr(+) Tr Rate Total Rate HSJA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='05 672 672 328 313 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='43% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='50% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='1 973 973 27 19 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='37% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='20% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='15 993 993 7 0 0% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='30% QEBA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='05 840 840 160 156 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='50% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='60% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='1 879 879 121 118 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='52% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='70% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='15 859 859 141 138 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='87% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='7% Table 4: The trace accuracy of different attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' In Table 4, NTr and Tr indicate the number of non- transferrable samples and transferrable samples respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' NTr(+) and Tr(+) indicate the number of successful tracing samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' We can see that for QEBA with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='1, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='15, the traceability to transferrable samples is all keep at a high level which is greater than 97%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' As for HSJA, when α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='05, 328 samples can be transferred, and the trace- ability of transferrable examples achieves 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='43%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' When α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='15, although the traceability of transferrable exam- ples decreases to 0%, only 7 samples are transferrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' So the total tracing rate is still at a high level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' In general, the pro- posed method either guarantees the high non-transferability or the high tracing accuracy for transferred samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='3 Limitations and adaptive attacks Although the proposed system maintains certain traceability in the buyers-seller setting, there are still some limitations that need to be addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' For example, once the attacker finds a way to attack C and bypass Ti, the tracing perfor- mance may degrade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' But we found that attacking such sys- tem could be a challenging topic itself (in our setting) as the attackers do not have access to all other copies and thus are unable to avoid the differences that our tracer exploits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Besides, it seems a more adaptive attack also comes with “cost”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' For instance, the approach of attacking C and by- passing Ti would degrade the visual quality of the attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' So future work may be paid on how to evade the attack by utilizing such “cost”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' 6 Conclusion This paper researches a new aspect of defending against ad- versarial attacks that is traceability of adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The techniques derived could aid forensic investigation of known attacks, and provide deterrence to future attacks in the buyers-seller setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' As for the mechanism, we de- sign a framework which contains two related components (model separation and origin tracing) to realize traceabil- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' For model separation, we propose a parallel network structure which pairs a unique tracer with the original classi- fier and a noise-sensitive training loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Tracer model injects the unique features and ensures the differences between dis- tributed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' As for origin tracing, we design an output- logits-based tracing mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' Based on this, the traceabil- ity of the attacked models can be realized when obtaining 400 Source 350 INon-Source 300 250 200 150 100 50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='5450 400 Source INon-Source 350 300 250 200 150 100 50500 450 Source INon-Source 400 350 300 250 200 150 100 50110 105 Tracing Accuracy ( 100 95 90 ResNet18-R +--ResNet18-S 85 VGG16-R +-- VGG16-S 80 Number of Distributed Models100 Tracing Accuracy (%) 66 98 96 95 94 ResNet18-R --ResNet18-S 93 VGG16-R +--VGG16-S 92 10 Number of Distributed Models110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='00 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='00 Tracing Accuracy 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='00 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='00 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='0 ResNet50-R ResNet50-S 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='0 —VGG19-R +-- VGG19-S 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content='00 2 3 4 5 D 10 Number of Distributed Modelsthe adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQfRfvj/content/2301.01218v1.pdf'} +page_content=' The experiment of multi-dataset and multi-network model shows that it is possible to achieve traceability through the adversarial examples.' metadata={'source': 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M. May1, Kevin N. +Ortiz Ceballos3, Sarah E. Moran4, Sarah Peacock5,6, Kevin +B. Stevenson1, Mercedes L´opez-Morales3, Ryan J. +MacDonald7,8, L. C. Mayorga1, David K. Sing2,9, Kristin S. +Sotzen1, Jeff A. Valenti10, Jea Adams3, Munazza K. +Alam, Natasha E. Batalha12, Katherine A. Bennett9, Junellie +Gonzalez-Quiles9, James Kirk13, Ethan Kruse5,6, Joshua D. +Lothringer14, Zafar Rustamkulov9 and Hannah R. Wakeford15 +1Johns Hopkins APL, Laurel, MD, 20723, USA. +2Department of Physics & Astronomy, Johns Hopkins University, +Baltimore, MD, USA. +3Center for Astrophysics | Harvard & Smithsonian, 60 Garden St, +Cambridge, MA 02138, USA. +4Lunar and Planetary Laboratory, University of Arizona, Tucson, +AZ, 85721, USA. +5University of Maryland, Baltimore County, MD 21250, USA. +6NASA Goddard Space Flight Center, Greenbelt, MD 20771, +USA. +7Department of Astronomy, University of Michigan, Ann Arbor, +MI, USA. +8NHFP Sagan Fellow. +9Department of Earth & Planetary Sciences, Johns Hopkins +University, Baltimore, MD, USA. +10Space Telescope Science Institute, Baltimore, MD 21218, USA. +11Carnegie Earth & Planets Laboratory, Washington, DC, 20015, +USA. +12NASA Ames Research Center, Moffett Field, CA, USA. +13Department of Physics, Imperial College London, Prince +Consort Road, London, SW7 2AZ, UK. +1 +arXiv:2301.04191v1 [astro-ph.EP] 10 Jan 2023 + +Springer Nature 2021 LATEX template +2 +Lustig-Yaeger & Fu et al. — LHS 475b with JWST +14Department of Physics, Utah Valley University, Orem, UT, +84058 USA. +15School of Physics, HH Wills Physics Laboratory, University of +Bristol, Bristol, UK. +*Corresponding author(s). E-mail(s): +jacob.lustig-yaeger@jhuapl.edu; guangweifu@gmail.com; +Abstract +The critical first step in the search for life on exoplanets over the next +decade [1, 2] is to determine whether rocky planets transiting small M- +dwarf stars possess atmospheres [3, 4] and, if so, what processes sculpt +them over time [5–7]. Because of its broad wavelength coverage and +improved resolution compared to previous methods, spectroscopy with +JWST offers a new capability to detect and characterize the atmo- +spheres of Earth-sized, M-dwarf planets [8, 9]. Here we use JWST to +independently validate the discovery of LHS 475b [10], a warm (586 +K), 0.99 Earth-radius exoplanet, interior to the habitable zone, and +report a precise 2.9 − 5.3 µm transmission spectrum. With two transit +observations, we rule out primordial hydrogen-dominated and cloudless +pure methane atmospheres. Thus far, the featureless transmission spec- +trum remains consistent with a planet that has a high-altitude cloud +deck (similar to Venus), a tenuous atmosphere (similar to Mars), or no +appreciable atmosphere at all (akin to Mercury). There are no signs +of stellar contamination due to spots or faculae [11]. Our observations +demonstrate that JWST has the requisite sensitivity to constrain the +secondary atmospheres of terrestrial exoplanets with absorption fea- +tures < 50 ppm, and that our current atmospheric constraints speak +to the nature of the planet itself, rather than instrumental limits. +Keywords: JWST, Terrestrial Exoplanet Atmospheres, Transmission +Spectroscopy + +Springer Nature 2021 LATEX template +Lustig-Yaeger & Fu et al. — LHS 475b with JWST +3 +The search for atmospheres on rocky exoplanets has only just begun. Prior +constraints on the presence of terrestrial exoplanet atmospheres using the Hub- +ble Space Telescope (HST) and the Spitzer Space Telescope (Spitzer) have +succeeded in ruling out primordial H2/He atmospheres [12–16] that would pro- +duce large and detectable absorption features in a transmission spectrum; thick +atmospheres that would produce shallow infrared secondary eclipse depths +[17]; and a tentative detection of a terrestrial atmosphere [18] that remains +controversial [19, 20]. JWST is expected to break new ground in the search +for atmospheres on rocky exoplanets that transit nearby M dwarfs [9, 21]. +However, theoretical modeling work predicts a tumultuous stellar environment +in these compact M-dwarf systems [22, 23] that raises the critical question +of whether or not small, rocky exoplanets can maintain thick and detectable +atmospheres in the face of significant atmospheric loss processes. +We observed two transits of LHS 475b (previously the planet candidate +TOI 910.01) on 31 August 2022 and 4 September 2022 with JWST’s Near +InfraRed Spectrograph (NIRSpec) [24, 25] G395H instrument mode as part +of the JWST Cycle 1 Guest Observing (GO) Program 1981 (PI: K. Steven- +son). This mode covers wavelengths 2.87 − 5.27 µm and has a native spectral +resolving power of R = λ/∆λ ≈ 2700. We used the Bright Object Time Series +(BOTS) mode with the NRSRAPID readout pattern, S1600A1 slit, and the +SUB2048 subarray. Each time-series observation lasted a total of 2.9 hours, +which captured the 39.98 ± 4.04 minute transit that occurred during this +window. This resulted in approximately 1.75 hours and 0.5 hours of stellar +baseline before and after transit, respectively. The Methods contains additional +information on the observations. +We selected the LHS 475 system as one of several nearby M-dwarf systems +with known or candidate rocky planets. Prior to validation, LHS 475b was +classified as a planet candidate first identified in Sector 12 by the Transiting +Exoplanet Survey Satellite (TESS) [26]. TESS observed subsequent transits of +the planet in Sectors 13, 27, and 39. LHS 475b transits a 3300 K, 0.2789 R⊙ +M3.5V dwarf star on a 2.029-day orbital period [27]. This planet is likely to +be tidally locked, with a permanent dayside facing its host star [28], and an +equilibrium temperature of 586 K. +We validated the discovery of LHS 475b by eliminating both instrumental +and astrophysical false positives. JWST detected two transit signals at the +predicted times that are consistent in depth and duration with the 45 TESS +transits (978 ± 73 ppm, 42 ± 13 minutes). Archival DSS images from 1999 +rule out the possibility of a background transiting star-planet system or an +eclipsing binary. LHS 475 is a high-proper-motion star (1.28 arcsec/year) and +no flux sources were identified in the archival data along its path from 1999 to +2022. See the Methods for more details about the archival imagery. In parallel +with this work, Ment et al. (in prep) performed an independent validation of +LHS 475b using ground-based follow-up observations. +We reduced the JWST data using three independent pipelines—Eureka! +[29], FIREFLy [30], and Tiberius [31–33]—that yielded consistent results + +Springer Nature 2021 LATEX template +4 +Lustig-Yaeger & Fu et al. — LHS 475b with JWST +Fig. 1 White light curves from both LHS 475b visits using the FIREFLy reduction (see +Methods). For each visit, we combined data from both the NRS1 and NRS2 detectors into +a single white light curve and applied a vertical offset for clarity. A transit model and visit- +long linear trend are sufficient to fit the raw white light curves (panel a). Residuals from the +best fit (panel b) highlight a small, ∼15-minute ramp at the start of each visit. Residuals +are shown on the same y-scale as panel a. Both histograms of the residuals (panel c) are +Gaussian distributed. +(within 1.1σ, see Methods for details on each analysis). For our final inter- +pretation, we utilize results from the FIREFLy pipeline as it is the most +representative of the three reductions. We generated white light curves across +the full G395H wavelength range covered by the two detectors, NRS1 from +2.884 - 3.720 µm and NRS2 from 3.820 - 5.177 µm. The planet transits are +clearly visible in the raw white light curves (see Figure 1). We note the presence +of a small ramp at the start of the observation; no additional structure is seen +in the residuals. There is also no evidence of starspot crossings during the tran- +sits. Our joint fit to the white light curves gives a planet-to-star radius ratio +of Rp/Rs = 0.03257 ± 0.00014, a mid-transit time of T0 = 59822.8762593 ± +0.000026 BMJDTDB, an orbital period of P = 2.029088 ± 0.000006 days, a +ratio of the semi-major axis to the stellar radius of a/Rs = 15.87235 ± 0.472, +and an inclination of i = 87.194◦ ± 1.39◦. Therefore, LHS 475b has a radius +of Rp = 0.99 ± 0.05 R⊕ (6319 ± 318 km). Although LHS 475b’s mass has not +been measured, assuming an interior composition that is consistent with the +small, rocky M-dwarf exoplanet population [34], we estimate a planet mass of +Mp = 0.914 ± 0.187 M⊕. +We fitted the spectroscopic light curves at the detector’s pixel resolution +to derive wavelength-dependent transit depths independently for the first and +second visit. The orbital parameters were fixed to the values from the joint +white light curve analysis, leaving the planet-to-star radius ratio, linear tem- +poral slope, and a constant offset as free parameters. We adopted stellar limb +darkening from a 3D stellar model grid [35]. We then performed a weighted +average of spectra from the two visits and binned the combined native-pixel- +resolution transit depths into 56 points (R ≈ 100). Our co-added and binned +transmission spectrum is shown in Figure 2. + +Data + Model +Residuals + offset +1.002 +Visit ++ +1.001 +relative flux +1.000 +Visit 2 +0.999 +(a) +(b) +-2.0 -1.5 -1.0 -0.5 0.0 +0.5 +2.0 +-1.5-1.0 +-0.5 +0.0 +0.5 +time from mid-transit fhoursSpringer Nature 2021 LATEX template +Lustig-Yaeger & Fu et al. — LHS 475b with JWST +5 +Fig. 2 +Final, binned spectrum (black points) compared to models (coloured lines). Top: +Our data strongly (> 10σ) rule out hydrogen-dominated atmospheres with compositions +from 1× – 100× solar metallicity, with reduced-χ2s reported in the legend for each model. +The blue shaded bar highlights the region detailed in the bottom panel. Bottom: Our data +also rule out, though to lower (2–5σ) significance, high mean molecular weight compositions +of 1000× solar metallicity or a pure methane atmosphere. We weakly disfavor a pure water +atmosphere or an Earth composition atmosphere. The data are consistent with a pure carbon +dioxide atmosphere or that of an airless body. Each model is plotted relative to the mean +transit depth. +The observed transmission spectrum is featureless. A flat line, representa- +tive of an airless-body or high mean molecular weight (MMW) atmosphere, +fitted to the binned data produces a reduced χ2 = 0.91. No evidence is seen for +stellar contamination from unocculted cool spots or hot faculae on the stellar +disk [e.g., 11, 15, see Methods]. Despite the featureless spectrum, the precision +is sufficiently high to rule out (> 5σ) several archetypal atmospheric composi- +tions, including primordial hydrogen-helium atmospheres with less than 100 × +solar metallicity, as well as pure CH4 atmospheres ≥ 1 bar. We can specifically +rule out this pure CH4 atmospheres due to the low mass of the CH4 molecule +and the presence of the strong 3.3 µm CH4 band in the G395H bandpass. Other + +Springer Nature 2021 LATEX template +6 +Lustig-Yaeger & Fu et al. — LHS 475b with JWST +secondary atmospheres are more challenging to rule out and distinguish from +one another, however. We only weakly disfavor (at ≳1σ; [36]) 1000 × solar +metallicity, pure steam, or warm Earth-like atmospheric compositions. Both +a pure ≥ 1 bar carbon dioxide atmosphere or no atmosphere are favored yet +are statistically indistinguishable from each other. In the context of Solar Sys- +tem terrestrial archetype atmospheres (see Methods Fig. 11), we also weakly +disfavor (≳1σ) clear, warm Venus-like and Titan-like atmospheres. We cannot +statistically distinguish between a thin Mars-like atmosphere, a hazy Titan-like +atmosphere, and a cloudy Venus, which are all consistent with the data. +Following previous analyses [37], we performed Bayesian retrievals to better +explore the range of atmospheres that remain consistent with our spectro- +scopic measurements. We assumed a five component atmospheric composition +consisting of the four most common and spectroscopically active molecules +(H2O, CO2, CH4, and CO) in the Solar System terrestrial atmospheres, plus +an unspecified gas that constitutes the bulk atmospheric composition but is +spectroscopically inactive at these wavelengths [e.g. 38]. We allow the mean +molecular weight of the bulk gas to vary between 2.5 g/mol and 50 g/mol. +Since a solid planetary surface and an optically thick gray cloud deck are +indistinguishable in the transit spectrum, we fit for the apparent surface pres- +sure (the pressure of an opaque surface above which the atmosphere extends). +We marginalize over the aforementioned planet mass, which is assumed to be +consistent with a rocky interior composition. The vertical extent of the atmo- +sphere is dictated by the scale height, which is implicitly controlled by varying +the atmospheric temperature, mean molecular weight, and planet gravity. See +Figure 3 for the retrieval results summarizing the range of allowed atmo- +spheres given our data and highlighting the degeneracies that persist among +the remaining atmospheric possibilities. +If the planet has an atmosphere, it is likely to be a high mean molecu- +lar weight secondary atmosphere that is tenuous (Mars-like) or cloudy/hazy +(Venus-like or Titan-like). Compact atmospheres with small scale heights are +preferred across the full range of apparent surface pressures. High mean molec- +ular weight atmospheres dominated by species heavier than 40 g/mol, like CO2 +or Argon, can be thicker while maintaining relatively flat spectra. Atmospheric +characteristics that increase the scale height, including high temperatures and +low mean molecular weight bulk atmospheric compositions, tend to be dis- +favored, particularly for apparent surface pressures ≳10 mbar. Models with +scale heights larger than 20 km strongly skew towards the low mean molecular +weight atmospheres (µ < 10 g/mol), make up ∼50% of the lowest apparent +surface pressure samples, and tend to have low abundances of all absorbing +molecules. These extended atmospheres with low apparent surface pressures +are unlikely to form clouds or hazes at such high altitudes and are the most sus- +ceptible to atmospheric loss, making them less physically plausible scenarios. +Although LHS 475 is typical of low-activity M dwarfs in the solar neighbor- +hood [39], atmospheric escape processes are still a concern for a primordial + +Springer Nature 2021 LATEX template +Lustig-Yaeger & Fu et al. — LHS 475b with JWST +7 +Fig. 3 Retrieval results showing preferred atmospheric properties for models containing +H2O, CO2, CH4, and CO, plus a variable bulk gas composition for LHS 475b given the trans- +mission spectrum measurements. Darker color shading indicates higher relative posterior +probability density as a function of the apparent surface pressure (P0), molecular weight (µ) +of the bulk atmospheric composition (left), and isothermal scale height (H, right). Dashed +contours denote the 1σ (white), 2σ (gray), and 3σ (black) Bayesian credible regions. The +red arrow depicts how the Jeans escape flux depends on the scale height and emphasizes the +region of the parameter space that is more susceptible to atmospheric escape. If the planet +possesses an atmosphere with at least 1 ppm CO2 or CH4, then the models prefer high mean +molecular weight, compact atmospheres (µ > 20 g/mol at 1.2σ; H < 25 km at 1.2σ) with +low apparent surface pressures (P0 < 0.01 bar at 1.3σ; P0 < 1 bar at 2σ). These scenarios +correspond to either a tenuous or cloudy secondary atmosphere. +extended atmosphere, and if LHS 475 b is indeed airless, such processes would +likely constitute the primary reason for this. +Our two transit observations demonstrate that JWST has the sensitivity +to detect and constrain the secondary atmospheres of terrestrial exoplanets, +and therefore our atmospheric non-detection reflects the nature of the target +itself. We place a 3σ constraint on the maximum size of absorption features +in our spectrum at 61 ppm for H2O at 2.8 µm, 38 ppm for CH4 at 3.3 µm, +49 ppm for CO2 at 4.3 µm, and 62 ppm for CO at 4.6 µm. These constraints +demonstrate JWST’s sensitivity to absorption features smaller than 50 ppm +for an Earth-sized exoplanet. We find no indication of a noise floor down to +5 ppm (See Methods Figure 8). These are critical benchmarks for forthcom- +ing rocky exoplanet observations with JWST. Furthermore, our non-detection +of starspot crossings during transit and the lack of stellar contamination in +the transmission spectrum are promising signs in this initial reconnaissance +of LHS 475b. These findings indicate that additional transit observations of +LHS 475b with JWST are likely to tighten the constraints on a possible atmo- +sphere. A third transit of LHS 475b is scheduled as part of this program (GO +1981) in 2023. An alternative path to break the degeneracy between a cloudy +planet and an airless body is to obtain thermal emission measurements of +LHS 475b during secondary eclipse because an airless body is expected to be +several hundred Kelvin hotter than a cloudy world and will therefore produce + +10-6 +Titan (haze-top) +increasing +atmospheric +10 +escape +)-4 +Venus (cloud-top) +10 +100 +Farth (clear) +Titan (clear) +O +a +Venus (clear) +10 +20 +30 +40 +0 +20 +40 +60 +80 +atmospheric mmw [g/mol] +scale height [km]Springer Nature 2021 LATEX template +8 +Lustig-Yaeger & Fu et al. — LHS 475b with JWST +large and detectable eclipse depths at JWST’s MIRI wavelengths [4, 40]. Our +findings only skim the surface of what is possible with JWST. +Acknowledgements +This work is based in part on observations made with the NASA/ESA/CSA +JWST. The data were obtained from the Mikulski Archive for Space Telescopes +at the Space Telescope Science Institute, which is operated by the Association +of Universities for Research in Astronomy, Inc., under NASA contract NAS +5-03127 for JWST. These observations are associated with program #1981. +Support for program #1981 was provided by NASA through a grant from +the Space Telescope Science Institute, which is operated by the Association +of Universities for Research in Astronomy, Inc., under NASA contract NAS +5-03127. + +Springer Nature 2021 LATEX template +Lustig-Yaeger & Fu et al. — LHS 475b with JWST +9 +Fig. 4 +2D light curves of LHS 475b as a function of time and wavelength for the first +visit, measured with NIRSpec/G395H. The horizontal stripe down the middle of each panel +corresponds to the gap between the NRS1 and NRS2 detectors. Left: data normalized by +the median stellar spectrum. Middle: Maximum probability transit models. Right: Residuals +from the model fit. Note, the Eureka! and FIREFLy reductions trim more blue columns from +NRS1 where there is minimal throughput than the Tiberius pipeline, which accounts for the +regions without data in those reductions. Similarly, the Eureka! reduction also trims off the +initial ramp that can be seen in Figure 1. +Methods. +1 Data Analysis +1.1 Observations +We observed two transits of LHS 475b with the NIRSpec G395H grating +covering the 2.87–5.14 µm wavelength range split over the NRS1 and NRS2 +detectors, with a detector gap between 3.72 and 3.82 µm. The first transit was +observed on the 31 August 2022 18:48 UTC and the second on 4 September +2022 20:09 UTC. Each visit lasted 4.4 hours in total with 2.9 hours of expo- +sure time. Both transits were executed with the same observing settings, using +the Bright Object Time Series (BOTS) mode with the NRSRAPID readout +pattern, S1600A1 slit, and the SUB2048 subarray from NIRSpec. We obtained +a total of 1158 integrations per visit, with 9 groups per integration and 0.902 +seconds per group. +We extracted and analyzed the data from each visit independently with the +Eureka!, FIREFLy and Tiberius pipelines as described below. 2D lightcurves, +models, and residuals from the three reductions are shown in Figure 4. + +Data +Model +Residuals +0.0015 +Eureka! +4.5 +4.0 +0.0010 +3.5 +3.0 - ++ +0.0005 +5.0 +FIREFLY +wavelength +residuals +4.0 ++ 0.0000 +3.5 +3.0 ++-0.0005 +5.0 +Tiberius +4.5 +4.0 +-0.0010 +3.5 +3.0 +-0.0015 +-2.0 -1.5 -1.0 -0.5 0.0 0.5-2.0 -1.5 -1.0 -0.5 0.0 0.5-2.0 -1.5 -1.0 -0.5 0.0 0.5 +time from mid-transit [hours]Springer Nature 2021 LATEX template +10 +Lustig-Yaeger & Fu et al. — LHS 475b with JWST +1.2 Spectral Extraction +1.2.1 Eureka! +Eureka! [29] is an end-to-end analysis pipeline for time series observations +(TSOs) of exoplanets. Eureka! serves as a wrapper for stages 1 and 2 of the +jwst pipeline [41], allowing the user to specify which steps are run in addition +to custom modules. In later stages, Eureka! performs spectroscopic extraction, +light curve generation, and light curve fitting. +In this work, we apply Eureka!’s custom group-level background subtrac- +tion (GLBS) in stage 1 prior to ramp fitting to remove 1/f noise which has +been found to impact the accuracy of ramp fits for data with a small numbers +of groups up the ramp [30, 42]. Due to G395H’s curved trace we first identify +the center of the trace, then mask all pixels within an aperture of 8 pixels. All +remaining pixels in a given column (cross-dispersion direction) were used to +calculate a median background/noise level for that column. We skip the jump +step detection, otherwise running all standard stage 1 steps for TSOs. +In stage 2, we skip the flat field step (at the time of writing, only pre-flight +flat fields were available, which are insufficient for the precision we require +and adds significant noise to the data) and the photom step. Because we are +interested in relative flux measurements, we do not require the absolute flux +calibration provided by these two steps. +A second round of background subtraction is done in stage 3 to capture +any remaining background or 1/f noise, using pixels more than 9 pixels away +from the center of the trace. The spectrum is extracted with an aperture of 5 +pixels for NRS1 and 4 pixels for NRS2 using median frame optimal spectral +extraction. To convert from DN/s to electrons, we apply a median of the gain +files. At the time of writing only pre-flight gain files were available, which are +insufficient for the precision we require and adds significant noise to the data +if applied on a per-pixel basis. For NRS1 we extract only columns 800 − 2047 +due to the negligible throughput outside of that region of the detector. For +NRS2 we extract the full dispersion direction, but note that the edges are less +reliable due to the trace approaching the top or bottom of the subarray. +White light curves are generated across the full wavelength range of the +extracted data: 2.884 - 3.720 µm for NRS1 and 3.820 - 5.177 µm for NRS2. +For transit 1 we reach a white light precision of 112 ppm and 162 ppm for +NRS1 and NRS2, respectively. For transit 2 we reach a white light precision +of 116 ppm and 149 ppm for NRS1 and NRS2, respectively. For each transit, +NRS1 and NRS2 are combined into a single white light curve prior to light +curve fitting. We extract spectroscopic light curves at the pixel-resolution fol- +lowing recommendations from [43], however we find that our GLBS routine +sufficiently removes the 1/f noise in our data set, with no improvement on the +final transmission spectrum precision between fitting light curves at the native +pixel resolution and then binning, or binning prior to fitting (see Section 1.3.1). +Figure 5 shows our spectroscopic precision compared to expected noise lev- +els. Bad columns are denoted by squares (flagged in both transits) or darker + +Springer Nature 2021 LATEX template +Lustig-Yaeger & Fu et al. — LHS 475b with JWST +11 +Fig. 5 Eureka! spectrophotometric precision at the native pixel resolution, compared to +expected noise levels for both events. The expected noise level, as well as 1.25× and 2× the +expected noise are shown as grey lines, these have been smoothed to the resolution of the +final transit spectrum for visualization purposes. Squares denote columns which are greater +than 1.5× the expected noise level in both transits, dark blue circles denote columns which +are greater than 1.5× the expected noise level in only one transit. These columns are flagged +and not used to generate the final transmission spectrum. +circles (flagged in only one transit). This corresponds to 1.13% and 1.56% of +columns in transit 1 NRS1 and NRS2, respectively, and 1.05% and 2.10% of +columns in transit 2 NRS1 and NRS2, respectively. Excluding these columns, +for transit 1 we achieve a median precision of 1.19× and 1.23× the expected +noise level for NRS1 and NRS2, respectively, while for transit 2 we achieve +1.19× and 1.24× the expected noise level for NRS1 and NRS2, respectively. +1.2.2 FIREFLy +We used the FIREFLy [Fast InfraRed Exoplanet Fitting for Lightcurves, 30, 44] +to analyse the JWST data. We started with the uncal.fits files and ran the +jwst pipeline for stage 1 and 2 with modified steps including group-level 1/f +and background subtraction and skipping the jump-step. Both changes were +shown to decrease the scattering in the extracted lightcurves. After obtaining +the rateints.fits files from the stage 2 output, we performed custom cosmic +rays and bad/hot pixels corrections. The spectral traces are then masked in +the cleaned 2D images before applying 1/f correction, which subtracts the +median value of the unmasked background pixels at each column. Next, we +measured the shifts of the spectral trace in x and y directions by using cross +correlation in the selected 2D spectral region. The measured shifts are less +than one hundredth of a pixel which illustrates the excellent pointing stability + +10000+ +Transit 1 +NRS1 +NRS2 +9000 +Native Pixel Resolution +[wdd] +8000 +7000 +precision [ +6000 +2x Expected Limit +5000 +4000 +3000 +2000 +Expected Precision Limit +10000 +Transit 2 +NRS1 +NRS2 +9000 +Native Pixel Resolution +precision [ppm] +8000 +7000 +6000 + 2x Expected Limit +5000 +4000 +3000 +Expected Precision Limit +2000 +3.0 +3.2 +3.4 +3.8 +4.0 +4.2 +4.4 +4.6 +5.0 +3.6 +4.8 +5.2 +wavelength [um]Springer Nature 2021 LATEX template +12 +Lustig-Yaeger & Fu et al. — LHS 475b with JWST +of JWST. After aligning each 2D spectrum, we determine the spectral trace by +first cross correlating a Gaussian profile at each column to obtain the spectrum +location in the y direction, and then fit a 4th order polynomial as a function of +the x direction. The spectrum is then extracted for each integration centered +at the fitted spectral trace to form the light curves. +1.2.3 Tiberius +The Tiberius pipeline is a spectral extraction and light-curve fitting code based +on the LRG-BEASTS pipeline [31–33]. We used Tiberius on the Eureka! stage +1 group-level background-subtracted product, which had 1/f noise removed, +to produce white and spectroscopic transit light curves. First we created bad- +pixel masks for NRS1 and NRS2 by manually selecting hot pixels in the data. +These hot pixels were combined with all pixels flagged as 3σ outliers from the +background, and were interpolated over using their nearest neighboring pixels. +We also interpolate the spatial dimension of the data on a 10x grid, which +improves flux extraction at the sub-pixel level, reducing noise. The spectra +were then traced by fitting Gaussian functions for each column of the detectors, +and then using a running median to smooth the trace centers. These centers +were fit with a 4th-order polynomial, 3σ outliers were removed, and the centers +were again refit with a 4th-order polynomial. +In addition to the background subtraction already performed in the cre- +ation of the stage 1 product, we perform an additional background subtraction +step here to remove residual background light or remaining 1/f noise. We +mask from the detector a defined aperture of 4 pixels plus 6 more pixels off- +set from it, and clipped 3σ outliers in the background pixels, with respect to +their specific column and frame. Finally, the background signal for each col- +umn was subtracted from it, and the spectra were then extracted using a 4 +pixel aperture. +1.3 Light Curve Fitting +1.3.1 Eureka! +We perform a joint fit on both white light curves to constrain the system +parameters. Limb darkening is calculated with the ExoTic-LD pacakge [45–47] +using a quadratic limb darkening [48, 49] and the 3D stellar grid from [35]. +Stellar parameters are adopted from [35], assuming Teff = 3312 K, log(g) = +4.94, Fe/H = 0.0. For all light curve fits both limb darkening parameters are +held constant. We find that the uncertainty induced in the light curve fits by +the stellar models is smaller than the uncertainty in individual transit depths, +with consistent transit spectra regardless of free or fixed limb darkening. We +trim the first 150 integrations prior to light curve fitting to remove a slight +ramp at the beginning of the data, which can be seen in Figure 1. +For all light curve fitting we consider a transit model [batman, 50] and a +linear ramp in time. We use emcee [51], running each chain to at least 10× +the auto-correlation time. The joint white light curve fit includes the planet + +Springer Nature 2021 LATEX template +Lustig-Yaeger & Fu et al. — LHS 475b with JWST +13 +Fig. 6 Comparison of uncertainty on planet radius derived from light curves fit at the native +pixel resolution and fitting of pre-binned light curves. The y-axis is in parts per thousand. We +find little to no difference in the uncertainty, suggesting that our 1/f correction is sufficient +to address the column-column variances. +radius, orbital period, center of transit, inclination, and scaled semi-major +axis as shared parameters, and independent temporal ramps for each white +light curve. Best fit orbital parameters are given in Table 1. The Eureka! +spectroscopic fits adopt the Eureka! white light best-fit orbital parameters +and only fit for planet radius and the linear temporal ramp. Following [43] we +extract and fit our light curves at the native pixel resolution of the detectors, +and later bin the data to our preferred resolution. +To test the robustness of our group-level 1/f noise correction, we also fit +a set of pre-binned light curves and compare the resulting uncertainty on the +planet radius. Figure 6 shows our uncertainty on planet radius for transit 1 for +both the native pixel resolution light curve fitting, and our pre-binned light +curve fitting. We find no significant improvement by fitting the full resolution +light curves, suggesting the 1/f noise has been sufficiently removed. We suggest +that this test should be run on all NIRSpec G395H TSOs to ensure that one +has sufficiently removed the 1/f noise. +1.3.2 FIREFLy +The extracted light curves are first summed in the wavelength direction includ- +ing both NRS1 and NRS2 to form the whitelight light curve for each visit. We +then used batman [50] and emcee [51] to joint fit the whitelight lightcurves +from the two visits with six free parameters including Rplanet/Rstar, a/Rstar, +orbital inclination, mid-transit time for both visits, and linear temporal slope +for both visits. The best-fit joint white light orbital parameters are listed in +Table 1. We fixed the limb darkening to the quadratic coefficients from the 3D +stellar model in the Stagger-grid [35] interpolated at Fe/H=0, Teff=3312K +and log(g)=4.94. +The orbital parameters and quadratic limb darkening coefficients are +then fixed to fit the lightcurve from each wavelength column. We used the +scipy.optimize.curvefit function with three free parameters including +linear temporal slope, constant offset and Rplanet/Rstar. + +Transit 1 +Native Pixel Light Curve Fitting +Pre-binned Light Curve Fitting +O +NRS1 +● NRS2 +3.0 +3.2 +3.4 +3.6 +3.8 +4.0 +4.2 +4.4 +4.6 +4.8 +5.0 +5.2 +wavelength [um]Springer Nature 2021 LATEX template +14 +Lustig-Yaeger & Fu et al. — LHS 475b with JWST +Parameter +Eureka! +FIREFLy +Tiberius +Rp/Rs +[unitless] +0.032756 ++1.44×10−4 +−1.44×10−4 +0.03257 ++1.40×10−4 +−1.43×10−4 +0.032226 ++4.86×10−4 +−4.86×10−4 +T0 +[BMJDT DB] +59822.8762805 ++2.92×10−5 +−2.91×10−5 +59822.8762593 ++2.62×10−5 +−2.62×10−5 +59822.8763396 ++5.85×10−5 +−5.85×10−5 +Period +[days] +2.02908843 ++5.65×10−6 +−5.66×10−6 +2.0290882 +(Fixed) +2.02909 +(Fixed) +a/Rs +[unitless] +15.223 ++4.62×10−1 +−4.37×10−1 +15.87235 ++4.88×10−1 +−4.56×10−1 +18.161 ++1.79 +−1.79 +i +[degrees] +86.991 ++1.41×10−1 +−1.39×10−1 +87.194 ++1.41×10−1 +−1.37×10−1 +88.237 ++4.80×10−1 +−4.80×10−1 +Table 1 Best fit orbital parameters from white light curve fitting. We adopt the FIREFLy +results as our system parameters. Eureka! and FIREFLy values are derived from joint fits to +both white light curves. Tiberius parameters are derived from a weighted mean of fits to +individual light curves. +1.3.3 Tiberius +We extracted a white light curve for each detector (NRS1 and NRS2), for +each transit. These white light curves were fit independently using a Lev- +enberg–Marquardt damped least squares routine with the Tiberius pipeline. +Limb darkening parameters were obtained with LDTK [52, 53] from assumed +stellar parameters of Teff = 3312 K, log(g) = 4.94, Fe/H = 0.0, and a quadratic +limb darkening law was used. The results of the white light curve fits were +used to fix the transit parameters for the spectroscopic light curve fits, which +were performed at pixel-level resolution using the same damped least squares +routine. Since the white light curves were fit independently, the best-fit param- +eters in Table 1 were obtained from a weighted average of the results of each +of the four white light curve fits (weighted by flux received on each detector). +1.4 Final Transmission Spectrum +All three independently reduced spectra from above are in agreement, showing +no atmospheric features and being statistically consistent with a flat line. The +findings reported in the study do not depend upon which reduction pipeline +is used. To select the final transmission spectrum for model interpretation, we +performed two tests. The first test computed the mean absolute deviation of +each spectrum relative to the averaged spectrum of the three reductions. The +purpose of this test was to identify the reduction that is the most representative +of the three reductions. The FIREFLy reduction was favored by this test. The +second test computed the reduced chi-squared relative to a flat line. This +test was meant to validate the size of the error bars. The unbinned FIREFLy +transmission spectrum had a reduced chi-squared of 1.015. +1.5 Planet Validation +The JWST detection of a transit at the same period, phase, and depth as the +TESS TOI eliminates the possibility of a TESS false positive due to a telescope +or instrument systematic effect. This leaves only astrophysical sources, such +as a background eclipsing binary, as the remaining false positive mechanism. + +Springer Nature 2021 LATEX template +Lustig-Yaeger & Fu et al. — LHS 475b with JWST +15 +Fig. 7 A 3.36” × 3.36” DSS image centered on LHS 475 taken 1999 June 20. The red +circle depicts the star’s J2000 position per Simbad, whereas the blue circle indicates the +star’s position for the JWST observations in September of 2022. We see no indication of a +background star at the 2022 position that could be the source of the observed transit signal. +For reference, NIRSpec’s field of view is 1.6×1.6 pixels on this image. +Using an archival DSS image of LHS 475, we leverage the star’s high +proper motion to rule out astrophysical false positives. The star moves 1.28 +arcsec/year [54, 0.3423 arcsec/year in RA, −1.2303 arcsec/year in Dec;], which +corresponds to ∼29 pixels in Figure 7 from the June 1999 DSS image to our +Sep 2022 JWST observation. The lack of measurable flux at LHS 475’s 2022 +position enables us to rule out all scenarios involving transits within a poten- +tial background system. Finally, we rule our a stellar binary companion due to +the precisely measured Gaia DR3 parallax of 80.1134 mas, which corresponds +to a distance of only 12.5 pc. +1.6 Implications for JWST/NIRSpec Noise Floor +In the interest of exploring the effects of correlated noise and constraining the +instrument noise floor, we concatenate residuals from both visits and com- +pute Allan variance plots for the white and spectroscopic light curve fits (see +Figure 8). Using the Eureka! white light curve data, we find no indication of a +noise floor down to 5 ppm; however, we identify correlated noise at timescales +of < 5 minutes. This timescale is consistent with the thermal cycling of heaters +in the ISIM Electronics Compartment, which induces small forces on the tele- +scopes backplane structure [55]. The effect is semi-periodic, the result of several +heaters cycling at different frequencies. + +200 +20000 +175 - +17500 +150 +15000 +125 +12500 + Pixel Number +2000 +10000 +100 +2022 +7500 +75 - +5000 +50 +2500 +25 +0 + 0 +0 +25 +50 +75 +100 +125 +150 +175 +200 +Pixel NumberSpringer Nature 2021 LATEX template +16 +Lustig-Yaeger & Fu et al. — LHS 475b with JWST +Fig. 8 Allan variance plots from the white and spectroscopic light curve fits. Panel (a) illus- +trates that the white light curve residuals from two of the analyses exhibit some correlated +noise at timescales of < 5 minutes (< 35 integrations). This is likely due to uncorrected 1/f +noise from the thermal cycling of on-board heaters [55, Section 4.5.3]. At longer timescales +(> 18 minutes), the Eureka! pipeline returns to the expected standard error with RMS val- +ues below 10 ppm. The Tiberius reduction did not sum the flux across both detectors and +was not used for this noise floor analysis. The spectroscopic RMS values in panels (b) – (d) +are more consistent with the standard error, thus confirming that the spectroscopic light +curves are dominated by white noise. +2 Modeling +With the reduced data and coadded transmission spectrum produced in the +previous section, we now use a variety of models to update the state of knowl- +edge on the LHS 475 system. We use archival photometry to update the +LHS 475 stellar parameters and assess the impact of stellar contamination on +the JWST transmission spectrum; we use empirical mass-radius relations to +estimate the planet mass given our precise radius measurement; and we fit +atmospheric models to the transmission spectrum to obtain constraints on the +possible atmospheric composition of LHS 475b. + +100 +Normalized RMS +10 +Eureka! Median RMS +Std. Err. (1/V N) +10-2 +100 +101 +102 +Bin Size [Number of Integrations]100 +Normalized RMS +10 +Tiberius Median RMS +Std. Err. (1/V N) +10-2 +100 +101 +102 +Bin Size [Number of Integrations]102 +RMS [ppm] +101 +ppm +Firefly RMS +Std. Err. (1/VN) +Eureka! RMS +Std. Err. (1/VN) +100 +100 +101 +102 +Bin Size [Number of Integrations]100 +Normalized RMS +10 +Firefly Median RMS +Std. Err. (1/V N) +10-2 +100 +101 +102 +Bin Size [Number of Integrations]Springer Nature 2021 LATEX template +Lustig-Yaeger & Fu et al. — LHS 475b with JWST +17 +2.1 Stellar Modeling and Transit Light Source +Contamination +We use PHOENIX spectra guided by archival photometry1 from the VizieR Pho- +tometry Viewer2 to improve constraints on the stellar parameters. Effective +temperature (Teff) is a primary driver of spectral shape and so we are able to +refine estimates by matching models to the observations at visible and near- +IR wavelengths. To do this, we computed a grid of synthetic spectra following +similar procedures to those outlined in [53] with Teff = 3200 – 3400 K (∆T += 10 K), log(g) = 4.5 – 5.2 dex (∆log(g) = 0.1 dex), and M⋆ = 0.262 M⊙. +This parameter space was chosen by expanding around the stellar parameters +published in the Tess Input Catalog [27]. For each model, we computed syn- +thetic visible and near-IR photometry over the same wavelengths as the filter +profiles for available measurements for LHS 475 and used a reduced χ2 test to +identify the model that most closely matched the observations. The reduced +χ2 test was conducted with both the observations and models normalized to +the 2MASS J band flux density value to isolate matching the spectral shape. +The fully explored grid yielded χ2 +ν values between 6.8 – 987.2, with 30 mod- +els returning similar values less than 50 (Figure 9). These models have Teff = +3300 (+80, -30) K, log(g) = 5.2 ± 0.5 g/cm3, and M⋆ = 0.262 M⊙. To deter- +mine the radius of the star we scaled all models with χ2 +ν < 50 by R2 +⋆/dist2 +until FJ2MASS,mod = FJ2MASS,obs, +R⋆ = +� +(FJ2MASS,obs/FJ2MASS,mod) × dist2 +(1) +We adopted the Gaia EDR3 distance of 12.481 ± 0.0065 pc [54], which +returned a radius of 0.2789 ± 0.0014 R⊙. +LHS 475 is typical of low-activity M dwarfs in the solar neighborhood. +TESS only detected two flares on LHS 475, both with energy below 1031 erg. +The inferred flare rate and other activity diagnostics are all consistent with the +general population of relatively inactive M dwarfs in a volume-limited sample +[39]. Hα and He I D3 are both in absorption, not emission. Ca II 8542 is +relatively deep. +Fitting models of the Transit Light Source (TLS) effect to the observed and +coadded transmission spectrum allows us to assess the degree to which stellar +contamination may impact and/or explain any characteristics of the planet’s +transmission spectrum. The TLS effect can impart slopes and features into +the transmission spectrum due to differences in the spot or faculae coverage +along the planet’s transit chord relative to the average coverage across the +visible stellar disk [11]. Following the formalism of [15], we calculate the TLS +1For NIRSpec observations, the jwst pipeline requires the flat field step be run for absolute flux +calibration. At the time of writing, only ground or dummy frames were available for the three +types of NIRSpec flat fields and for the correction applied in the photom step of the jwst Stage 2 +pipeline. These ground and dummy frames do not provide high accuracy absolute flux calibration, +so we choose to not use our new data for Stellar Modeling at this time. +2http://vizier.cds.unistra.fr/vizier/sed/ + +Springer Nature 2021 LATEX template +18 +Lustig-Yaeger & Fu et al. — LHS 475b with JWST +Fig. 9 Comparison of our 30 closest matching PHOENIX models (χ2 +ν < 50) to all available +archival photometry of LHS 475 from the VizieR Photometry Viewer. These models have +Teff = 3380 – 3320 K, log(g) = 4.7 – 5.7 g/cm2, M = 0.262 M⊙. +contamination spectrum +ϵλ = (1 − fspot − ffac)Sλ,phot + fspotSλ,spot + ffacSλ,fac +(1 − Fspot − Ffac)Sλ,phot + FspotSλ,spot + FfacSλ,fac +(2) +where Sλ,phot, Sλ,spot, and Sλ,fac refer to the spectrum of the stellar photo- +sphere, spots, and faculae, respectively, fspot and ffac refer to the spot and +faculae projected area covering fractions along the transit chord, and similarly +Fspot and Ffac refer to the spot and faculae projected area covering fractions +across the entire visible stellar disk. Thus, ϵλ is the ratio of the stellar spec- +trum along the transit chord to the spectrum of the whole disk, and a general +model for how the TLS effect contaminates the observed transmission spec- +trum. Given Equation 2 the observed drop in flux that we refer to as the +transmission spectrum is simply +∆Fλ,obs = ϵλ +�Rp +Rs +�2 +λ +(3) +where the TLS contamination spectrum is multiplied by the wavelength- +dependent “true” planet transmission spectrum. We use the Dynesty nested +sampling code [56] to infer posterior distributions for the TLS contamination +model parameters under the assumption of a wavelength independent planet +transmission spectrum. We run the standard nested sampling algorithm [57] + +PHOENIX Models with x<50 +1.2 +LHS 475 Observations +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +5000 +10000 +15000 +20000 +25000 +30000 +Wavelength (A)Springer Nature 2021 LATEX template +Lustig-Yaeger & Fu et al. — LHS 475b with JWST +19 +Fig. 10 Corner plot comparing the prior (orange) and posterior (dark blue) PDFs for +a subset of the fitting parameters in the TLS contamination retrieval. The flat spectrum +reveals a consistent spot (and faculae) coverage along the transit chord compared to the full +stellar disk. +with 1000 live points until the estimated contribution to the total evidence +from the remaining prior volume drops below the threshold of dlogz=0.075. +In general, no evidence of TLS contamination is observed in the flat +transmission spectrum and the TLS model readily reproduces the featureless +spectrum. Figure 10 compares the prior and posterior probability distributions +for a subset of the TLS model parameters. The inferred posterior distribution +for the TLS contamination model generally reproduce the prior distributions, +with the exception of the covariance between the spot (faculae) area cover- +ing fraction along the transit chord compared to the spot (faculae) covering +fraction on the full disk. These two convariance are constrained along a line + +disk spot +-0.15 +Posterior PDF +Prior PDF +disk faculae +-0.19 +0.8 +disk faculae +fraction +0.6 +chord spot +0.2 +chord spot +fraction +chord faculae +0.2 +-0.21 +chord faculae +fraction +0.6 +disk spot +disk faculae +chord spot +chord faculae +fraction +fraction +fraction +fractionSpringer Nature 2021 LATEX template +20 +Lustig-Yaeger & Fu et al. — LHS 475b with JWST +with a slope of approximately unity, such that the ratio of spot (faculae) cov- +ering fraction on the full stellar disk to spot (faculae) covering fraction along +the transit chord is 1.005 ± 0.003 (0.948 ± 0.006). This implies that—although +the exact area covered by spots (and faculae) is not well constrained—at the +observed precision there is no evidence of differing spot (or faculae) cover- +age along the transit chord compared to the average stellar disk. We repeated +the same TLS contamination retrieval with the addition of the transit depth +measured by TESS in the optical (978±73 ppm) and obtained the same result. +2.2 Planet Radius, Mass, and Equilibrium Temperature +From the constraint on the white light curve transit depth (1060 ± 9 ppm) +and the stellar radius (Rs = 0.279 ± 0.014 R⊙), we calculate the planet radius +to be Rp = 0.991 ± 0.050 R⊕ (6319 ± 318 km). The 5% radius precision is +dominated by uncertainty in the stellar radius. For reference, the preexisting +radius constraint from TESS sectors 12-39 was 0.93 ± 0.70 R⊕ [10]. +Despite the lack of a mass measurement for LHS 475b, we use three different +methods to estimate the mass: 1) from the transmission spectrum [13], 2) from +probabilistic mass-radius-relation [58], and 3) from probabilistic bulk density +arguments. We use atmospheric models over a range of masses compared to +the spectroscopic data from NIRSpec/G395H to infer conservative upper and +lower limits of the possible mass for LHS 475b. To do so, we employ the +forward model framework discussed in the following section. First, we find the +uppermost mass limit by finding the densest planet that could stably support +a hydrogen-helium envelope and fit the data. To obtain a reduced-χ2 ≤ 1, we +determine that we must consider a mass of 24 M⊕. Combined with the precise +radius constraint, this upper limit mass results in a planetary density of 119 +g/cm−3, or 6× that of pure uranium. Given this unrealistic density, we can +clearly reject a hydrogen-helium atmosphere around a very dense planet. On +the other hand, to find the lowest mass that is consistent with the NIRSpec +data, we instead consider a planet with a very high mean molecular weight +atmosphere, but from a reasonably abundant molecule – that of pure CO2 +– and scale the mass down until we obtain a reduced-χ2 ≤ 1. Under this +atmospheric assumption, we find that masses consistent with the data extend +down to 0.78 M⊕. Together this method gives us a range of masses consistent +with the observed atmosphere between 0.78 - 24 M⊕. +Next, we use the mass-radius relationship gleaned from the existing pop- +ulation of small M dwarf exoplanets to estimate the planet’s mass given our +precise radius constraint. Using the Forcaster code’s probabilistic mass-radius +relationship and mass prediction tool [58], we estimate the mass of LHS 475b +to be Mp = 0.980+0.632 +−0.359 M⊕. +Our third mass estimate leverages recent results on the interior bulk densi- +ties among the M dwarf small planet population. Given the radius constraint +for LHS 475b, the planet is consistent with the population of M dwarf plan- +ets having rocky interior compositions (1.21 ± 0.28 R⊕) [34]. Therefore, if we +assume that LHS 475b is indeed a rocky planet with a mean bulk density + +Springer Nature 2021 LATEX template +Lustig-Yaeger & Fu et al. — LHS 475b with JWST +21 +consistent with the M dwarf rocky planet population (0.94 ± 0.13 ρ⊕) [34], +then we find the planet mass to be Mp = 0.914 ± 0.187 M⊕. If we consider +that instead the planet were in the population of lower density water worlds +(with 50% water, 50% rock interiors), then this would ultimately have ram- +ifications for the scale height and water content of the atmosphere [e.g., 59], +that are inconsistent with the featureless transmission spectrum that we mea- +sured. Therefore, it is likely that LHS 475b has a mass that is consistent with +a rocky mean bulk density. In the atmospheric models that follow, we assume +the planet is consistent with the population with rocky interiors and use the +corresponding mass Mp = 0.914 ± 0.187 M⊕, which is consistent with our +previous estimates, albeit with a tighter constraint. +We update LHS 475b’s zero bond albedo equilibrium temperature to +586 ± 12 K (assuming uniform heat redistribution). In the limit of instant +re-radiation expected from a planet with a tenuous or nonexistent atmo- +sphere, the estimated day side brightness temperature is 748 ± 16 K. These +updates may aid in the planning of any future secondary eclipse observations +of LHS 475b. +2.3 Atmospheric Modeling +We use atmospheric radiative transfer models to simulate the transmission +spectrum of LHS 475b for comparison with our JWST observations. In the next +section, forward models of single-composition end-member atmospheres and +archetypal atmospheres are used to illustrate the atmospheric compositions +that are consistent with our observed data. Then, retrieval models are used +to simulate a broad range of atmospheric compositions to place constraints +on key atmospheric parameters given the precise, yet featureless transmission +spectrum. +2.3.1 Forward Modeling +We use the forward modeling capabilities of two different open-source atmo- +spheric radiative transfer codes, PICASO [60] and CHIMERA [61, 62], to explore +the plausibility of various atmospheric archetypes. We compute each model +atmosphere for a planet mass consistent with a rocky mean bulk density, +Mp = 0.914 M⊕, a planetary radius of Rp = 0.991 R⊕, and a stellar radius +Rs = 0.279 R⊙, and a planetary equilibrium temperature of Teq = 600 K, +as above. In each case, we compare the modeled transmission spectrum to +the NIRSpec/G395H data for LHS 475b from 2.9 – 5.3 µm and compute the +reduced-χ2 between the modeled spectrum and the data. +For the CHIMERA models, we compute chemically consistent atmospheric +mixing ratios for 1×, 10×, 100×, and 1000× solar metalicities, with a solar +C/O ratio. CHIMERA uses a preset grid of atmospheric molecular abundances +along temperature-pressure profiles, metallicity, and C/O ratio generated from +the NASA CEA code [63]. For the temperature-pressure profile, the code uses +the five-parameter, double gray, one-dimensional parametrization of [64], where + +Springer Nature 2021 LATEX template +22 +Lustig-Yaeger & Fu et al. — LHS 475b with JWST +we input a planetary equilibrium temperature of 600 K. For these CHIMERA +models, we include opacity from H2O, CH4, CO, CO2, NH3, N2, HCN, H2S, +H2/He CIA [65, 66], and Rayleigh scattering from H2. We consider simplistic +cloudy hydrogen-dominated atmospheric models with CHIMERA by computing +a cloud-top pressure for a grey absorbing cloud. We generate atmospheric +transmission models with the correlated-k method of radiative transfer and +bin the resulting model to the data before calculating our reduced-χ2. +For the PICASO models, we generate simplified end-member atmospheric +compositions with isothermal temperature-pressure profiles. We set a pressure +grid which ranges from 1 µbar to 100 bar, and then set an isothermic tem- +perature at the equilibrium temperature of 600 K. For the models shown in +Figure 2, each atmosphere consists solely of either H2O, CO2, CH4, or as in +the case of the Earth-like atmosphere, follows the atmospheric abundances of +Earth above the water cold-trap, with 78% N2, 21% O2, 0.9% Ar, 416 ppm +CO2, 524 ppm He, and 187 ppm CH4. For the models shown in Figure 11, +we generate individual pressure grids with an upper bound according to the +terrestrial body’s surface pressure (e.g., the Earth-like model has an upper +atmospheric pressure bound of 1 bar; the Venus-like model has an upper pres- +sure bound of 90 bar). We assume isothermal temperature profiles (at 600 K) +with atmospheric abundances fixed to the composition of each Solar System +body above any cold trap. For the cloudy Venus and hazy Titan cases, we +implement a simple grey absorbing cloud at the pressure level according to the +Venus cloud-top (1 mbar) and the Titan haze-top (0.01 mbar). The opacity +database is resampled to R=10,000 and is taken from [67]. Models are then +binned to the data for reduced-χ2 comparison. +We strongly rule out clear atmospheres of 1× to 100× solar, with reduced- +χ2s ≥ 9, or over 10σ. Given the mass estimate analysis above, even with +the uncertain planetary mass, we are able to reject low (≤100× solar) atmo- +spheres. To obtain a reduced-χ2 ∼ 1 in cloudy low-metallicity atmospheres, +we must insert an opaque cloud deck with cloud-top pressure between 0.5 and +1 µbar, which can be discarded as unrealistic given the lack of cloud-forming +material at such low pressures. Each of these cases represents a hydrogen-rich +atmosphere around a rocky, 600 K planet, which would not be stable against +escape over the lifetime of the system, and thus our ability to reject them is +not unexpected. +For the 1000× atmosphere, we calculate a reduced-χ2 to the data of 1.5, +which weakly rules out this scenario to 2.5σ. The pure methane atmosphere +is rejected with a reduced-χ2=2.3, or 5σ. Both the end-member atmospheric +compositions of a pure steam or Earth-like atmospheric abundances are weakly +disfavored at ≳1σ. A pure 1 bar carbon dioxide atmosphere or no atmosphere +at all are preferred but not statistically distinguishable from each other. For +the Solar System terrestrial archetype atmospheres shown in Figure 11, we +also weakly disfavor (≳1σ) clear Venus, Titan, or Earth-like atmospheres, but +cannot statistically distinguish between a thin Mars-like atmosphere, a hazy + +Springer Nature 2021 LATEX template +Lustig-Yaeger & Fu et al. — LHS 475b with JWST +23 +3.0 +3.5 +4.0 +4.5 +5.0 +Wavelength ( m) +200 +150 +100 +50 +0 +50 +100 +150 +200 +Relative Transit Depth (ppm) +Earth-like: +2=1.1 +Mars-like: +2=1.0 +Titan-like, clear: +2=1.3 +Titan-like, hazy: +2=0.95 +Venus-like, clear: +2=1.1 +Venus-like, cloudy: +2=0.97 +Mercury-like: +2=0.91 +JWST/NIRSpec G395H +Fig. 11 +Final, binned spectrum (black points) compared to atmospheric models with +compositions of the Solar System terrestrial planets (coloured lines). Our data, to weakly rule +out Earth composition (blue solid), clear Titan composition (orange solid), and clear Venus +composition atmospheres (yellow solid). However, the data are all consistent within error to +that of a hazy Titan composition with a haze-top at 0.01 mbar (dotted orange), a cloudy +Venus composition with a cloud-top at 1 mbar (dotted yellow), and a Mars composition +atmosphere (red solid), as well as that of an airless body, like Mercury (grey dotted line). +Titan-like atmosphere, or a cloudy Venus, as consistent with the retrieval +modeling shown in Figure 3 and discussed below. +2.3.2 Retrieval Modeling +We use two different atmospheric retrieval codes—smarter and POSEIDON—to +explore the range of atmospheric properties that are consistent with, or ruled +out, by LHS 475b’s transmission spectrum. +Retrievals with smarter +The smarter retrieval code [68, 69] couples line-by-line radiative transfer cal- +culations from the Spectral Mapping Atmospheric Radiative Transfer forward +model (smart [70]) to the dynesty nested sampling Bayesian inference code +[56] to retrieve planetary and atmospheric parameters that are consistent with +the JWST observations. We assume an isothermal temperature-pressure profile +and evenly-mixed gas volume mixing ratios. We calculate line absorption coef- +ficients for gaseous molecules using the lblabc code [70] with inputs from the +HITRAN2016 line list [71]. To speed up the retrieval calculations, absorption +coefficients are produced for an isothermal temperature of 550 K and resam- +pled to a fixed wavenumber resolution of 0.25 cm−1. Our tests that relaxed +these assumptions on the line absorption coefficients resulted in negligible +errors relative to the measurement uncertainties. +Our nominal smarter retrieval setup uses 9 free parameters that include +the log10volume mixing ratios for the molecules H2O, CH4, CO2, and CO, + +Springer Nature 2021 LATEX template +24 +Lustig-Yaeger & Fu et al. — LHS 475b with JWST +along with the reference radius of the planet (Rp,ref) at the spectral con- +tinuum (which is interpreted as either a cloud-top or the solid-surface), the +atmospheric pressure at the reference radius (P0), the isothermal temperature +(T0), the planet mass (Mp), and the mean molecular weight (MMW) of the +bulk atmospheric composition (µ). We impose uninformative flat priors on the +gases within the interval U(−12, 0) log10(VMR), the radius within ±10% of +the white light radius constraint, the apparent surface pressure P0 ∼ U(−6, 1) +log10(bar), and the isothermal temperature T0 ∼ U(200, 900) K. The total +atmospheric MMW is calculated self-consistently from the gases included in +the retrieval plus an unknown, agnostic background gas that fills the remain- +ing volume of the atmosphere after the other gases are accounted for. The +agnostic background gas has a molecular weight sampled from a flat prior dis- +tribution µ ∼ U(2.5, 50.0) g/mol. This covers a range in MMW from a low +mass solar composition mixture of H2+He to high mass, simple molecules such +as CO2 and O3. While this model construction is similar to other retrievals +that assume a known background gas such as H2+He or N2, using a flat prior +on the molecular weight of the background gas eliminates a strong implicit +prior on the total atmospheric MMW (which is strongly biased to that of the +assumed background gas). We assume that the planet possesses a rocky inte- +rior composition, as previously discussed, and sample planet masses from a +normal distribution Mp ∼ N(0.914, 0.187) M⊕. +We run smarter retrievals using the dynesty code with the standard nested +sampling algorithm [57] and fit the final coadded transmission spectrum from +the FIREFLy reduction binned to a fixed resolution of ∆λ = 10 nm. We use +600 live points and run the model until the estimated contribution to the +total evidence from the remaining prior volume drops below the threshold of +dlogz=0.075. To obtain additional posterior samples that effectively reduces +the numerical sampling errors in the final visualization of the posteriors, we +run an MCMC chain using emcee [51]. The MCMC is run with 135 walkers for +1000 steps and is initialized using points from the equally weighted dynesty +posterior. The resulting MCMC chain requires no iterations to be removed +for the burn-in and the emcee posteriors agree with the dynesty posteriors to +within the finite sampling uncertainty. Our final posteriors are constructed by +combining the list of samples obtained with the two inference codes. +Figure 12 shows the posterior PDFs from our nominal smarter retrieval +along with an overview of spectral models sampled from the posterior. +Although a large swath of terrestrial atmospheric parameter space remains +allowed given the observations, a non-negligible subset of models are disfa- +vored and provide us with insights into the nature of the planet. Figure 3 +highlights these constraints and includes the isothermal scale height for atmo- +spheres contained in the posterior distribution. Scale height calculations are +performed in a post-processing step after running the retrieval to compress the +degeneracies between planet gravity (fit in terms of planet radius and mass), +isothermal temperature, and mean molecular weight into a single representa- +tive value for the atmosphere’s vertical extensiveness. In general, atmospheric + +Springer Nature 2021 LATEX template +Lustig-Yaeger & Fu et al. — LHS 475b with JWST +25 +Fig. 12 Corner plot showing the 1D and 2D marginalized posterior probability distribution +for a subset of the smarter model parameters. The upper right axis shows the 1σ and 3σ +envelope around the median retrieved spectrum, which corresponds to the multidimensional +posterior PDF projected onto the observed spectrum. Disfavored atmospheres are thick +(large P0), hot (large T0), and composed primarily of light molecules (low µ). +characteristics that tend towards increasing the scale height of the atmosphere +are disfavored, including high temperatures and low mean molecular weight +bulk atmospheric compositions, particularly for atmospheres with apparent +surface pressures ≳10 mbar (1000 Pa). Conversely, compact atmospheres with +small scale heights—due to high mean molecular weight molecules or cool +temperatures—are allowed across the full range of apparent surface pressures +explored. These two general characteristics yield a preference for extremely low +apparent surface pressures of ∼1 µbar. High abundances of CO2 and CH4 can +be ruled out in the thick and extended atmospheric scenarios. While CH4 can +be ruled out in a relatively low MMW CH4-dominated atmosphere (16 g/mol), + +NIRSpec G395H +-1.70 +1300 +Atmospheric Model (± lo, 3o) + 218.27; x2 += 0.97 +transit depth [ppm] +1200 +1100 +1000 +[] L +900 +095 +800 +3.5 +4.0 +4.5 +5.0 +[g/mol] +15.86 +wavelength [μm] +[g/ mol] +3. +-4.79 +H20 +-2.87 +92- +001 +5.0 +8 +To [K] +CH4 +Po [Pa] +μ [g/mol] +H20 +CO2 +COSpringer Nature 2021 LATEX template +26 +Lustig-Yaeger & Fu et al. — LHS 475b with JWST +CO2 is more difficult to rule out in a heavier CO2-dominated atmosphere (44 +g/mol). +The H2O marginalized posterior shows a slight uptick towards large VMRs +due to a small rise in the spectrum at the blue end (<3 µm) where there is +a H2O band. However, we caution that since a flat line model fit provides a +χ2 ≈ 1, the retrieval is inherently overfitting and cannot lead to a statistically +significant detection of molecular absorption from these data. To emphasize +this point we fit the spectrum with a generalized Gaussian model (plus a flat +transit depth component) [e.g. 42] as a minimally parametric stand-in model +for any molecular absorbers not included in the retrieval. This model recognizes +the same blue end slope in its maximum likelihood solution, but is disfavored +relative to the best fitting flat line at 3.1σ, further indicating that the “feature” +is consistent with noise. +We also run a series of smarter retrieval models with the same setup as +previously described except with single gas compositions. From the posterior +distributions, we derive the maximum size of molecular absorption features +such that any larger and they would have been detected in the spectrum. At +3σ confidence, we rule out H2O absorption features larger than 61 ppm at 2.8 +µm, CH4 features larger than 38 ppm at 3.3 µm, CO2 features larger than 49 +ppm at 4.3 µm, and CO features larger than 62 ppm at 4.6 µm. +Retrievals with POSEIDON +POSEIDON [72] is an atmospheric retrieval code that has been widely applied +to interpret transmission spectra of giant exoplanets. POSEIDON also supports +retrievals of terrestrial exoplanets [73, 74], which we here apply to LHS 475b’s +transmission spectrum. The most up-to-date description of POSEIDON’s radia- +tive transfer technique, forward atmospheric model, and opacity sources is +contained in [75]. We explore the range of possible atmospheres for LHS 475b +using the nested sampling algorithm PyMultiNest [76, 77]. +We employ a 9-parameter POSEIDON retrieval configuration. We compute +transmission spectra at a spectral resolution of R = 20,000 from 2.6–5.3 µm +(using cross sections resampled from a high-resolution wavenumber grid with +0.01 cm−1 spacing). Our model atmospheres cover 10−7–10 bar with 100 layers +spaced uniformly in log-pressure. We assume 1D plane-parallel atmospheres +with an isothermal pressure-temperature profile, uniform-in-altitude gas vol- +ume mixing ratios, and that hydrostatic equilibrium and the ideal gas law hold +throughout the atmosphere. The stellar radius is fixed to Rs = 0.279 R⊙. The +atmospheric structure and composition are thus described by 7 quantities: the +isothermal temperature, T, the atmospheric radius at the 1 bar apparent sur- +face pressure level, Rp, ref, and the volume mixing ratios of H2, H2O, CH4, CO2, +and CO. We prescribe N2 as a spectrally inactive filler gas, which allows the +mean molecular weight to vary in a similar manner to the smarter retrievals, +except bounded within the simplex of gas weights included in the model. We +also fit for the pressure of an opaque surface (or cloud), Psurf, and the gravita- +tional field strength at the pressure level corresponding to the observed planet + +Springer Nature 2021 LATEX template +Lustig-Yaeger & Fu et al. — LHS 475b with JWST +27 +Fig. 13 Retrieved volume mixing ratios from the POSEIDON retrievals of LHS 475b’s +transmission spectrum. Two retrievals with different prior treatments for the atmospheric +composition are overplotted: centered log-ratio (CLR) transformed abundances with a pri- +ori unknown composition (green); and log-uniform abundances assuming an N2-dominated +atmosphere (orange). Statistical 2σ upper and lower limits are annotated (or ‘N/A’ if uncon- +strained). Both retrievals rule out H2-dominated atmospheres. The log-uniform retrieval +finds upper limits on H2O, CH4, CO2, and CO due to the assumption that N2 dominates +the atmosphere, while the agnostic CLR treatment does not find upper limits for their abun- +dances. For clarity in viewing upper limits, we switch from a logarithmic to linear x-axis at a +mixing ratio of 10%. The probability densities for the linear histogram bins are renormalized +to match the probability density of the nearest logarithmic bin left of the 10% boundary. +radius (r = 0.991 R⊕), g. Our priors for the non-mixing ratio parameters are +as follows: T ∼ U (200 K, 900 K), Rp, ref ∼ U (0.9 Rp, 1.1 Rp), log10 Psurf ∼ U +(-7, 1) (units of bar), and log10 g ∼ N (2.960, 0.0992) (units of cm s−2). The +Gaussian prior on log10 g arises from error propagation from the uncertainties +on Rp and Mp — with the latter uncertainty assuming the same rocky interior +assumption as the other models. We use 4,000 PyMultiNest live points during +each retrieval. +We explore two distinct prior treatments for the atmospheric gas mix- +ing ratios during our POSEIDON retrievals. Our first approach parameterizes +the mixing ratios of H2, H2O, CH4, CO2, and CO with priors uniform-in- +the-logarithm, log10 Xi ∼ U (-12, 0), with the remainder of the atmosphere +filled with N2 (XN2 = 1 − � +i Xi). Any samples requiring negative N2 mix- +ing ratios are rejected. This ‘log-uniform’ mixing ratio prior is the standard +method used for giant exoplanet retrievals, albeit with H2 + He assumed as +the filler gas. However, this approach implicitly places a strong prior favouring +high abundances for the filler gas [78]. For small planets such as LHS 475b, +where we do not know a priori which gas dominates the atmosphere, one + +2α limits +2o limits +2o limits +N/A +H2 < 27% +N/A +N² > 37% +H0 < 19% +H2 < 12% +CLR Prior +log-uniform Prior +10-610-410-2 0.2 +1.010-610-410-2 0.20.4 +0.4 +0.6 +0.8 +1.0 +10-610-410-2 0.2　0.4　0.6 +0.8 +0.60.81.0 +N2 +H2 +H20 +2o limits +2o limits +2o limits +Probability density +N/A +N/A +N/A +CH4 < 9% +CO2 < 14% +CO < 44% +10-610-410-2 0.2 +10-610-410-2 0.2 +10-610-410-2 0.2 +0.4 +0.6 +0.8 +1.0 +0.4 +0.6 +0.8 +1.0 +0.4 +0.6 +0.8 +1.0 +CH4 +CO2 +COSpringer Nature 2021 LATEX template +28 +Lustig-Yaeger & Fu et al. — LHS 475b with JWST +may prefer an agnostic prior that treats all n gases equally. Instead of the +agnostic background gas employed by smarter to resolve this assumption, +our second approach with POSEIDON uses the centred log-ratio transformation +(CLR) [79] of the mixing ratios as free parameters: ξi = ln(Xi/g(X)), where +g(X) = +��n +j=0 Xj +�1/n +. For n = 6 gases with a minimum mixing ratio of +Xmin = 10−12, we ascribe a uniform prior on the 5 CLR variables: ξi ∼ U +(-20.996, 22.105) — for our POSEIDON model, these correspond to H2, H2O, +CH4, CO2, and CO. The upper limit corresponds to the ith gas (i = 1...5) +dominating the atmosphere and all other gases having Xj̸=i = Xmin, while the +lower limit corresponds to Xi = Xmin and the other gases equally filling the +remainder of the atmosphere. Since �n +i=0 ξi = 0 (which automatically ensures +�n +i=0 Xi = 1), we use a numerical rejection scheme to ensure that ξ0 (corre- +sponding here to N2) falls within the allowed prior range for the other ξi. The +results for the CLR approach are permutation invariant, so switching which +gas corresponds to i = 0 does not alter the results. +We find that the derived constraints on LHS 475b’s atmosphere are sen- +sitive to the choice of mixing ratio prior. Figure 13 compares the retrieved +abundances from POSEIDON for the CLR and log-uniform mixing ratio pri- +ors. Both approaches rule out H2 dominated atmospheres: log (H2) < 27% for +CLR priors vs. log (H2) < 12% for log-uniform priors (both 2σ upper limits). +However, the log-uniform retrieval also infers upper limits on the abundances +of H2O, CH4, CO2, and CO — ranging from < 9% to < 44% — which arise +from the built in prior bias towards N2 being the background gas. The CLR +prior, in contrast, recognizes that these heavier gases all provide reasonable +explanations for LHS 475b’s flat transmission spectrum due to their high mean +molecular weight — consistent with the smarter retrieval which also ruled out +low mean molecular weights. However, even with the CLR prior, certain atmo- +spheric scenarios are still disfavored. By examining Figure 14, which shows the +full POSEIDON posterior for the CLR prior, one can see that CH4 dominated +atmospheres with surface pressures ≳ 10 mbar are ruled out to 3σ confidence. +In other words, our retrieval accounting for the a priori unknown background +gas confirms the result from our forward modelling analysis that thick, pure +CH4 atmospheres with Psurf ≥ 1 bar are strongly ruled by our LHS 475b +transmission spectrum. +Acknowledgments. +Data Availability: +The data used in this paper are from the JWST Cycle 1 General Observer +program 1981 and are publicly available on the Mikulski Archive for Space +Telescopes (https://mast.stsci.edu). Fully reduced data products from this +paper will posted on the Zenodo long term public archive upon acceptance. + +Springer Nature 2021 LATEX template +Lustig-Yaeger & Fu et al. — LHS 475b with JWST +29 +Rp, ref = 0.97+0.01 +−0.02 +2.70 +2.85 +3.00 +3.15 +3.30 +log g +log g = 2.98+0.09 +−0.09 +6 +4 +2 +0 +log Psurf +log Psurf = −3.96+2.70 +−2.05 +300 +450 +600 +750 +900 +T +T = 393+264 +−138 +10.0 +7.5 +5.0 +2.5 +0.0 +log H2 +log H2 = −6.20+3.89 +−3.75 +10.0 +7.5 +5.0 +2.5 +0.0 +log H2O +log H2O = −5.21+4.67 +−4.46 +10.0 +7.5 +5.0 +2.5 +0.0 +log CH4 +log CH4 = −5.88+4.04 +−3.96 +10.0 +7.5 +5.0 +2.5 +0.0 +log CO2 +log CO2 = −5.85+5.62 +−4.10 +0.90 +0.95 +1.00 +1.05 +Rp, ref +10.0 +7.5 +5.0 +2.5 +0.0 +log CO +2.70 +2.85 +3.00 +3.15 +3.30 +log g +6 +4 +2 +0 +log Psurf +300 +450 +600 +750 +900 +T +10.0 +7.5 +5.0 +2.5 +0.0 +log H2 +10.0 +7.5 +5.0 +2.5 +0.0 +log H2O +10.0 +7.5 +5.0 +2.5 +0.0 +log CH4 +10.0 +7.5 +5.0 +2.5 +0.0 +log CO2 +10.0 +7.5 +5.0 +2.5 +0.0 +log CO +log CO = −2.00+2.00 +−6.40 +LHS 475b +Fig. 14 Corner plot showing the 1D and 2D marginalized posterior probability distribu- +tions from the POSEIDON retrieval using CLR mixing ratio parameters. The units are: Rp, ref +(R⊕), g (cm s−2), Psurf (bar), and T (K). The inset shows the corresponding retrieved +transmission spectrum model (1σ and 2σ confidence regions) compared to the NIRSpec +G395H observations. The solution rules out H2-dominated atmospheres (to > 5σ) and thick +atmospheres (Psurf ≳ 10 mbar) dominated by CH4 (to 3σ). +Code Availability: +The codes used throughout this work for data analysis, atmospheric mod- +eling, and manuscript preparation are as follows: Astropy [80, 81], Batman +[50], CHIMERA [61, 62], Dynesty [56], emcee [51], Eureka! 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Lothringer14, Zafar Rustamkulov9 and Hannah R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Wakeford15 1Johns Hopkins APL, Laurel, MD, 20723, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 2Department of Physics & Astronomy, Johns Hopkins University, Baltimore, MD, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 3Center for Astrophysics | Harvard & Smithsonian, 60 Garden St, Cambridge, MA 02138, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 4Lunar and Planetary Laboratory, University of Arizona, Tucson, AZ, 85721, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 5University of Maryland, Baltimore County, MD 21250, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 6NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 7Department of Astronomy, University of Michigan, Ann Arbor, MI, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 8NHFP Sagan Fellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 9Department of Earth & Planetary Sciences, Johns Hopkins University, Baltimore, MD, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 10Space Telescope Science Institute, Baltimore, MD 21218, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 11Carnegie Earth & Planets Laboratory, Washington, DC, 20015, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 12NASA Ames Research Center, Moffett Field, CA, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 13Department of Physics, Imperial College London, Prince Consort Road, London, SW7 2AZ, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='04191v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='EP] 10 Jan 2023 Springer Nature 2021 LATEX template 2 Lustig-Yaeger & Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' — LHS 475b with JWST 14Department of Physics, Utah Valley University, Orem, UT, 84058 USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 15School of Physics, HH Wills Physics Laboratory, University of Bristol, Bristol, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Corresponding author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' E-mail(s): jacob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='lustig-yaeger@jhuapl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' guangweifu@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Abstract The critical first step in the search for life on exoplanets over the next decade [1, 2] is to determine whether rocky planets transiting small M- dwarf stars possess atmospheres [3, 4] and, if so, what processes sculpt them over time [5–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Because of its broad wavelength coverage and improved resolution compared to previous methods, spectroscopy with JWST offers a new capability to detect and characterize the atmo- spheres of Earth-sized, M-dwarf planets [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Here we use JWST to independently validate the discovery of LHS 475b [10], a warm (586 K), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='99 Earth-radius exoplanet, interior to the habitable zone, and report a precise 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='9 − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='3 µm transmission spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' With two transit observations, we rule out primordial hydrogen-dominated and cloudless pure methane atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Thus far, the featureless transmission spec- trum remains consistent with a planet that has a high-altitude cloud deck (similar to Venus), a tenuous atmosphere (similar to Mars), or no appreciable atmosphere at all (akin to Mercury).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' There are no signs of stellar contamination due to spots or faculae [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Our observations demonstrate that JWST has the requisite sensitivity to constrain the secondary atmospheres of terrestrial exoplanets with absorption fea- tures < 50 ppm, and that our current atmospheric constraints speak to the nature of the planet itself, rather than instrumental limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Keywords: JWST, Terrestrial Exoplanet Atmospheres, Transmission Spectroscopy Springer Nature 2021 LATEX template Lustig-Yaeger & Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' — LHS 475b with JWST 3 The search for atmospheres on rocky exoplanets has only just begun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Prior constraints on the presence of terrestrial exoplanet atmospheres using the Hub- ble Space Telescope (HST) and the Spitzer Space Telescope (Spitzer) have succeeded in ruling out primordial H2/He atmospheres [12–16] that would pro- duce large and detectable absorption features in a transmission spectrum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' thick atmospheres that would produce shallow infrared secondary eclipse depths [17];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' and a tentative detection of a terrestrial atmosphere [18] that remains controversial [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' JWST is expected to break new ground in the search for atmospheres on rocky exoplanets that transit nearby M dwarfs [9, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' However, theoretical modeling work predicts a tumultuous stellar environment in these compact M-dwarf systems [22, 23] that raises the critical question of whether or not small, rocky exoplanets can maintain thick and detectable atmospheres in the face of significant atmospheric loss processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We observed two transits of LHS 475b (previously the planet candidate TOI 910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='01) on 31 August 2022 and 4 September 2022 with JWST’s Near InfraRed Spectrograph (NIRSpec) [24, 25] G395H instrument mode as part of the JWST Cycle 1 Guest Observing (GO) Program 1981 (PI: K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Steven- son).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' This mode covers wavelengths 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='87 − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='27 µm and has a native spectral resolving power of R = λ/∆λ ≈ 2700.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We used the Bright Object Time Series (BOTS) mode with the NRSRAPID readout pattern, S1600A1 slit, and the SUB2048 subarray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Each time-series observation lasted a total of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='9 hours, which captured the 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='98 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='04 minute transit that occurred during this window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' This resulted in approximately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='75 hours and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5 hours of stellar baseline before and after transit, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The Methods contains additional information on the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We selected the LHS 475 system as one of several nearby M-dwarf systems with known or candidate rocky planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Prior to validation, LHS 475b was classified as a planet candidate first identified in Sector 12 by the Transiting Exoplanet Survey Satellite (TESS) [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' TESS observed subsequent transits of the planet in Sectors 13, 27, and 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' LHS 475b transits a 3300 K, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='2789 R⊙ M3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5V dwarf star on a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='029-day orbital period [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' This planet is likely to be tidally locked, with a permanent dayside facing its host star [28], and an equilibrium temperature of 586 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We validated the discovery of LHS 475b by eliminating both instrumental and astrophysical false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' JWST detected two transit signals at the predicted times that are consistent in depth and duration with the 45 TESS transits (978 ± 73 ppm, 42 ± 13 minutes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Archival DSS images from 1999 rule out the possibility of a background transiting star-planet system or an eclipsing binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' LHS 475 is a high-proper-motion star (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='28 arcsec/year) and no flux sources were identified in the archival data along its path from 1999 to 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' See the Methods for more details about the archival imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' In parallel with this work, Ment et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' (in prep) performed an independent validation of LHS 475b using ground-based follow-up observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We reduced the JWST data using three independent pipelines—Eureka!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' [29], FIREFLy [30], and Tiberius [31–33]—that yielded consistent results Springer Nature 2021 LATEX template 4 Lustig-Yaeger & Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' — LHS 475b with JWST Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 1 White light curves from both LHS 475b visits using the FIREFLy reduction (see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' For each visit, we combined data from both the NRS1 and NRS2 detectors into a single white light curve and applied a vertical offset for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' A transit model and visit- long linear trend are sufficient to fit the raw white light curves (panel a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Residuals from the best fit (panel b) highlight a small, ∼15-minute ramp at the start of each visit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Residuals are shown on the same y-scale as panel a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Both histograms of the residuals (panel c) are Gaussian distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' (within 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='1σ, see Methods for details on each analysis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' For our final inter- pretation, we utilize results from the FIREFLy pipeline as it is the most representative of the three reductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We generated white light curves across the full G395H wavelength range covered by the two detectors, NRS1 from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='884 - 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='720 µm and NRS2 from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='820 - 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='177 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The planet transits are clearly visible in the raw white light curves (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We note the presence of a small ramp at the start of the observation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' no additional structure is seen in the residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' There is also no evidence of starspot crossings during the tran- sits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Our joint fit to the white light curves gives a planet-to-star radius ratio of Rp/Rs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='03257 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='00014, a mid-transit time of T0 = 59822.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='8762593 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='000026 BMJDTDB, an orbital period of P = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='029088 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='000006 days, a ratio of the semi-major axis to the stellar radius of a/Rs = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='87235 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='472, and an inclination of i = 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='194◦ ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='39◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Therefore, LHS 475b has a radius of Rp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='05 R⊕ (6319 ± 318 km).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Although LHS 475b’s mass has not been measured, assuming an interior composition that is consistent with the small, rocky M-dwarf exoplanet population [34], we estimate a planet mass of Mp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='914 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='187 M⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We fitted the spectroscopic light curves at the detector’s pixel resolution to derive wavelength-dependent transit depths independently for the first and second visit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The orbital parameters were fixed to the values from the joint white light curve analysis, leaving the planet-to-star radius ratio, linear tem- poral slope, and a constant offset as free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We adopted stellar limb darkening from a 3D stellar model grid [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We then performed a weighted average of spectra from the two visits and binned the combined native-pixel- resolution transit depths into 56 points (R ≈ 100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Our co-added and binned transmission spectrum is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Data + Model Residuals offset 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='002 Visit + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='001 relative flux 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='000 Visit 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='999 (a) (b) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5 time from mid-transit fhoursSpringer Nature 2021 LATEX template Lustig-Yaeger & Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' — LHS 475b with JWST 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 2 Final, binned spectrum (black points) compared to models (coloured lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Top: Our data strongly (> 10σ) rule out hydrogen-dominated atmospheres with compositions from 1× – 100× solar metallicity, with reduced-χ2s reported in the legend for each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The blue shaded bar highlights the region detailed in the bottom panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Bottom: Our data also rule out, though to lower (2–5σ) significance, high mean molecular weight compositions of 1000× solar metallicity or a pure methane atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We weakly disfavor a pure water atmosphere or an Earth composition atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The data are consistent with a pure carbon dioxide atmosphere or that of an airless body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Each model is plotted relative to the mean transit depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The observed transmission spectrum is featureless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' A flat line, representa- tive of an airless-body or high mean molecular weight (MMW) atmosphere, fitted to the binned data produces a reduced χ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' No evidence is seen for stellar contamination from unocculted cool spots or hot faculae on the stellar disk [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=', 11, 15, see Methods].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Despite the featureless spectrum, the precision is sufficiently high to rule out (> 5σ) several archetypal atmospheric composi- tions, including primordial hydrogen-helium atmospheres with less than 100 × solar metallicity, as well as pure CH4 atmospheres ≥ 1 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We can specifically rule out this pure CH4 atmospheres due to the low mass of the CH4 molecule and the presence of the strong 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='3 µm CH4 band in the G395H bandpass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Other Springer Nature 2021 LATEX template 6 Lustig-Yaeger & Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' — LHS 475b with JWST secondary atmospheres are more challenging to rule out and distinguish from one another, however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We only weakly disfavor (at ≳1σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' [36]) 1000 × solar metallicity, pure steam, or warm Earth-like atmospheric compositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Both a pure ≥ 1 bar carbon dioxide atmosphere or no atmosphere are favored yet are statistically indistinguishable from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' In the context of Solar Sys- tem terrestrial archetype atmospheres (see Methods Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 11), we also weakly disfavor (≳1σ) clear, warm Venus-like and Titan-like atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We cannot statistically distinguish between a thin Mars-like atmosphere, a hazy Titan-like atmosphere, and a cloudy Venus, which are all consistent with the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Following previous analyses [37], we performed Bayesian retrievals to better explore the range of atmospheres that remain consistent with our spectro- scopic measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We assumed a five component atmospheric composition consisting of the four most common and spectroscopically active molecules (H2O, CO2, CH4, and CO) in the Solar System terrestrial atmospheres, plus an unspecified gas that constitutes the bulk atmospheric composition but is spectroscopically inactive at these wavelengths [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We allow the mean molecular weight of the bulk gas to vary between 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5 g/mol and 50 g/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Since a solid planetary surface and an optically thick gray cloud deck are indistinguishable in the transit spectrum, we fit for the apparent surface pres- sure (the pressure of an opaque surface above which the atmosphere extends).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We marginalize over the aforementioned planet mass, which is assumed to be consistent with a rocky interior composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The vertical extent of the atmo- sphere is dictated by the scale height, which is implicitly controlled by varying the atmospheric temperature, mean molecular weight, and planet gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' See Figure 3 for the retrieval results summarizing the range of allowed atmo- spheres given our data and highlighting the degeneracies that persist among the remaining atmospheric possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' If the planet has an atmosphere, it is likely to be a high mean molecu- lar weight secondary atmosphere that is tenuous (Mars-like) or cloudy/hazy (Venus-like or Titan-like).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Compact atmospheres with small scale heights are preferred across the full range of apparent surface pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' High mean molec- ular weight atmospheres dominated by species heavier than 40 g/mol, like CO2 or Argon, can be thicker while maintaining relatively flat spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Atmospheric characteristics that increase the scale height, including high temperatures and low mean molecular weight bulk atmospheric compositions, tend to be dis- favored, particularly for apparent surface pressures ≳10 mbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Models with scale heights larger than 20 km strongly skew towards the low mean molecular weight atmospheres (µ < 10 g/mol), make up ∼50% of the lowest apparent surface pressure samples, and tend to have low abundances of all absorbing molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' These extended atmospheres with low apparent surface pressures are unlikely to form clouds or hazes at such high altitudes and are the most sus- ceptible to atmospheric loss, making them less physically plausible scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Although LHS 475 is typical of low-activity M dwarfs in the solar neighbor- hood [39], atmospheric escape processes are still a concern for a primordial Springer Nature 2021 LATEX template Lustig-Yaeger & Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' — LHS 475b with JWST 7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 3 Retrieval results showing preferred atmospheric properties for models containing H2O, CO2, CH4, and CO, plus a variable bulk gas composition for LHS 475b given the trans- mission spectrum measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Darker color shading indicates higher relative posterior probability density as a function of the apparent surface pressure (P0), molecular weight (µ) of the bulk atmospheric composition (left), and isothermal scale height (H, right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Dashed contours denote the 1σ (white), 2σ (gray), and 3σ (black) Bayesian credible regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The red arrow depicts how the Jeans escape flux depends on the scale height and emphasizes the region of the parameter space that is more susceptible to atmospheric escape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' If the planet possesses an atmosphere with at least 1 ppm CO2 or CH4, then the models prefer high mean molecular weight, compact atmospheres (µ > 20 g/mol at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='2σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' H < 25 km at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='2σ) with low apparent surface pressures (P0 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='01 bar at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='3σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' P0 < 1 bar at 2σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' These scenarios correspond to either a tenuous or cloudy secondary atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' extended atmosphere, and if LHS 475 b is indeed airless, such processes would likely constitute the primary reason for this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Our two transit observations demonstrate that JWST has the sensitivity to detect and constrain the secondary atmospheres of terrestrial exoplanets, and therefore our atmospheric non-detection reflects the nature of the target itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We place a 3σ constraint on the maximum size of absorption features in our spectrum at 61 ppm for H2O at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='8 µm, 38 ppm for CH4 at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='3 µm, 49 ppm for CO2 at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='3 µm, and 62 ppm for CO at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='6 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' These constraints demonstrate JWST’s sensitivity to absorption features smaller than 50 ppm for an Earth-sized exoplanet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We find no indication of a noise floor down to 5 ppm (See Methods Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' These are critical benchmarks for forthcom- ing rocky exoplanet observations with JWST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Furthermore, our non-detection of starspot crossings during transit and the lack of stellar contamination in the transmission spectrum are promising signs in this initial reconnaissance of LHS 475b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' These findings indicate that additional transit observations of LHS 475b with JWST are likely to tighten the constraints on a possible atmo- sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' A third transit of LHS 475b is scheduled as part of this program (GO 1981) in 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' An alternative path to break the degeneracy between a cloudy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='planet and an airless body is to obtain thermal emission measurements of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='LHS 475b during secondary eclipse because an airless body is expected to be ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='several hundred Kelvin hotter than a cloudy world and will therefore produce ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='10-6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='Titan (haze-top) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='increasing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='atmospheric ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='escape ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=')-4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='Venus (cloud-top) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='Farth (clear) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='Titan (clear) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='Venus (clear) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='40 ' metadata={'source': 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template ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='Lustig-Yaeger & Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' — LHS 475b with JWST large and detectable eclipse depths at JWST’s MIRI wavelengths [4, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Our findings only skim the surface of what is possible with JWST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Acknowledgements This work is based in part on observations made with the NASA/ESA/CSA JWST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The data were obtained from the Mikulski Archive for Space Telescopes at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=', under NASA contract NAS 5-03127 for JWST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' These observations are associated with program #1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Support for program #1981 was provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=', under NASA contract NAS 5-03127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Springer Nature 2021 LATEX template Lustig-Yaeger & Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' — LHS 475b with JWST 9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 4 2D light curves of LHS 475b as a function of time and wavelength for the first visit, measured with NIRSpec/G395H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The horizontal stripe down the middle of each panel corresponds to the gap between the NRS1 and NRS2 detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Left: data normalized by the median stellar spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Middle: Maximum probability transit models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Right: Residuals from the model fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Note, the Eureka!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' and FIREFLy reductions trim more blue columns from NRS1 where there is minimal throughput than the Tiberius pipeline, which accounts for the regions without data in those reductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Similarly, the Eureka!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' reduction also trims off the initial ramp that can be seen in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 1 Data Analysis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='1 Observations We observed two transits of LHS 475b with the NIRSpec G395H grating covering the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='87–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='14 µm wavelength range split over the NRS1 and NRS2 detectors, with a detector gap between 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='72 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='82 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The first transit was observed on the 31 August 2022 18:48 UTC and the second on 4 September 2022 20:09 UTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Each visit lasted 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='4 hours in total with 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='9 hours of expo- sure time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Both transits were executed with the same observing settings, using the Bright Object Time Series (BOTS) mode with the NRSRAPID readout pattern, S1600A1 slit, and the SUB2048 subarray from NIRSpec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We obtained a total of 1158 integrations per visit, with 9 groups per integration and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='902 seconds per group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We extracted and analyzed the data from each visit independently with the Eureka!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=', FIREFLy and Tiberius pipelines as described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 2D lightcurves, models, and residuals from the three reductions are shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Data Model Residuals 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0015 Eureka!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0010 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 - + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0005 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 FIREFLY wavelength residuals 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0000 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 +-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0005 5.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='1 Eureka!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Eureka!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' [29] is an end-to-end analysis pipeline for time series observations (TSOs) of exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Eureka!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' serves as a wrapper for stages 1 and 2 of the jwst pipeline [41], allowing the user to specify which steps are run in addition to custom modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' In later stages, Eureka!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' performs spectroscopic extraction, light curve generation, and light curve fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' In this work, we apply Eureka!’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='s custom group-level background subtrac- tion (GLBS) in stage 1 prior to ramp fitting to remove 1/f noise which has been found to impact the accuracy of ramp fits for data with a small numbers of groups up the ramp [30, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Due to G395H’s curved trace we first identify the center of the trace, then mask all pixels within an aperture of 8 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' All remaining pixels in a given column (cross-dispersion direction) were used to calculate a median background/noise level for that column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We skip the jump step detection, otherwise running all standard stage 1 steps for TSOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' In stage 2, we skip the flat field step (at the time of writing, only pre-flight flat fields were available, which are insufficient for the precision we require and adds significant noise to the data) and the photom step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Because we are interested in relative flux measurements, we do not require the absolute flux calibration provided by these two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' A second round of background subtraction is done in stage 3 to capture any remaining background or 1/f noise, using pixels more than 9 pixels away from the center of the trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The spectrum is extracted with an aperture of 5 pixels for NRS1 and 4 pixels for NRS2 using median frame optimal spectral extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' To convert from DN/s to electrons, we apply a median of the gain files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' At the time of writing only pre-flight gain files were available, which are insufficient for the precision we require and adds significant noise to the data if applied on a per-pixel basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' For NRS1 we extract only columns 800 − 2047 due to the negligible throughput outside of that region of the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' For NRS2 we extract the full dispersion direction, but note that the edges are less reliable due to the trace approaching the top or bottom of the subarray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' White light curves are generated across the full wavelength range of the extracted data: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='884 - 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='720 µm for NRS1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='820 - 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='177 µm for NRS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' For transit 1 we reach a white light precision of 112 ppm and 162 ppm for NRS1 and NRS2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' For transit 2 we reach a white light precision of 116 ppm and 149 ppm for NRS1 and NRS2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' For each transit, NRS1 and NRS2 are combined into a single white light curve prior to light curve fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We extract spectroscopic light curves at the pixel-resolution fol- lowing recommendations from [43], however we find that our GLBS routine sufficiently removes the 1/f noise in our data set, with no improvement on the final transmission spectrum precision between fitting light curves at the native pixel resolution and then binning, or binning prior to fitting (see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Figure 5 shows our spectroscopic precision compared to expected noise lev- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Bad columns are denoted by squares (flagged in both transits) or darker Springer Nature 2021 LATEX template Lustig-Yaeger & Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' — LHS 475b with JWST 11 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 5 Eureka!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' spectrophotometric precision at the native pixel resolution, compared to expected noise levels for both events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The expected noise level, as well as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='25× and 2× the expected noise are shown as grey lines, these have been smoothed to the resolution of the final transit spectrum for visualization purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Squares denote columns which are greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5× the expected noise level in both transits, dark blue circles denote columns which are greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5× the expected noise level in only one transit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' These columns are flagged and not used to generate the final transmission spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' circles (flagged in only one transit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' This corresponds to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='13% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='56% of columns in transit 1 NRS1 and NRS2, respectively, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='05% and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='10% of columns in transit 2 NRS1 and NRS2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Excluding these columns, for transit 1 we achieve a median precision of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='19× and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='23× the expected noise level for NRS1 and NRS2, respectively, while for transit 2 we achieve 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='19× and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='24× the expected noise level for NRS1 and NRS2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='2 FIREFLy We used the FIREFLy [Fast InfraRed Exoplanet Fitting for Lightcurves, 30, 44] to analyse the JWST data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We started with the uncal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='fits files and ran the jwst pipeline for stage 1 and 2 with modified steps including group-level 1/f and background subtraction and skipping the jump-step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Both changes were shown to decrease the scattering in the extracted lightcurves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' After obtaining the rateints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='fits files from the stage 2 output, we performed custom cosmic rays and bad/hot pixels corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The spectral traces are then masked in the cleaned 2D images before applying 1/f correction, which subtracts the median value of the unmasked background pixels at each column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Next, we measured the shifts of the spectral trace in x and y directions by using cross correlation in the selected 2D spectral region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The measured shifts are less than one hundredth of a pixel which illustrates the excellent pointing stability 10000+ Transit 1 NRS1 NRS2 9000 Native Pixel Resolution [wdd] 8000 7000 precision [ 6000 2x Expected Limit 5000 4000 3000 2000 Expected Precision Limit 10000 Transit 2 NRS1 NRS2 9000 Native Pixel Resolution precision [ppm] 8000 7000 6000 2x Expected Limit 5000 4000 3000 Expected Precision Limit 2000 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='2 wavelength [um]Springer Nature 2021 LATEX template 12 Lustig-Yaeger & Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' — LHS 475b with JWST of JWST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' After aligning each 2D spectrum, we determine the spectral trace by first cross correlating a Gaussian profile at each column to obtain the spectrum location in the y direction, and then fit a 4th order polynomial as a function of the x direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The spectrum is then extracted for each integration centered at the fitted spectral trace to form the light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='3 Tiberius The Tiberius pipeline is a spectral extraction and light-curve fitting code based on the LRG-BEASTS pipeline [31–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We used Tiberius on the Eureka!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' stage 1 group-level background-subtracted product, which had 1/f noise removed, to produce white and spectroscopic transit light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' First we created bad- pixel masks for NRS1 and NRS2 by manually selecting hot pixels in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' These hot pixels were combined with all pixels flagged as 3σ outliers from the background, and were interpolated over using their nearest neighboring pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We also interpolate the spatial dimension of the data on a 10x grid, which improves flux extraction at the sub-pixel level, reducing noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The spectra were then traced by fitting Gaussian functions for each column of the detectors, and then using a running median to smooth the trace centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' These centers were fit with a 4th-order polynomial, 3σ outliers were removed, and the centers were again refit with a 4th-order polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' In addition to the background subtraction already performed in the cre- ation of the stage 1 product, we perform an additional background subtraction step here to remove residual background light or remaining 1/f noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We mask from the detector a defined aperture of 4 pixels plus 6 more pixels off- set from it, and clipped 3σ outliers in the background pixels, with respect to their specific column and frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Finally, the background signal for each col- umn was subtracted from it, and the spectra were then extracted using a 4 pixel aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='3 Light Curve Fitting 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='1 Eureka!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We perform a joint fit on both white light curves to constrain the system parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Limb darkening is calculated with the ExoTic-LD pacakge [45–47] using a quadratic limb darkening [48, 49] and the 3D stellar grid from [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Stellar parameters are adopted from [35], assuming Teff = 3312 K, log(g) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='94, Fe/H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' For all light curve fits both limb darkening parameters are held constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We find that the uncertainty induced in the light curve fits by the stellar models is smaller than the uncertainty in individual transit depths, with consistent transit spectra regardless of free or fixed limb darkening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We trim the first 150 integrations prior to light curve fitting to remove a slight ramp at the beginning of the data, which can be seen in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' For all light curve fitting we consider a transit model [batman, 50] and a linear ramp in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We use emcee [51], running each chain to at least 10× the auto-correlation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The joint white light curve fit includes the planet Springer Nature 2021 LATEX template Lustig-Yaeger & Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' — LHS 475b with JWST 13 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 6 Comparison of uncertainty on planet radius derived from light curves fit at the native pixel resolution and fitting of pre-binned light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The y-axis is in parts per thousand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We find little to no difference in the uncertainty, suggesting that our 1/f correction is sufficient to address the column-column variances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' radius, orbital period, center of transit, inclination, and scaled semi-major axis as shared parameters, and independent temporal ramps for each white light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Best fit orbital parameters are given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The Eureka!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' spectroscopic fits adopt the Eureka!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' white light best-fit orbital parameters and only fit for planet radius and the linear temporal ramp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Following [43] we extract and fit our light curves at the native pixel resolution of the detectors, and later bin the data to our preferred resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' To test the robustness of our group-level 1/f noise correction, we also fit a set of pre-binned light curves and compare the resulting uncertainty on the planet radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Figure 6 shows our uncertainty on planet radius for transit 1 for both the native pixel resolution light curve fitting, and our pre-binned light curve fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We find no significant improvement by fitting the full resolution light curves, suggesting the 1/f noise has been sufficiently removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We suggest that this test should be run on all NIRSpec G395H TSOs to ensure that one has sufficiently removed the 1/f noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='2 FIREFLy The extracted light curves are first summed in the wavelength direction includ- ing both NRS1 and NRS2 to form the whitelight light curve for each visit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We then used batman [50] and emcee [51] to joint fit the whitelight lightcurves from the two visits with six free parameters including Rplanet/Rstar, a/Rstar, orbital inclination, mid-transit time for both visits, and linear temporal slope for both visits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The best-fit joint white light orbital parameters are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We fixed the limb darkening to the quadratic coefficients from the 3D stellar model in the Stagger-grid [35] interpolated at Fe/H=0, Teff=3312K and log(g)=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The orbital parameters and quadratic limb darkening coefficients are then fixed to fit the lightcurve from each wavelength column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We used the scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='optimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='curvefit function with three free parameters including linear temporal slope, constant offset and Rplanet/Rstar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Transit 1 Native Pixel Light Curve Fitting Pre-binned Light Curve Fitting O NRS1 NRS2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='2 wavelength [um]Springer Nature 2021 LATEX template 14 Lustig-Yaeger & Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' — LHS 475b with JWST Parameter Eureka!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' FIREFLy Tiberius Rp/Rs [unitless] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='032756 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='44×10−4 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='44×10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='03257 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='40×10−4 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='43×10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='032226 +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='86×10−4 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='86×10−4 T0 [BMJDT DB] 59822.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='8762805 +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='92×10−5 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='91×10−5 59822.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='8762593 +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='62×10−5 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='62×10−5 59822.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='8763396 +5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='85×10−5 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='85×10−5 Period [days] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='02908843 +5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='65×10−6 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='66×10−6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0290882 (Fixed) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='02909 (Fixed) a/Rs [unitless] 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='223 +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='62×10−1 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='37×10−1 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='87235 +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='88×10−1 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='56×10−1 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='161 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='79 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='79 i [degrees] 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='991 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='41×10−1 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='39×10−1 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='194 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='41×10−1 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='37×10−1 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='237 +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='80×10−1 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='80×10−1 Table 1 Best fit orbital parameters from white light curve fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We adopt the FIREFLy results as our system parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Eureka!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' and FIREFLy values are derived from joint fits to both white light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Tiberius parameters are derived from a weighted mean of fits to individual light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='3 Tiberius We extracted a white light curve for each detector (NRS1 and NRS2), for each transit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' These white light curves were fit independently using a Lev- enberg–Marquardt damped least squares routine with the Tiberius pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Limb darkening parameters were obtained with LDTK [52, 53] from assumed stellar parameters of Teff = 3312 K, log(g) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='94, Fe/H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0, and a quadratic limb darkening law was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The results of the white light curve fits were used to fix the transit parameters for the spectroscopic light curve fits, which were performed at pixel-level resolution using the same damped least squares routine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Since the white light curves were fit independently, the best-fit param- eters in Table 1 were obtained from a weighted average of the results of each of the four white light curve fits (weighted by flux received on each detector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='4 Final Transmission Spectrum All three independently reduced spectra from above are in agreement, showing no atmospheric features and being statistically consistent with a flat line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The findings reported in the study do not depend upon which reduction pipeline is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' To select the final transmission spectrum for model interpretation, we performed two tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The first test computed the mean absolute deviation of each spectrum relative to the averaged spectrum of the three reductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The purpose of this test was to identify the reduction that is the most representative of the three reductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The FIREFLy reduction was favored by this test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The second test computed the reduced chi-squared relative to a flat line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' This test was meant to validate the size of the error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The unbinned FIREFLy transmission spectrum had a reduced chi-squared of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5 Planet Validation The JWST detection of a transit at the same period, phase, and depth as the TESS TOI eliminates the possibility of a TESS false positive due to a telescope or instrument systematic effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' This leaves only astrophysical sources, such as a background eclipsing binary, as the remaining false positive mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Springer Nature 2021 LATEX template Lustig-Yaeger & Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' — LHS 475b with JWST 15 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 7 A 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='36” × 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='36” DSS image centered on LHS 475 taken 1999 June 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The red circle depicts the star’s J2000 position per Simbad, whereas the blue circle indicates the star’s position for the JWST observations in September of 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We see no indication of a background star at the 2022 position that could be the source of the observed transit signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' For reference, NIRSpec’s field of view is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='6×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='6 pixels on this image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Using an archival DSS image of LHS 475, we leverage the star’s high proper motion to rule out astrophysical false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The star moves 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='28 arcsec/year [54, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='3423 arcsec/year in RA, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='2303 arcsec/year in Dec;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='], which corresponds to ∼29 pixels in Figure 7 from the June 1999 DSS image to our Sep 2022 JWST observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The lack of measurable flux at LHS 475’s 2022 position enables us to rule out all scenarios involving transits within a poten- tial background system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Finally, we rule our a stellar binary companion due to the precisely measured Gaia DR3 parallax of 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='1134 mas, which corresponds to a distance of only 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='6 Implications for JWST/NIRSpec Noise Floor In the interest of exploring the effects of correlated noise and constraining the instrument noise floor, we concatenate residuals from both visits and com- pute Allan variance plots for the white and spectroscopic light curve fits (see Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Using the Eureka!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' white light curve data, we find no indication of a noise floor down to 5 ppm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' however, we identify correlated noise at timescales of < 5 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' This timescale is consistent with the thermal cycling of heaters in the ISIM Electronics Compartment, which induces small forces on the tele- scopes backplane structure [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The effect is semi-periodic, the result of several heaters cycling at different frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 200 20000 175 - 17500 150 15000 125 12500 Pixel Number 2000 10000 100 2022 7500 75 - 5000 50 2500 25 0 0 0 25 50 75 100 125 150 175 200 Pixel NumberSpringer Nature 2021 LATEX template 16 Lustig-Yaeger & Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' — LHS 475b with JWST Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 8 Allan variance plots from the white and spectroscopic light curve fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Panel (a) illus- trates that the white light curve residuals from two of the analyses exhibit some correlated noise at timescales of < 5 minutes (< 35 integrations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' This is likely due to uncorrected 1/f noise from the thermal cycling of on-board heaters [55, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' At longer timescales (> 18 minutes), the Eureka!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' pipeline returns to the expected standard error with RMS val- ues below 10 ppm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The Tiberius reduction did not sum the flux across both detectors and was not used for this noise floor analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The spectroscopic RMS values in panels (b) – (d) are more consistent with the standard error, thus confirming that the spectroscopic light curves are dominated by white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 2 Modeling With the reduced data and coadded transmission spectrum produced in the previous section, we now use a variety of models to update the state of knowl- edge on the LHS 475 system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We use archival photometry to update the LHS 475 stellar parameters and assess the impact of stellar contamination on the JWST transmission spectrum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' we use empirical mass-radius relations to estimate the planet mass given our precise radius measurement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' and we fit atmospheric models to the transmission spectrum to obtain constraints on the possible atmospheric composition of LHS 475b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 100 Normalized RMS 10 Eureka!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Median RMS Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' (1/V N) 10-2 100 101 102 Bin Size [Number of Integrations]100 Normalized RMS 10 Tiberius Median RMS Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' (1/V N) 10-2 100 101 102 Bin Size [Number of Integrations]102 RMS [ppm] 101 ppm Firefly RMS Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' (1/VN) Eureka!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' RMS Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' (1/VN) 100 100 101 102 Bin Size [Number of Integrations]100 Normalized RMS 10 Firefly Median RMS Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' (1/V N) 10-2 100 101 102 Bin Size [Number of Integrations]Springer Nature 2021 LATEX template Lustig-Yaeger & Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' — LHS 475b with JWST 17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='1 Stellar Modeling and Transit Light Source Contamination We use PHOENIX spectra guided by archival photometry1 from the VizieR Pho- tometry Viewer2 to improve constraints on the stellar parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Effective temperature (Teff) is a primary driver of spectral shape and so we are able to refine estimates by matching models to the observations at visible and near- IR wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' To do this, we computed a grid of synthetic spectra following similar procedures to those outlined in [53] with Teff = 3200 – 3400 K (∆T = 10 K), log(g) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5 – 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='2 dex (∆log(g) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='1 dex), and M⋆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='262 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' This parameter space was chosen by expanding around the stellar parameters published in the Tess Input Catalog [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' For each model, we computed syn- thetic visible and near-IR photometry over the same wavelengths as the filter profiles for available measurements for LHS 475 and used a reduced χ2 test to identify the model that most closely matched the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The reduced χ2 test was conducted with both the observations and models normalized to the 2MASS J band flux density value to isolate matching the spectral shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The fully explored grid yielded χ2 ν values between 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='8 – 987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='2, with 30 mod- els returning similar values less than 50 (Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' These models have Teff = 3300 (+80, -30) K, log(g) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5 g/cm3, and M⋆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='262 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' To deter- mine the radius of the star we scaled all models with χ2 ν < 50 by R2 ⋆/dist2 until FJ2MASS,mod = FJ2MASS,obs, R⋆ = � (FJ2MASS,obs/FJ2MASS,mod) × dist2 (1) We adopted the Gaia EDR3 distance of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='481 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0065 pc [54], which returned a radius of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='2789 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0014 R⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' LHS 475 is typical of low-activity M dwarfs in the solar neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' TESS only detected two flares on LHS 475, both with energy below 1031 erg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The inferred flare rate and other activity diagnostics are all consistent with the general population of relatively inactive M dwarfs in a volume-limited sample [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Hα and He I D3 are both in absorption, not emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Ca II 8542 is relatively deep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Fitting models of the Transit Light Source (TLS) effect to the observed and coadded transmission spectrum allows us to assess the degree to which stellar contamination may impact and/or explain any characteristics of the planet’s transmission spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The TLS effect can impart slopes and features into the transmission spectrum due to differences in the spot or faculae coverage along the planet’s transit chord relative to the average coverage across the visible stellar disk [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Following the formalism of [15], we calculate the TLS 1For NIRSpec observations, the jwst pipeline requires the flat field step be run for absolute flux calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' At the time of writing, only ground or dummy frames were available for the three types of NIRSpec flat fields and for the correction applied in the photom step of the jwst Stage 2 pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' These ground and dummy frames do not provide high accuracy absolute flux calibration, so we choose to not use our new data for Stellar Modeling at this time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 2http://vizier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='unistra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='fr/vizier/sed/ Springer Nature 2021 LATEX template 18 Lustig-Yaeger & Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' — LHS 475b with JWST Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 9 Comparison of our 30 closest matching PHOENIX models (χ2 ν < 50) to all available archival photometry of LHS 475 from the VizieR Photometry Viewer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' These models have Teff = 3380 – 3320 K, log(g) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='7 – 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='7 g/cm2, M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='262 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' contamination spectrum ϵλ = (1 − fspot − ffac)Sλ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='phot + fspotSλ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='spot + ffacSλ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='fac (1 − Fspot − Ffac)Sλ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='phot + FspotSλ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='spot + FfacSλ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='fac (2) where Sλ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='phot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Sλ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='spot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' and Sλ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='fac refer to the spectrum of the stellar photo- sphere,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' spots,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' and faculae,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' fspot and ffac refer to the spot and faculae projected area covering fractions along the transit chord,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' and similarly Fspot and Ffac refer to the spot and faculae projected area covering fractions across the entire visible stellar disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Thus, ϵλ is the ratio of the stellar spec- trum along the transit chord to the spectrum of the whole disk, and a general model for how the TLS effect contaminates the observed transmission spec- trum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Given Equation 2 the observed drop in flux that we refer to as the transmission spectrum is simply ∆Fλ,obs = ϵλ �Rp Rs �2 λ (3) where the TLS contamination spectrum is multiplied by the wavelength- dependent “true” planet transmission spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We use the Dynesty nested sampling code [56] to infer posterior distributions for the TLS contamination model parameters under the assumption of a wavelength independent planet transmission spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We run the standard nested sampling algorithm [57] PHOENIX Models with x<50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='2 LHS 475 Observations 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 5000 10000 15000 20000 25000 30000 Wavelength (A)Springer Nature 2021 LATEX template Lustig-Yaeger & Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' — LHS 475b with JWST 19 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 10 Corner plot comparing the prior (orange) and posterior (dark blue) PDFs for a subset of the fitting parameters in the TLS contamination retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The flat spectrum reveals a consistent spot (and faculae) coverage along the transit chord compared to the full stellar disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' with 1000 live points until the estimated contribution to the total evidence from the remaining prior volume drops below the threshold of dlogz=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='075.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' In general, no evidence of TLS contamination is observed in the flat transmission spectrum and the TLS model readily reproduces the featureless spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Figure 10 compares the prior and posterior probability distributions for a subset of the TLS model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The inferred posterior distribution for the TLS contamination model generally reproduce the prior distributions, with the exception of the covariance between the spot (faculae) area cover- ing fraction along the transit chord compared to the spot (faculae) covering fraction on the full disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' These two convariance are constrained along a line disk spot 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='15 Posterior PDF Prior PDF disk faculae 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='8 disk faculae fraction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='6 chord spot 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='2 chord spot fraction chord faculae 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='21 chord faculae fraction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='6 disk spot disk faculae chord spot chord faculae fraction fraction fraction fractionSpringer Nature 2021 LATEX template 20 Lustig-Yaeger & Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' — LHS 475b with JWST with a slope of approximately unity, such that the ratio of spot (faculae) cov- ering fraction on the full stellar disk to spot (faculae) covering fraction along the transit chord is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='005 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='003 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='948 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' This implies that—although the exact area covered by spots (and faculae) is not well constrained—at the observed precision there is no evidence of differing spot (or faculae) cover- age along the transit chord compared to the average stellar disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We repeated the same TLS contamination retrieval with the addition of the transit depth measured by TESS in the optical (978±73 ppm) and obtained the same result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='2 Planet Radius, Mass, and Equilibrium Temperature From the constraint on the white light curve transit depth (1060 ± 9 ppm) and the stellar radius (Rs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='279 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='014 R⊙), we calculate the planet radius to be Rp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='991 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='050 R⊕ (6319 ± 318 km).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The 5% radius precision is dominated by uncertainty in the stellar radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' For reference, the preexisting radius constraint from TESS sectors 12-39 was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='70 R⊕ [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Despite the lack of a mass measurement for LHS 475b, we use three different methods to estimate the mass: 1) from the transmission spectrum [13], 2) from probabilistic mass-radius-relation [58], and 3) from probabilistic bulk density arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We use atmospheric models over a range of masses compared to the spectroscopic data from NIRSpec/G395H to infer conservative upper and lower limits of the possible mass for LHS 475b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' To do so, we employ the forward model framework discussed in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' First, we find the uppermost mass limit by finding the densest planet that could stably support a hydrogen-helium envelope and fit the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' To obtain a reduced-χ2 ≤ 1, we determine that we must consider a mass of 24 M⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Combined with the precise radius constraint, this upper limit mass results in a planetary density of 119 g/cm−3, or 6× that of pure uranium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Given this unrealistic density, we can clearly reject a hydrogen-helium atmosphere around a very dense planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' On the other hand, to find the lowest mass that is consistent with the NIRSpec data, we instead consider a planet with a very high mean molecular weight atmosphere, but from a reasonably abundant molecule – that of pure CO2 – and scale the mass down until we obtain a reduced-χ2 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Under this atmospheric assumption, we find that masses consistent with the data extend down to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='78 M⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Together this method gives us a range of masses consistent with the observed atmosphere between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='78 - 24 M⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Next, we use the mass-radius relationship gleaned from the existing pop- ulation of small M dwarf exoplanets to estimate the planet’s mass given our precise radius constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Using the Forcaster code’s probabilistic mass-radius relationship and mass prediction tool [58], we estimate the mass of LHS 475b to be Mp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='980+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='632 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='359 M⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Our third mass estimate leverages recent results on the interior bulk densi- ties among the M dwarf small planet population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Given the radius constraint for LHS 475b, the planet is consistent with the population of M dwarf plan- ets having rocky interior compositions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='28 R⊕) [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Therefore, if we assume that LHS 475b is indeed a rocky planet with a mean bulk density Springer Nature 2021 LATEX template Lustig-Yaeger & Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' — LHS 475b with JWST 21 consistent with the M dwarf rocky planet population (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='13 ρ⊕) [34], then we find the planet mass to be Mp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='914 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='187 M⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' If we consider that instead the planet were in the population of lower density water worlds (with 50% water, 50% rock interiors), then this would ultimately have ram- ifications for the scale height and water content of the atmosphere [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=', 59], that are inconsistent with the featureless transmission spectrum that we mea- sured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Therefore, it is likely that LHS 475b has a mass that is consistent with a rocky mean bulk density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' In the atmospheric models that follow, we assume the planet is consistent with the population with rocky interiors and use the corresponding mass Mp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='914 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='187 M⊕, which is consistent with our previous estimates, albeit with a tighter constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We update LHS 475b’s zero bond albedo equilibrium temperature to 586 ± 12 K (assuming uniform heat redistribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' In the limit of instant re-radiation expected from a planet with a tenuous or nonexistent atmo- sphere, the estimated day side brightness temperature is 748 ± 16 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' These updates may aid in the planning of any future secondary eclipse observations of LHS 475b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='3 Atmospheric Modeling We use atmospheric radiative transfer models to simulate the transmission spectrum of LHS 475b for comparison with our JWST observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' In the next section, forward models of single-composition end-member atmospheres and archetypal atmospheres are used to illustrate the atmospheric compositions that are consistent with our observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Then, retrieval models are used to simulate a broad range of atmospheric compositions to place constraints on key atmospheric parameters given the precise, yet featureless transmission spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='1 Forward Modeling We use the forward modeling capabilities of two different open-source atmo- spheric radiative transfer codes, PICASO [60] and CHIMERA [61, 62], to explore the plausibility of various atmospheric archetypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We compute each model atmosphere for a planet mass consistent with a rocky mean bulk density, Mp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='914 M⊕, a planetary radius of Rp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='991 R⊕, and a stellar radius Rs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='279 R⊙, and a planetary equilibrium temperature of Teq = 600 K, as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' In each case, we compare the modeled transmission spectrum to the NIRSpec/G395H data for LHS 475b from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='9 – 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='3 µm and compute the reduced-χ2 between the modeled spectrum and the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' For the CHIMERA models, we compute chemically consistent atmospheric mixing ratios for 1×, 10×, 100×, and 1000× solar metalicities, with a solar C/O ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' CHIMERA uses a preset grid of atmospheric molecular abundances along temperature-pressure profiles, metallicity, and C/O ratio generated from the NASA CEA code [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' For the temperature-pressure profile, the code uses the five-parameter, double gray, one-dimensional parametrization of [64], where Springer Nature 2021 LATEX template 22 Lustig-Yaeger & Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' — LHS 475b with JWST we input a planetary equilibrium temperature of 600 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' For these CHIMERA models, we include opacity from H2O, CH4, CO, CO2, NH3, N2, HCN, H2S, H2/He CIA [65, 66], and Rayleigh scattering from H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We consider simplistic cloudy hydrogen-dominated atmospheric models with CHIMERA by computing a cloud-top pressure for a grey absorbing cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We generate atmospheric transmission models with the correlated-k method of radiative transfer and bin the resulting model to the data before calculating our reduced-χ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' For the PICASO models, we generate simplified end-member atmospheric compositions with isothermal temperature-pressure profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We set a pressure grid which ranges from 1 µbar to 100 bar, and then set an isothermic tem- perature at the equilibrium temperature of 600 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' For the models shown in Figure 2, each atmosphere consists solely of either H2O, CO2, CH4, or as in the case of the Earth-like atmosphere, follows the atmospheric abundances of Earth above the water cold-trap, with 78% N2, 21% O2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='9% Ar, 416 ppm CO2, 524 ppm He, and 187 ppm CH4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' For the models shown in Figure 11, we generate individual pressure grids with an upper bound according to the terrestrial body’s surface pressure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=', the Earth-like model has an upper atmospheric pressure bound of 1 bar;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' the Venus-like model has an upper pres- sure bound of 90 bar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We assume isothermal temperature profiles (at 600 K) with atmospheric abundances fixed to the composition of each Solar System body above any cold trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' For the cloudy Venus and hazy Titan cases, we implement a simple grey absorbing cloud at the pressure level according to the Venus cloud-top (1 mbar) and the Titan haze-top (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='01 mbar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The opacity database is resampled to R=10,000 and is taken from [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Models are then binned to the data for reduced-χ2 comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We strongly rule out clear atmospheres of 1× to 100× solar, with reduced- χ2s ≥ 9, or over 10σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Given the mass estimate analysis above, even with the uncertain planetary mass, we are able to reject low (≤100× solar) atmo- spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' To obtain a reduced-χ2 ∼ 1 in cloudy low-metallicity atmospheres, we must insert an opaque cloud deck with cloud-top pressure between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5 and 1 µbar, which can be discarded as unrealistic given the lack of cloud-forming material at such low pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Each of these cases represents a hydrogen-rich atmosphere around a rocky, 600 K planet, which would not be stable against escape over the lifetime of the system, and thus our ability to reject them is not unexpected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' For the 1000× atmosphere, we calculate a reduced-χ2 to the data of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5, which weakly rules out this scenario to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The pure methane atmosphere is rejected with a reduced-χ2=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='3, or 5σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Both the end-member atmospheric compositions of a pure steam or Earth-like atmospheric abundances are weakly disfavored at ≳1σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' A pure 1 bar carbon dioxide atmosphere or no atmosphere at all are preferred but not statistically distinguishable from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' For the Solar System terrestrial archetype atmospheres shown in Figure 11, we also weakly disfavor (≳1σ) clear Venus, Titan, or Earth-like atmospheres, but cannot statistically distinguish between a thin Mars-like atmosphere, a hazy Springer Nature 2021 LATEX template Lustig-Yaeger & Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' — LHS 475b with JWST 23 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 Wavelength ( m) 200 150 100 50 0 50 100 150 200 Relative Transit Depth (ppm) Earth-like: 2=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='1 Mars-like: 2=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 Titan-like, clear: 2=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='3 Titan-like, hazy: 2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='95 Venus-like, clear: 2=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='1 Venus-like, cloudy: 2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='97 Mercury-like: 2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='91 JWST/NIRSpec G395H Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 11 Final, binned spectrum (black points) compared to atmospheric models with compositions of the Solar System terrestrial planets (coloured lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Our data, to weakly rule out Earth composition (blue solid), clear Titan composition (orange solid), and clear Venus composition atmospheres (yellow solid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' However, the data are all consistent within error to that of a hazy Titan composition with a haze-top at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='01 mbar (dotted orange), a cloudy Venus composition with a cloud-top at 1 mbar (dotted yellow), and a Mars composition atmosphere (red solid), as well as that of an airless body, like Mercury (grey dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Titan-like atmosphere, or a cloudy Venus, as consistent with the retrieval modeling shown in Figure 3 and discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='2 Retrieval Modeling We use two different atmospheric retrieval codes—smarter and POSEIDON—to explore the range of atmospheric properties that are consistent with, or ruled out, by LHS 475b’s transmission spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Retrievals with smarter The smarter retrieval code [68, 69] couples line-by-line radiative transfer cal- culations from the Spectral Mapping Atmospheric Radiative Transfer forward model (smart [70]) to the dynesty nested sampling Bayesian inference code [56] to retrieve planetary and atmospheric parameters that are consistent with the JWST observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We assume an isothermal temperature-pressure profile and evenly-mixed gas volume mixing ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We calculate line absorption coef- ficients for gaseous molecules using the lblabc code [70] with inputs from the HITRAN2016 line list [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' To speed up the retrieval calculations, absorption coefficients are produced for an isothermal temperature of 550 K and resam- pled to a fixed wavenumber resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='25 cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Our tests that relaxed these assumptions on the line absorption coefficients resulted in negligible errors relative to the measurement uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Our nominal smarter retrieval setup uses 9 free parameters that include the log10volume mixing ratios for the molecules H2O, CH4, CO2, and CO, Springer Nature 2021 LATEX template 24 Lustig-Yaeger & Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' — LHS 475b with JWST along with the reference radius of the planet (Rp,ref) at the spectral con- tinuum (which is interpreted as either a cloud-top or the solid-surface), the atmospheric pressure at the reference radius (P0), the isothermal temperature (T0), the planet mass (Mp), and the mean molecular weight (MMW) of the bulk atmospheric composition (µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We impose uninformative flat priors on the gases within the interval U(−12, 0) log10(VMR), the radius within ±10% of the white light radius constraint, the apparent surface pressure P0 ∼ U(−6, 1) log10(bar), and the isothermal temperature T0 ∼ U(200, 900) K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The total atmospheric MMW is calculated self-consistently from the gases included in the retrieval plus an unknown, agnostic background gas that fills the remain- ing volume of the atmosphere after the other gases are accounted for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The agnostic background gas has a molecular weight sampled from a flat prior dis- tribution µ ∼ U(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5, 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0) g/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' This covers a range in MMW from a low mass solar composition mixture of H2+He to high mass, simple molecules such as CO2 and O3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' While this model construction is similar to other retrievals that assume a known background gas such as H2+He or N2, using a flat prior on the molecular weight of the background gas eliminates a strong implicit prior on the total atmospheric MMW (which is strongly biased to that of the assumed background gas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We assume that the planet possesses a rocky inte- rior composition, as previously discussed, and sample planet masses from a normal distribution Mp ∼ N(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='914, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='187) M⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We run smarter retrievals using the dynesty code with the standard nested sampling algorithm [57] and fit the final coadded transmission spectrum from the FIREFLy reduction binned to a fixed resolution of ∆λ = 10 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We use 600 live points and run the model until the estimated contribution to the total evidence from the remaining prior volume drops below the threshold of dlogz=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='075.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' To obtain additional posterior samples that effectively reduces the numerical sampling errors in the final visualization of the posteriors, we run an MCMC chain using emcee [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The MCMC is run with 135 walkers for 1000 steps and is initialized using points from the equally weighted dynesty posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The resulting MCMC chain requires no iterations to be removed for the burn-in and the emcee posteriors agree with the dynesty posteriors to within the finite sampling uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Our final posteriors are constructed by combining the list of samples obtained with the two inference codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Figure 12 shows the posterior PDFs from our nominal smarter retrieval along with an overview of spectral models sampled from the posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Although a large swath of terrestrial atmospheric parameter space remains allowed given the observations, a non-negligible subset of models are disfa- vored and provide us with insights into the nature of the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Figure 3 highlights these constraints and includes the isothermal scale height for atmo- spheres contained in the posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Scale height calculations are performed in a post-processing step after running the retrieval to compress the degeneracies between planet gravity (fit in terms of planet radius and mass), isothermal temperature, and mean molecular weight into a single representa- tive value for the atmosphere’s vertical extensiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' In general, atmospheric Springer Nature 2021 LATEX template Lustig-Yaeger & Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' — LHS 475b with JWST 25 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 12 Corner plot showing the 1D and 2D marginalized posterior probability distribution for a subset of the smarter model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The upper right axis shows the 1σ and 3σ envelope around the median retrieved spectrum, which corresponds to the multidimensional posterior PDF projected onto the observed spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Disfavored atmospheres are thick (large P0), hot (large T0), and composed primarily of light molecules (low µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' characteristics that tend towards increasing the scale height of the atmosphere are disfavored, including high temperatures and low mean molecular weight bulk atmospheric compositions, particularly for atmospheres with apparent surface pressures ≳10 mbar (1000 Pa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Conversely, compact atmospheres with small scale heights—due to high mean molecular weight molecules or cool temperatures—are allowed across the full range of apparent surface pressures explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' These two general characteristics yield a preference for extremely low apparent surface pressures of ∼1 µbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' High abundances of CO2 and CH4 can be ruled out in the thick and extended atmospheric scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' While CH4 can be ruled out in a relatively low MMW CH4-dominated atmosphere (16 g/mol), NIRSpec G395H 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='70 1300 Atmospheric Model (± lo, 3o) 218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' x2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='97 transit depth [ppm] 1200 1100 1000 [] L 900 095 800 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 [g/mol] 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='86 wavelength [μm] [g/ mol] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='79 H20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='87 92- 001 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 8 To [K] CH4 Po [Pa] μ [g/mol] H20 CO2 COSpringer Nature 2021 LATEX template 26 Lustig-Yaeger & Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' — LHS 475b with JWST CO2 is more difficult to rule out in a heavier CO2-dominated atmosphere (44 g/mol).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The H2O marginalized posterior shows a slight uptick towards large VMRs due to a small rise in the spectrum at the blue end (<3 µm) where there is a H2O band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' However, we caution that since a flat line model fit provides a χ2 ≈ 1, the retrieval is inherently overfitting and cannot lead to a statistically significant detection of molecular absorption from these data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' To emphasize this point we fit the spectrum with a generalized Gaussian model (plus a flat transit depth component) [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 42] as a minimally parametric stand-in model for any molecular absorbers not included in the retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' This model recognizes the same blue end slope in its maximum likelihood solution, but is disfavored relative to the best fitting flat line at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='1σ, further indicating that the “feature” is consistent with noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We also run a series of smarter retrieval models with the same setup as previously described except with single gas compositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' From the posterior distributions, we derive the maximum size of molecular absorption features such that any larger and they would have been detected in the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' At 3σ confidence, we rule out H2O absorption features larger than 61 ppm at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='8 µm, CH4 features larger than 38 ppm at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='3 µm, CO2 features larger than 49 ppm at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='3 µm, and CO features larger than 62 ppm at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='6 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Retrievals with POSEIDON POSEIDON [72] is an atmospheric retrieval code that has been widely applied to interpret transmission spectra of giant exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' POSEIDON also supports retrievals of terrestrial exoplanets [73, 74], which we here apply to LHS 475b’s transmission spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The most up-to-date description of POSEIDON’s radia- tive transfer technique, forward atmospheric model, and opacity sources is contained in [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We explore the range of possible atmospheres for LHS 475b using the nested sampling algorithm PyMultiNest [76, 77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We employ a 9-parameter POSEIDON retrieval configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We compute transmission spectra at a spectral resolution of R = 20,000 from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='6–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='3 µm (using cross sections resampled from a high-resolution wavenumber grid with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='01 cm−1 spacing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Our model atmospheres cover 10−7–10 bar with 100 layers spaced uniformly in log-pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We assume 1D plane-parallel atmospheres with an isothermal pressure-temperature profile, uniform-in-altitude gas vol- ume mixing ratios, and that hydrostatic equilibrium and the ideal gas law hold throughout the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The stellar radius is fixed to Rs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='279 R⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The atmospheric structure and composition are thus described by 7 quantities: the isothermal temperature, T, the atmospheric radius at the 1 bar apparent sur- face pressure level, Rp, ref, and the volume mixing ratios of H2, H2O, CH4, CO2, and CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We prescribe N2 as a spectrally inactive filler gas, which allows the mean molecular weight to vary in a similar manner to the smarter retrievals, except bounded within the simplex of gas weights included in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We also fit for the pressure of an opaque surface (or cloud), Psurf, and the gravita- tional field strength at the pressure level corresponding to the observed planet Springer Nature 2021 LATEX template Lustig-Yaeger & Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' — LHS 475b with JWST 27 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 13 Retrieved volume mixing ratios from the POSEIDON retrievals of LHS 475b’s transmission spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Two retrievals with different prior treatments for the atmospheric composition are overplotted: centered log-ratio (CLR) transformed abundances with a pri- ori unknown composition (green);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' and log-uniform abundances assuming an N2-dominated atmosphere (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Statistical 2σ upper and lower limits are annotated (or ‘N/A’ if uncon- strained).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Both retrievals rule out H2-dominated atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The log-uniform retrieval finds upper limits on H2O, CH4, CO2, and CO due to the assumption that N2 dominates the atmosphere, while the agnostic CLR treatment does not find upper limits for their abun- dances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' For clarity in viewing upper limits, we switch from a logarithmic to linear x-axis at a mixing ratio of 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The probability densities for the linear histogram bins are renormalized to match the probability density of the nearest logarithmic bin left of the 10% boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' radius (r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='991 R⊕), g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Our priors for the non-mixing ratio parameters are as follows: T ∼ U (200 K, 900 K), Rp, ref ∼ U (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='9 Rp, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='1 Rp), log10 Psurf ∼ U (-7, 1) (units of bar), and log10 g ∼ N (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='960, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0992) (units of cm s−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The Gaussian prior on log10 g arises from error propagation from the uncertainties on Rp and Mp — with the latter uncertainty assuming the same rocky interior assumption as the other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We use 4,000 PyMultiNest live points during each retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We explore two distinct prior treatments for the atmospheric gas mix- ing ratios during our POSEIDON retrievals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Our first approach parameterizes the mixing ratios of H2, H2O, CH4, CO2, and CO with priors uniform-in- the-logarithm, log10 Xi ∼ U (-12, 0), with the remainder of the atmosphere filled with N2 (XN2 = 1 − � i Xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Any samples requiring negative N2 mix- ing ratios are rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' This ‘log-uniform’ mixing ratio prior is the standard method used for giant exoplanet retrievals, albeit with H2 + He assumed as the filler gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' However, this approach implicitly places a strong prior favouring high abundances for the filler gas [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' For small planets such as LHS 475b, where we do not know a priori which gas dominates the atmosphere, one 2α limits 2o limits 2o limits N/A H2 < 27% N/A N² > 37% H0 < 19% H2 < 12% CLR Prior log-uniform Prior 10-610-410-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='010-610-410-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 10-610-410-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 N2 H2 H20 2o limits 2o limits 2o limits Probability density N/A N/A N/A CH4 < 9% CO2 < 14% CO < 44% 10-610-410-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='2 10-610-410-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='2 10-610-410-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 CH4 CO2 COSpringer Nature 2021 LATEX template 28 Lustig-Yaeger & Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' — LHS 475b with JWST may prefer an agnostic prior that treats all n gases equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Instead of the agnostic background gas employed by smarter to resolve this assumption, our second approach with POSEIDON uses the centred log-ratio transformation (CLR) [79] of the mixing ratios as free parameters: ξi = ln(Xi/g(X)), where g(X) = ��n j=0 Xj �1/n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' For n = 6 gases with a minimum mixing ratio of Xmin = 10−12, we ascribe a uniform prior on the 5 CLR variables: ξi ∼ U (-20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='996, 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='105) — for our POSEIDON model, these correspond to H2, H2O, CH4, CO2, and CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The upper limit corresponds to the ith gas (i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5) dominating the atmosphere and all other gases having Xj̸=i = Xmin, while the lower limit corresponds to Xi = Xmin and the other gases equally filling the remainder of the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Since �n i=0 ξi = 0 (which automatically ensures �n i=0 Xi = 1), we use a numerical rejection scheme to ensure that ξ0 (corre- sponding here to N2) falls within the allowed prior range for the other ξi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The results for the CLR approach are permutation invariant, so switching which gas corresponds to i = 0 does not alter the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' We find that the derived constraints on LHS 475b’s atmosphere are sen- sitive to the choice of mixing ratio prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Figure 13 compares the retrieved abundances from POSEIDON for the CLR and log-uniform mixing ratio pri- ors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Both approaches rule out H2 dominated atmospheres: log (H2) < 27% for CLR priors vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' log (H2) < 12% for log-uniform priors (both 2σ upper limits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' However, the log-uniform retrieval also infers upper limits on the abundances of H2O, CH4, CO2, and CO — ranging from < 9% to < 44% — which arise from the built in prior bias towards N2 being the background gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The CLR prior, in contrast, recognizes that these heavier gases all provide reasonable explanations for LHS 475b’s flat transmission spectrum due to their high mean molecular weight — consistent with the smarter retrieval which also ruled out low mean molecular weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' However, even with the CLR prior, certain atmo- spheric scenarios are still disfavored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' By examining Figure 14, which shows the full POSEIDON posterior for the CLR prior, one can see that CH4 dominated atmospheres with surface pressures ≳ 10 mbar are ruled out to 3σ confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' In other words, our retrieval accounting for the a priori unknown background gas confirms the result from our forward modelling analysis that thick, pure CH4 atmospheres with Psurf ≥ 1 bar are strongly ruled by our LHS 475b transmission spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Data Availability: The data used in this paper are from the JWST Cycle 1 General Observer program 1981 and are publicly available on the Mikulski Archive for Space Telescopes (https://mast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Fully reduced data products from this paper will posted on the Zenodo long term public archive upon acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Springer Nature 2021 LATEX template Lustig-Yaeger & Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' — LHS 475b with JWST 29 Rp, ref = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='97+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='70 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='85 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='30 log g log g = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='98+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='09 6 4 2 0 log Psurf log Psurf = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='96+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='70 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='05 300 450 600 750 900 T T = 393+264 −138 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 log H2 log H2 = −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='20+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='89 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='75 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 log H2O log H2O = −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='21+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='67 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='46 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 log CH4 log CH4 = −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='88+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='04 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='96 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 log CO2 log CO2 = −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='85+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='62 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='05 Rp, ref 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 log CO 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='70 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='85 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='30 log g 6 4 2 0 log Psurf 300 450 600 750 900 T 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 log H2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 log H2O 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 log CH4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 log CO2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='0 log CO log CO = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='00+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='00 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='40 LHS 475b Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' 14 Corner plot showing the 1D and 2D marginalized posterior probability distribu- tions from the POSEIDON retrieval using CLR mixing ratio parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The units are: Rp, ref (R⊕), g (cm s−2), Psurf (bar), and T (K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The inset shows the corresponding retrieved transmission spectrum model (1σ and 2σ confidence regions) compared to the NIRSpec G395H observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' The solution rules out H2-dominated atmospheres (to > 5σ) and thick atmospheres (Psurf ≳ 10 mbar) dominated by CH4 (to 3σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Code Availability: The codes used throughout this work for data analysis, atmospheric mod- eling, and manuscript preparation are as follows: Astropy [80, 81], Batman [50], CHIMERA [61, 62], Dynesty [56], emcee [51], Eureka!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' [29], ExoCTK [82], Forecaster [58], IPython [83], jwst [41], Matplotlib [84], NumPy [85, 86], PICASO [60], POSEIDON [72], PyMC3[87], SciPy [88], smarter [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' References [1] Fujii, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=', Angerhausen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=', Deitrick, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=', Domagal-Goldman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=', Grenfell, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=', Hori, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=', Kane, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=', Pall´e, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=', Rauer, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=', Siegler, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=', Stapelfeldt, ×10-3 Retrieved Spectrum (Median) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='24 G 4102b Retrieved Spectrum (lo) Retrieved Spectrum (2o) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='20 Retrieved Spectrum (Binned) JWST NIRSpec G395H 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='04 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' — LHS 475b with JWST K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=', Stevenson, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' : Exoplanet Biosignatures: Observational Prospects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' Astrobiology 18(6), 739–778 (2018) arXiv:1705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='07098 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='EP].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='1089/ast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='1733 [2] of Sciences Engineering, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=', Medicine: Exoplanet Science Strategy, (2018) [3] Seager, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=', Deming, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=': On the Method to Infer an Atmosphere on a Tidally Locked Super Earth Exoplanet and Upper Limits to GJ 876d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content=' ApJ 703(2), 1884–1889 (2009) arXiv:0910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE2T4oBgHgl3EQf5gh4/content/2301.04191v1.pdf'} +page_content='1505 [astro-ph.' metadata={'source': 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Science, Carnegie Mellon University +2Machine Learning Department, Carnegie Mellon University +Abstract +On observing a sequence of i.i.d. data with distribution P on Rd, we ask the question of how one can +test the null hypothesis that P has a log-concave density. This paper proves one interesting negative and +positive result: the non-existence of test (super)martingales, and the consistency of universal inference. +To elaborate, the set of log-concave distributions L is a nonparametric class, which contains the set G of +all possible Gaussians with any mean and covariance. Developing further the recent geometric concept of +fork-convexity, we first prove that there do no exist any nontrivial test martingales or test supermartingales +for G (a process that is simultaneously a nonnegative supermartingale for every distribution in G), and +hence also for its superset L. Due to this negative result, we turn our attention to constructing an e- +process — a process whose expectation at any stopping time is at most one, under any distribution in L +— which yields a level-α test by simply thresholding at 1/α. We take the approach of universal inference, +which avoids intractable likelihood asymptotics by taking the ratio of a nonanticipating likelihood over +alternatives against the maximum likelihood under the null. Despite its conservatism, we show that the +resulting test is consistent (power one), and derive its power against Hellinger alternatives. To the best of +our knowledge, there is no other e-process or sequential test for L. +1 +Introduction +Log-concavity is an important and prevalent modelling assumption in the modern study of shape-constrained +nonparametrics [Sam18]. +Log-concave distributions include many common families of densities, including +normal, exponential, extreme-value, and logistic distributions, and further are frequently justified in diverse +application domains including economics, reliability theory and filtering in engineering, and survival analysis +in medicine [BB06]. At the same time, the family is technically amenable, and admits a unique maximum +likelihood estimate with a well developed minimax theory and computationally efficient estimators [CS10; +CDSS18; KDR19; Axe+19; DR11; RS19; CSS10]. +As a result, log-concave densities offer practitioners a +broadly applicable and usable structure. +Given the attractive properties of estimation within the log-concave family, tests for membership in the +same are an important and necessary line of investigation. We note that along with the applications mentioned +above, such tests also have theoretical interest; for instance, in much of computational learning theory, efficient +learning algorithms are only known when covariates are sampled according to a log-concave distribution [e.g. +KKMS08]. While the estimation of log-concave densities has seen significant advances over the past decade +or two (see, e.g., the citations above, and the survey by Samworth [Sam18]), testing for log-concavity has +been relatively poorly developed. Indeed, prior to 2021, there were no valid and powerful tests for the same— +both theoretically and practically—outside of certain restricted one-dimensional settings. +In a significant +development, recent work of Dunn et al. +[DGWR21] has developed such a test, based on the Universal +Inference strategy of Wasserman et al. [WRB20]. +Our work is concerned with testing log-concavity in a sequential setting. Concretely, we assume that we +are given streaming access to a sequence {Xt} that are drawn independently and identically from some d- +dimensional density p, and we wish to test the membership of p within the family of log-concave densities. +Such a sequential test can be identified with a stopping time τ, where stoppage indicates rejection of the null +1 + +hypothesis, and the test is α-valid if under the null, the probability that τ < ∞ is bounded by α. The principal +attractiveness of such sequential tests arises from their adaptivity: rather than fixing a number of samples a +priori, the test may adapt to the difficulty of the underlying instance, rejecting earlier in easier settings, and +allowing for a greater number of samples to detect subtle deviations from the null hypothesis. +Below, we first set up some notation, and then proceed to contextualise our study, and give a brief overview +of the contributions of our paper. +1.1 +Problem setup and background +We begin by describing the notation needed for our discussion, the testing problem under consideration, and +the fundamental notions of test martingales and e-processes. We shall give further definitions and details in +§2, as well as later in the text as the context arises. +Spaces and measures. Let {Xt} = (X1, X2, . . . ) denote a sequence of d-dimensional random vectors with entries +indexed by t, which are measurable maps from Ω := (Rd)N to Rd, endowed with the cylindrical Borel sigma- +algebra B(Rd)N. We use typewriter style fonts, e.g. P, to denote laws of random processes (i.e. probability +measures on (Ω, B(Rd)N)), and standard fonts, e.g. P to denote laws on (Rd, B(Rd)). We use F = {Ft} +to denote the natural filtration of the process {Xt}, where Ft := σ(X1, . . . , Xt), for each t. For a Borel +probability measure P on Rd, we use P ∞ to denote the law of an i.i.d. process drawn according to P. We +use D to denote the set of probability measures on Rd with Lebesgue densities, and D∞ = {P ∞ : P ∈ D}. +For P ∈ D, we use p to denote its Lebesgue density. For technical convenience we define D1 := {P ∈ D : +E[max(0, log p(X))] < ∞, E[∥X∥] < ∞}. A set of laws P is said to be mutually absolutely continuous (m.a.c.) +if for all P, Q ∈ P, P ≪ Q ≪ P. Finally, we frequently use Xt +1 := (X1, . . . , Xt) to denote finite prefixes of +{Xt}. +Log-concave measures. A function f : Rd → R is said to be log-concave if there exists a concave function +g such that f = eg. If f is further a density with respect to the Lebesgue measure, then it is said to be a +log-concave density. We denote the set of measures with log-concave densties as L, and use L∞ to denote the +set of i.i.d. log-concave measures on euclidean sequences, i.e. L∞ := {P ∞ : P ∈ L}. +Sequential test for log-concavity. +The testing problem of interest is formulated as follows: Let Xt +i.i.d. +∼ P +for some unknown P ∈ D. We wish to test the null hypothesis H0 : P ∈ L. +A sequential test corresponds to {Ft}-adapted stopping time, representing the (possibly infinite) time at +which the test stops and rejects the null hypothesis. We shall refer to this stopping time as the rejection time +of the sequential test. A test is said to be α-valid if its rejection time, τ satisfies that +sup +P ∈L +P ∞(τ < ∞) ≤ α, +meaning that, under the null, the probability of ever rejecting, i.e. of incurring a Type I error, is at most α. +Similarly, a test is said to be asymptotically (1 − β)-powerful against Q ⊂ D \ L if the probability of failing to +reject the null under any distribution in the alternative Q (also know as a type II error) is uniformly bounded +by β: +inf +Q∈Q Q∞(τ = ∞) ≤ β. +A test is said to be consistent against Q if it is asymptotically 1-powerful against the same. Note that, when +consistent, these tests are typically called ‘power-one tests’ (following Robbins) to differentiate them from the +traditional Waldian sequential testing paradigm for which stopping does not imply rejection of the null. +Test martingales, test supermartingales and e-processes. +We briefly survey key notions underlying +our discussion, namely test martingales, and e-processes, leaving details to §3 and §4 respectively. +Definition A process {Mt} is a nonnegative supermartingale (NSM) with respect to a filtration {Ft} and +a law P if it is adapted, nonnegative, and EP[Mt|Ft−1] ≤ Mt−1 for each t. If the inequality is further an +equality at each t, then {Mt} is a nonnegative martingale (NM). We shall succinctly say that such a process +is a P-NSM or P-NM respectively. +2 + +Obviously, every P-NM is also a P-NSM. An important basic inequality of Ville [Vil39] controls the tail +behaviour of NSMs: if {Mt} is a P-NSM such that M0 = 1, then for every α ∈ (0, 1], +P(∃t ≥ 1 : Mt ≥ 1/α) ≤ α. +The result above is a sequential (time-uniform) analogue of Markov’s inequality. Equivalently, one can make +claims at arbitrary stopping times: for all stopping times τ, P(Mτ ≥ 1/α) ≤ α. This can be seen by applying +the optional stopping theorem for NSMs [Mey66, Ch. V, Thm. 28] and Markov’s inequality. +We now extend the above notions to composite families of sequential laws. Throughout this paper we shall +take the filtration to be the natural filtration of the data, and will leave it implicit in our definitions below. +Definition For a set of sequential laws P, we say that a process {Mt} is a P-NSM if {Mt} is a P-NSM for +every P ∈ P. Similarly, {Mt} is a P-NM if it is a P-NM for every P ∈ P. A P-NSM such that M0 = 1 is +called a test supermartingale for P, and a P-NM such that M0 = 1 is called a test martingale for P. +Observe that test supermartingales satisfy Ville’s inequality for each P ∈ P, i.e., if {Mt} is a test super- +martingale for P, then for every α ∈ (0, 1], +∀P ∈ P, P(∃t ≥ 1 : Mt ≥ 1/α) ≤ α. +(1) +Test supermartingales are so named because they form the canonical path to sequentially testing composite +hypotheses, which is encapsulated entirely by the above relation, in that valid tests can be derived by rejecting +only when a test supermartingale crosses a threshold. They are particularly interesting in nonparametric +settings; for example one can use them to sequentially test the mean of a bounded random variable [WR23], +for testing symmetry [RRLK20], for two-sample testing [SR21], independence testing [PBKR22], and testing +calibration [AHZ21], to mention only a few interesting sequential nonparametric problems. We shall discuss +test supermartingales extensively in this paper. +E-Processes [RGVS22; RRLK20; GHK19; HRMS20] are a recently defined class of processes that will also +play a central role in this paper. +Definition A process {Et} is called an e-process with respect to a sequential law P if it is non-negative, and +for every stopping time τ, we have EP[Eτ] ≤ 1. Similarly, {Et} is an e-process for a class of sequential laws P +if it is an e-process with respect to every P ∈ P. +E-Processes have a variety of equivalent definitions [RRLK20, Lem. 6]. In particular it is sufficient for the +process to satisfy EP[Eτ] ≤ 1 for only bounded stopping times. +By the optional stopping theorem (which holds without restriction on stopping times for nonnegative +supermartingales), notice that every test supermartingale for a class P is also an e-process for this class. +Thus, e-processes generalise the notion of test supermartingales. We observe that a Ville-type relation also +holds for e-processes, simply due to Markov’s inequality: if Et is an e-process for P, then for every α ∈ (0, 1], +∀P ∈ P, stopping times τ, P(Eτ ≥ 1/α) ≤ αE[Eτ] ≤ α. +(2) +Much as Ville’s inequality over the class (1) captures the relevance of test supermartingales to sequential +testing, the above inequality captures the relevance of e-processes to the same. The notion of e-processes, +along with the non-sequential analogue of e-values, is gaining vogue in recent work in statistics due to this key +property, along with the fact that e-processes exist for many composite and nonparametric testing problems +for which test supermartingales do not exist (see, e.g., the recent survey by Ramdas et al. [RGVS22]). We +will also encounter this situation in the current paper. +It is important to note that test supermartingales or e-processes can directly be interpreted as evidence +against the null hypothesis: since we expect them to be less than one under the null, the larger their realized +value, the more evidence we have that the null hypothesis is wrong. +Thus, there is no explicit need to +threshold them at 1/α for some prespecified α; one can alternatively simply report the final value at the final +stopping time of the experiment (which can itself be arbitrarily chosen). Nevertheless, we present this paper +in the language of level-α tests because that is far more popular, and we refer the interested reader to the +aforementioned references for further discussion on e-processes. +3 + +1.2 +Inadequacy of Test (Super)Martingales, and the Power of E-Processes +One dominant (but sometimes hidden) principle behind sequential testing of composite hypotheses is the use +of nonnegative martingales (NMs), or nonnegative supermartingales (NSMs). Concretely, to test a composite +hypothesis P ∈ P, one attempts to construct a P-test supermartingale {Mt}, which was defined earlier. By +Ville’s inequality (1), the chance that Mt ever exceeds 1/α under any null law is bounded by α. Thus, these +test supermartingales immediately yield a valid test: reject the null when Mt ≥ 1/α. The associated rejection +time, of course, is the Mt-hitting time of 1/α. Such tests have game-theoretic interpretations, through the fact +that nonnegative (super)martingales represent wealth processes in betting games [RGVS22]. For example, a +P-test martingale is the wealth process of a gambler who bets against the hypothesis that the sequence {Xt} +is drawn according some law in P. The game is designed so that the gambler cannot hope to reliably (in +expectation) make money if the null hypothesis is true; this is imposed by a restriction that under any law in +P, the expected wealth multiplier in each round should be at most unity. +However, for sufficiently rich classes P, such a game leaves the gambler powerless; the gambler is so +constrained by the aforementioned restriction that the only option is to not bet at all (or throw away money). +This phenomenon was first observed in work on testing exchangability in discrete time binary processes by +Ramdas et al. [RRLK22], who demonstrated that any process {Mt} that is an NSM for all exchangable binary +laws is, almost surely, a strictly decreasing process (the wealth starts at one and can only possibly go down). +As a result, any test based on thresholding such processes must be powerless against any alternative. Our +first technical contribution demonstrates an anlogous phenomenon in the setting of log-concave distributions. +Specifically, we show that the smaller class of i.i.d. Gaussian processes is not testable using NMs (or NSMs), +since all such processes are trivial in the sense of being almost surely constant (or decreasing). The claim is +summarised below, where G∞ denotes the set of all i.i.d. Gaussian laws (of any mean and variance). +Theorem. (Informal) There are no nontrivial G∞-NSMs or G∞-NMs. A fortiori, there are also no nontrivial +L∞-NSMs or L∞-NMs. +Thus, log-concave densities represent a natural class of distributions that cannot be tested via martingales. +Testing via E-Processes. +Given that one cannot test for log-concavity (or indeed, Gaussianity) using +nonegative (super)martingales, we are left in a situation where the prevalent design paradigm for sequential +testing is neutralised. There are two contrasting lines of attack that can be employed instead. +The first of these involves designing a restricted filtration Gt, distinct from the natural filtration, under +which there might exist nontrivial test supermartingales. Ramdas et al. [RRLK22; RGVS22] highlight the +remarkable fact that shrinking a filtration could introduce new nontrivial (composite) test martingales when +none existed in the original filtration. Such a strategy was notably used by Vovk et al. [VNG03; FGNV12] to +develop a sequential test for exchangeability, where as mentioned above, no nontrivial test supermartingales +exist in the data filtration. There are two main disadvantages to such an approach. First, such test martingales +only yield an e-process for a restricted set of stopping times (those under the restricted filtration). From an +applied point of view, the use of such an e-process demands discipline from a practitioner—they cannot look +at the raw data to decide when to adaptively stop (a predefined stopping rule, like the hitting time of 1/α +is okay, but it may never be reached, in which case we may still wish to present the obtained evidence at +the stopping time). Second, from a design point of view, the construction of appropriate filtrations is itself a +subtle task that is heavily problem-dependent, and thus designing such tests is more of an art than a science. +In particular, no such construction is known or obvious for sequential log-concavity testing. +In contrast, we follow the alternative strategy of testing via an e-process. Recall that a process {Et} is +an e-process for a set of sequential laws P if, for every stopping time τ and every P ∈ P, EP[Eτ] ≤ 1. Such +processes bear a deep relationship to the aforementioned test martingales. Indeed, it has been argued that +(admissible) e-processes must take the form infP∈P M P +t , where each {M P +t } is a P-NM [RRLK20]. The same +observation lends e-processes a gambling interpretation as the wealth process of a gambler against a ‘family of +games’, wherein the gambler simultaneously plays a game against each P ∈ P, and their wealth is taken as the +smallest wealth amongst these games. The gambler can then make money only if each of these games makes +money, i.e., if ∀P ∈ P, M P +t grows without bound, which would then indicate that every P ∈ P can be rejected. +E-Processes offer a similar testing approach as the previously discussed test supermartingales, as elucidated +4 + +by the inequality (2). Indeed, given an e-process {Et} for P, we can construct an α-valid test of membership +in P by rejecting only if Et ≥ 1/α. Indeed, in this case, the rejection time is +τα := inf{t ≥ 1 : Et ≥ 1/α}, +and using the inequality (2), we may conclude that +∀P ∈ P, P(τα < ∞) ≤ α, +i.e. this test is valid for the composite null P. Note further that the validity extends beyond this: let σ be any +other stopping time with respect to the natural filtration of the data. We further have that P(Eσ ≥ 1/α) ≤ α, +and thus no extraneous stopping criterion can affect the validity of the test, as long as rejection occurs only +if Eσ ≥ 1/α. +The theory and applications of e-processes have seen considerable development in the recent literature +on sequential analysis (along with the more basic notion of e-variables in batched settings). The concept is +attractive thanks to its flexibility and simplicity (despite generalizing nonnegative martingales), but construct- +ing powerful e-processes is partly science and partly art [RGVS22]. In composite testing, e-processes are of +central importance since they do not encounter the same pitfalls as NSMs and NMs, and there do indeed exist +nontrivial e-processes even on classes where no such NSMs exist. Indeed, in some sense, e-processes can be +shown to lie at the very core of sequential composite testing [RLKR22]. +1.3 +Test Using Universal Likelihood Ratios: A simple E-Process +The universal inference strategy [WRB20] gives a simple and generic construction of e-processes when a +maximum likelihood estimate can be easily computed. +To contextualise this approach, we first consider the case of a point null and alternative P ∞ and Q∞. In +this case, classical sequential testing theory posits that the sequential likelihood ratio +Lt = +t� +s=1 +q(Xs) +p(Xs) +yields a valid and powerful test upon thresholding at 1/α. Indeed, under the null, {Lt} is an e-process, since +it is an NM. +Against simple nulls but composite alternatives, likelihood ratios such as the above are typically adjusted +to account for the variety of possible alternatives. One way to do this is to replace the above numerator with an +estimate ˆqs(Xs). Importantly, as long as this ˆqs is nonanticipating, i.e., is Ft−1-measurable (depending only +on the first t − 1 datapoints), the martingale property continues to hold. To highlight this nonanticipation, +we shall denote these estimators as ˆqs−1. A second option is to mix over alternatives, perhaps using some +non-informative “prior”, but we will go with the first option in this paper because we are dealing with a highly +nonparametric alternative (essentially the complement of all log-concave laws, or the unspecified subset of +those against which one may hope to have power) — it is easy to use kernel density estimates for ˆqs, but not +so easy to mix over such a loosely specified nonparametric alternative. +The sequential universal likelihood ratio statistic (ULR) extends the above to composite nulls when a +maximum likelihood estimator (MLE) is computable. +Concretely, the statistic is as follows: let ˆqt−1 be +any predictable probability density, that is ˆqt−1 may be expressed as a function of only {X1, . . . , Xt−1} and +additional independent randomness. As before, we should think of ˆq as trying to estimate the underlying law +p. Let ˆpt be the MLE over the null class L with the data Xt +1, i.e., +ˆpt = arg max +ˆp∈L +� +s≤t +log ˆp(Xs). +Notice that, unlike ˆqt−1, the MLE ˆpt makes use of Xt. The sequential ULR statistic is the process +Rt := +� +s≤t +ˆqs−1(Xs) +ˆpt(Xs) . +5 + +(Of course, if the numerator was simply � +s≤t ˆqt(Xs), where ˆqt is an MLE over a larger class calculated using +{X1, . . . , Xt}, then we would get the usual generalized likelihood ratio process. +However, we will handle +very rich nonparametric alternatives over which computing the MLE is for all practical purposes impossible, +and further, for irregular models like log-concave distributions, such generalized likelihood ratios are very +ill-behaved and not well understood.) +The principal factor underlying the utility of Rt is that it is an e-process. Indeed, for any P ∈ L, and t, +Rt is dominated by Ft(P) = � +s≤t ˆqs−1(Xs)/p(Xs). Further, {Ft(P)} is a P ∞-martingale started at 1, due to +the predictability of ˆq, and thus for any stopping time τ, +EP ∞[Rτ] ≤ EP ∞[Fτ(P)] ≤ 1. +Notice in the argument above that while the e-process is dominated by a P ∞-martingale, it is not itself a +martingale. Indeed, this property is crucial to the existence of nontrivial e-processes even when there are no +such test martingales. We note that this property of domination by a P ∞-NM for every P ∈ L (or in general +P-NM for P ∈ P) is equivalent to the e-process property itself, and can be taken as an alternate definition of +the same [RRLK20, Lem. 6]. +Due to the above observation, the ULR e-process yields a valid test upon thresholding at 1/α. The power +of any such test relies on the two aspects of how well ˆpt and {ˆqs}s≤t estimate the underlying law p. Indeed, +we argue in §4 that if p ̸∈ L, then � +s≤t p(Xs)/ˆpt(Xs) must grow exponentially with t. Thus, as long as +the sequential estimates ˆqt approximate p well in a cumulative regret sense, the procedure above must be +consistent. Concretely, define the regret of prediction using {ˆqt} as +ρt(ˆq; P) := +� +s≤t +(− log ˆqs−1(Xs)) − +� +s≤t +(− log p(Xs)), +so that better estimation results in lower regret, and define the ‘well-estimable’ class +Q(ˆq) := {P : ρt(ˆq; P)/t → 0 P-a.s. as t ր ∞}. +In Section 4 we show the following: +Theorem. (Informal) Let Rt denote the ULR e-process with the sequential estimator E . Then the test that +rejects when Rt ≥ 1/α is α-valid, and consistent against Q(ˆq). +In fact, in §4, we demonstrate a more refined version of the above statement, which allows ρt to grow +linearly, but at a rate bounded by the distance of p from log-concavity. In any case, we comment that the +class Q(·) above is quite rich. For instance, using sieve estimators yields low-regret estimation in the above +log-loss sense for nonparametric classes such as laws on compact intervals with smooth and bounded densities. +The ULR e-process thus gives a powerful test for log-concavity against a rich set of alternates, even though +no test martingale can deliver such properties. Our work thus offers further insight into the sequential testing +of rich composite nulls, and the primacy that e-processes must take in the modern study of the same. +Along with the above asymptotic consistency result, we further derive finite rejection rate bounds by +controlling the typical rejection time of the ULR e-process in terms of the Hellinger distance of the alternaive +law from log-concavity. In particular, we show explicit bounds on typical rejection times against Lipschitz and +bounded laws on the unit box. The above theoretical exploration is augmented with simulation studies on a +simple parametric family comprising a mixture of two Gaussians to empirically evaluate the validity and power +of the test. We find that in small dimensions d ≤ 3, the tests show excellent validity, as well as reasonable +power. We further use this simulation study to highlight the role of the quality of the estimators ˆqt in the +power of the test. +Summary of Contributions +To summarise, this paper is concerned with the theoretical and methodolog- +ical aspects of sequential testing for log-concavity. We first show a negative result that demonstrates that the +approach of constructing test (super)martingales is powerless for testing this class of laws, and along the way +also offering simple characterisations of the fork-convex hull of i.i.d. sequential laws. In the positive direction, +we propose using the Universal Inference based e-process as a way to test log-concavity in the absence of test +martingales. We theoretically demonstrate both the consistency of the resulting sequential test, along with +concrete adaptive bounds on typical rejection time under a wide class of alternatives, and illustrate the same +via simulation studies. +6 + +2 +Definitions, and Background on Log-Concave Distributions +We begin with basic background on log-concave distributions, and necessary notation. We refer the reader to +the survey of Samuard and Wellner for further details [SW14]. +Log-Concave Laws. +A distribution P on (Rd, B(Rd)) is called logarithmically concave (henceforth log- +concave) if for every pair of compact sets A, B and λ ∈ (0, 1), +P(λA + (1 − λ)B) ≥ P(A)λP(B)1−λ, +λA + (1 − λ)B is the Minkowski sum {λx + (1 − λ)y : x ∈ A, y ∈ B}. It is well known that a distribution +that admits a density with respect to the Lebesgue measure is log-concave if and only if P(dx) = eg(x)dx for +a concave function g. Recall that L denotes the class of log-concave distributions with density on Rd, while +L∞ denote the set of i.i.d. sequential laws P ∞ for P ∈ L. +Log-Concave M-projection. +Recall that D denotes the set of laws on (Rd, B(Rd)) that admit densities +with respect to the Lebesgue measure, and that +D1 := {P ∈ D : E[max(0, log p(X))] < ∞, E[∥X∥] < ∞}, +where p(·) is the density of P. For every P ∈ D1, there exists a unique law +LP := arg min +L∈L +KL(P∥L), +where KL(·∥·) is the KL-divergence, called its log-concave M-projection. We shall abuse notation and use Lp +to denote the Lebesgue density of LP (one is admitted as long as P ∈ D1). For a set of points {xt +1}, t ≥ d + 1, +the log-concave maximum likelihood estimator (MLE) is the log-concave M-projection of the empirical law +Pt = � +s≤t δxs/t, denoted ˆPt. Most commonly, we shall refer to its Lebesgue density, ˆpt, which may equivalently +be defined as +ˆpt := +arg max +log f is a concave function +f≥0, +� +f=1 +� +s≤t +log f(xs). +The log-concave MLE has extremely favourable theoretical properties when Xt +1 +i.i.d. +∼ P for some P ∈ D1. For +instance ˆpt → Lp in the strong sense that ∃a > 0 : +� +ea∥x∥|ˆpt(x) − Lp(x)| → 0 almost surely [CS10]. +Locally Absolutely Continuous Sequential Measures. +Let Γ denote the standard Gaussian law on Rd, +γ its density, and let Γ = Γ∞. Notice that Γ is the law of a white noise. A sequential law P is said to be +locally absolutely continuous (l.a.c.) with respect to Γ, denoted P ≪loc. Γ, if for all t, the law of the finite +prefix P|t(·) := P(Xt +1 ∈ ·) is absolutely continuous with respect to Γ|t. Such l.a.c. laws admit a density process, +denoted +ZP +t := dP|t +dΓ|t +. +As an example, if P = P ∞ for some law P with Lebesgue density p, then P ≪loc. Γ, and ZP +t = � +s≤t p(Xs)/γ(Xs). +Of course, we may specify sequential laws (that are ≪loc. Γ) by specifying their density processes. Note that +ZP +t is a likelihood ratio process with respect to Γ, and so is a Γ-martingale. We shall henceforth use Γ as a +reference measure for sequential laws, and almost entirely work under laws that are ≪loc. Γ. +Notice that if ZP +t−1 > 0 then for {Xt} ∼ P, ZP +t /ZP +t−1 is the Γ-conditional density of Xt given Xt−1 +1 +. Further, +ZP +t−1 = 0 =⇒ ZP +t = 0. As a result, we may write for any adapted process {Mt} that +ZP +t−1EP[Mt(Xt +1)|Ft−1] = ZP +t−1EP[Mt(Xt +1)1{ZP +t > 0}|Ft−1] = EΓ[MtZP +t 1{ZP +t > 0}|Ft−1] = EΓ[MtZP +t |Ft−1]. +From this, we observe that a process {Mt} is a P-NSM if and only if {ZP +t Mt} is a Γ-NSM. Indeed, if the +former is true, then we conclude from the above that EΓ[MtZP +t ] ≤ ZP +t−1Mt−1, while if the latter is true, then we +can conclude that ZP +t−1EP[Mt|Ft−1] ≤ ZP +t−1Mt−1 ⇐⇒ 1{ZP +t−1 > 0}EP[Mt|Ft−1] ≤ 1{ZP +t−1 > 0}Mt−1. Since +1{ZP +t−1 > 0} holds P-a.s., it follows that EP[Mt|Ft−1] ≤ Mt−1 P-a.s., and so Mt is a P-NSM. By maintaining +equalities in the above analysis, the analogous statement also holds for NMs. These facts are quite useful in +our later study of fork-convex hulls. +7 + +3 +There Are No Nontrivial Test Supermartingales for Log-concavity +We begin with defining a natural notion of triviality. +Definition An NSM {Mt} is said to be trivial if Γ(∃t : Mt > Mt−1) = 0. An NM {Mt} is said to be trivial if +Γ(∃t : Mt ̸= Mt−1) = 0. +In words, a NSM (NM) is trivial if, almost surely, it is a non-increasing (constant) process. For the remain- +der of this section, we will set {Ft} to be the natural filtration. We recall the notion of test supermartingales +for a class of laws P, which we shall refer to as just nonnegative supermartingales. +Definition For a set sequential laws P, we say that a process {Mt} is a P-NSM if {Mt} is a P-NSM for every +P ∈ P. Similarly, {Mt} is a P-NM is it is a P-NM for every P ∈ P. +With these definitions in hand, we state the main result of this section, the proof of which is left to §3.3. +Theorem 1. There are no nontrivial G∞-NSMs or G∞-NMs under the natural filtration, and a fortiori, there +are no nontrivial L∞-NSMs or L∞-NMs under the natural filtration. +As discussed by Ramdas et al. [RRLK22], the above result implies that any valid level-α sequential test +for log-concavity based on thresholding L∞-NSMs or L∞-NMs must be powerless. Indeed, in the former case, +such a test against any law that is locally absolutely continuous with respect to Γ will almost surely never +exceed its starting value, and thus will almost surely never reject. +Intuition behind the proof. The result arises from a contradiction. To illustrate this, suppose {Mt} is a G∞- +NSM, and that for some time t, given Ft−1, it increases on the event {Xt ∈ O} for some open ball O, i.e., +conditionally on Xt−1 +1 += xt−1 +1 +, {Xt ∈ O} ⊂ {Mt > Mt−1}. Notice that due to nonnegativity, at worst it could +be zero outside the ball. Now, consider a Gaussian GO of such a small variance that GO(O) ≈ 1. By tuning +this variance, we can ensure that Mt > Mt−1 with probability arbitrarily close to 1 given the history, and +since the drop in the Mt remains bounded outside of the ball, this ensures that the conditional expectation of +Mt strictly increases. Since this violates the supermartingale property against G∞ +B ∈ G∞, we must conclude +that no such ball O exists. +Of course, the set on which {Mt} increases need not contain any ball, but still be of nontrivial mass, not +to mention that this set may vary with the history in a complex way. We address such gaps by exploiting the +notion of fork-convexity [RRLK22] which serves as a sequential analogue of convexity especially germane to +(super)martingale properties, and is treated in the following section. In particular, it holds that any process +{Mt} that is a G∞-NSM (or NM) is also a NSM (or NM) with respect to any sequential law in the ‘fork-convex +hull’ of G∞. The main argument then demonstrates that the fork-convex hull of G∞ is incredibly rich, and +contains the laws of arbitrary independent processes with density (i.e., processes of jointly independent {Xt} +such that Xt ∼ pt ∈ D). This large set of laws entirely obstructs the NSM (or NM) property from holding +in any nontrivial manner, essentially using a robust version of the previous intuitive example. Schematically, +we take the following route to establish this result, where the forward direction of each implication exploits +fork-convex combinations (and the reverse is trivial). +G∞-NSM +G∞ +∗ -NSM +( +� G∗)-NSM +( +� G∗)-NSM +( +� D)-NSM +Trivial +⇐⇒ +⇐⇒ +⇐⇒ +⇐⇒ +⇐⇒ +Figure 1: Schematic view of the argument. G∗ is the set of all finite mixtures of Gaussians and G∗ denotes its +L1 closure. For any set P, the class � P consists of independent sequential laws with marginals in P. See +§3.2 for definitions. +3.1 +Fork-convex Combinations +In an algorithmic sense, for two laws P, Q, an α-convex combination R = αP +(1−α)Q is the law of the output +of the following procedure: independently sample U ∼ P and V ∼ Q, and output X = U or V according to +the outcome of an independent α-coin. Fork-convex combinations are the natural sequential extension of such +8 + +a procedure. Concretely, we sample two trajectories {Ut} ∼ P and {Vt} ∼ Q, release Xt = Ut for t ≤ s for +some time s, and then flip a h-coin (where h can depend on the history) to decide whether the subsequent tail +is Xt = Ut or Xt = Vt for t > s. Notice that this is a much richer notion than convex combinations: firstly, +the decision to release Ut or Vt only needs to be made for a tail of the output sequence, and secondly, the +mixture proportion can depend on the history. Formally, this is defined as follows. +Definition ([RRLK22]) Let P, Q ≪loc. Γ be sequential laws. Let s ∈ N, and let h ∈ [0, 1] be an Fs-measurable +random variable such that Γ(h < 1, ZQ +s = 0) = 0. Then the (s, h)-fork-convex combination of P with Q is the +sequential law R with density process +ZR +t := ZP +t 1{t ≤ s} + +� +hZP +t + (1 − h)ZQ +t +ZP +s +ZQs +� +1{t > s}. +We shall denote this succinctly as R = +� +P +s,h +−→ Q +� +. +Notice that fork-convex combinations probabilistically allow single data-dependent change-points, or ‘switches’, +from a law P to Q. The ratio ZP +s/ZQ +s accounts for the fact that the prefix up to time s was drawn according to +P in the case of a switch, and the condition on h ensures that ZQ +s ̸= 0 when we switch to Q (informally meaning +that the initial segment of data was not impossible under Q). +The importance of the above definition lies in the fact that fork-convex combinations preserve (super)martingale +properties. Recall from §2 that {Mt} is a P-NSM if and only if {ZP +t Mt} is a Γ-NSM. Now suppose {Mt} is +both a P-NSM and Q-NSM, let R be a (s, h)-fork-convex combination of P and Q. For t ≥ s + 1, and we have +EΓ[ZR +t Mt|Ft−1] = hEΓ[ZP +t Mt|Ft−1] + (1 − h)ZP +s +ZQs +EΓ[ZQ +t Mt] ≤ hZP +t−1Mt−1 + (1 − h)ZP +s +ZQs +ZQ +t−1Mt−1 = ZR +t−1Mt−1, +where we have utilized the fact that h and Z· +s are Ft−1-measurable. The same calculation is trivial for t ≤ s, +and follows similarly for the martingale property. +The same property extends considerably beyond finite +combinations to closed fork-convex hulls, which generalise the standard notion of closed convex hull of sets. +Definition ([RRLK22]) A set is said to be fork-convex if it contains all fork-convex combinations of its +elements. +Let P be a set of sequential laws that are locally absolutely continuous with respect to Γ. +The +fork-convex hull of P, denoted f-conv(P) is the intersection of all fork-convex sets containing P. The closed +fork-convex hull of P, denoted f-conv(P) is the closure of its fork-convex hull with respect to L1(Γ) convergence +of the likelihood ratio processes at every fixed time t. +Explicitly, the closure in the definition includes all processes Q such that there exists a sequence Qn with +density process {ZQn +t } such that ∀t, ZQn +t +→ ZQ +t in L1(Γ). We shall refer to this as the local L1(Γ) closure. This +closure induces considerable flexibility into closed fork-convex hulls, making the notion a powerful concept +in light of the following phenomenon, observed by Ramdas et al. [RRLK22, Thm. 13] whose argument we +reproduce below. +Proposition 2. For a set of sequential laws P, a process is a P-NSM if and only if it is a f-conv(P)-NSM. +Proof. The result is evident for the fork-convex hull as an extension of the previous two-point calculation. +This extends to closures as follows. Let {Mt} be the process in question, and suppose Pn → P in the sense +above for Pn ∈ f-conv(P). Let Zn +t := ZPn +t +and Zt := ZP +t . We know that for each t, Zn +t → Zt in L1(Γ). We need +to show that ZtMt is a Γ-NSM. To this end, fix a t, and, by passing to a subspace, assume that Zn +t → Zt and +Zn +t−1 → Zt−1 pointwise a.s. Now, since Zn +t Mt is a Γ-martingale, using Fatou’s lemma yields +EΓ[ZtMt|Ft−1] = EΓ[lim inf Zn +t Mt|Ft−1] ≤ lim inf EΓ[Zn +t Mt|Ft−1] ≤ lim inf Zn +t−1Mt−1 = Zt−1Mt−1. +It is worth noting that while the NSM property is preserved under closures above, the same is not necessarily +true of the martingale property due to the use of Fatou’s Lemma when handling closures in the above proof. +Nevertheless, the NM (and indeed the martingale property without appeal to non-negativity) persists under +fork-convex hulls, without the closure, giving us the following characterisation. +Proposition 3. For a set of sequential laws P, a process is a P-NM if and only if it is a f-conv(P)-NM. +9 + +3.2 +The Fork-Convex Hull of Independent Sequential Laws +Proposition 2 gives us a concrete attack to showing the triviality of G∞-NSMs: we shall show that the fork- +convex hull of this set is far too rich to allow the existence of nontrivial NSMs. The bulk of our argument +develops simple structural characterisations of fork-convex hulls of independent sequential laws. This section +describes this characterisation using three properties, whose proof we leave to §A.2. We begin with a key +definition that sets notation for ‘independent sequential laws’ from a set. +Definition Let P be a set of distribution on Rd. For a sequence of distributions {Pt}t∈N, we define �{Pt} +as the sequential distribution of a stochastic process {Xt}t∈N such that all Xt are jointly independent, and for +each t ∈ N, Xt ∼ Pt. We further define � P := {�{Pt} : Pt ∈ P ∀t}, i.e. the set of laws of independent +stochastic processes with laws at each time lying in P. +Note that � P is a much richer set than the i.i.d. sequential laws, which we denote P∞ := {P ∞ : P ∈ P}. +In light of this, the following result demonstrates the richness of fork-convex hulls. Recall that a set of laws is +mutually absolutely continuous (m.a.c.) if every pair of laws contained in it is mutually absolutely continuous. +Lemma 4. Let P ⊂ D be a m.a.c. set of laws with density on Rd. Then, f-conv(P∞) ⊃ � P ⊃ P∞. +To sketch the argument underlying the above, fix any P = �{Pt}. It suffices to demonstrate a sequence of +laws {RT}T ∈N, each generated by finite fork-convex combinations of P∞-laws (and their fork-convex combina- +tions) such that for t ≤ T, the density process of RT and P agree. The conclusion then follows under closure, +since RT → P in the appropriate sense. The concrete witness for the above Lemma is the following sequence +R1 := P ∞ +1 , RT = +� +RT −1 T −1,0 +−→ P ∞ +T +� +, +where each fork-convex combination is valid since P is m.a.c. In essence, this exploits the fact that fork-convex +combinations let one switch between laws after a time of our choosing. See A.2 for details. +Next, we exploit the convex combination properties of fork-convex combinations to demonstrate that fork- +convex combinations of i.i.d. laws includes i.i.d. products over mixtures as well. To this end, let us define the +mixture classes as below. +Definition Let P be a set of distributions on Rd. For k ∈ N, we let Pk be the class of laws formed by k-fold +mixtures of laws in P, and denote P∗ = � +k∈N Pk as the class of laws formed by finite mixtures of laws in P. +Note that P∗ is well defined since Pk form an increasing set. The second key result shown in §A.2 is +Lemma 5. Let P ⊂ D be a m.a.c. set of laws on Rd. Then f-conv(P∞) ⊃ P∞ +∗ . +The key observation underlying the above is already demonstrated in showing that f-conv(P∞) ⊃ P∞ +2 . To +see this, fix any P, Q ∈ P, and α ∈ [0, 1]. We need to demonstrate a sequence of laws RT constructed via +repeated fork-convex combinations that match the density process of R := (αP + (1 − α)Q)∞ for times up to +T . This is realised as follows: +R0 := P ∞, ST := +� +RT −1 T −1,α +−→ Q∞� +, RT := +� +ST +T,0 +−→ P ∞� +. +In the above, ST matches the density process of R up to time T by mixing between RT −1 (whose tail behaves +as P ∞) and Q∞ appropriately. RT then switches the tail of ST to behave as P ∞ to enable the recursion. This +argument extends to P∞ +k +for any arbitrary k by inducting over k (which is possible since a member of Pk is a +mixture of a Pk−1 law and a P law). Since k is arbitrary, this immediately extends to P∗. +Finally, we exploit the closure properties of fork-convex hulls under L1(Γ) to extend fork-convex hulls from +product measures over a set to product measures over closures of that set. +Lemma 6. Let P be a set of distributions on Rd that have densities. Then f-conv(� P) ⊃ � P, where P is +the L1(Γ)-closure of P. +The above lemma is a straightforward consequence of the closure properties as detailed in §A.2. +10 + +3.3 +Proof of the Absence of Nontrivial Test Martingales +The previous section demonstrates that taking closed fork-convex hulls can yield significant expansion to +product laws over sequences. This section exploits these properties to demonstrate the triviality of G∞-NSMs. +The key observation underlying this is the following standard fact about the richness of Gaussian mixtures. +Recall that P denotes the L1(Γ)-closure of P. +Lemma 7. G∗ is L1(Γ)-dense in the set of all distributions with densities, i.e., G∗ = D. +The L1(Leb)-denseness of mixtures of Gaussians in D is a classical fact; for instance see the work of +Alspach and Sorenson [AS72] or Lo [Lo72]. More recently, a considerably more robust result was presented by +Bacharoglu [Bac10], who shows that Gaussian mixtures are dense in nonnegative simple functions in both an +L1 and an L∞ sense. This also suffices for our purposes since nonnegative simple functions are themselves L1- +dense in nonnegative integrable functions. The L1(Γ)-denseness follows since Γ admits a uniformly bounded +density with respect to the Lebesgue measure. +We note that our argument extends to any such set, i.e., to any P such that P = D. The Gaussians serve +as a convenient witness within L for which this property holds. With this in hand, we proceed as below. +Proof of Theorem 1. Let {Mt} be a G∞-NSM. First observe by Lemma 5 and Proposition 2 that as a conse- +quence, {Mt} is also a G∞ +∗ -NSM. Next, by Lemma 4 and Proposition 2, it is further a (� G∗)-NSM. Similarly, +by Lemma 6 and Proposition 2, we conclude that {Mt} is also a (� G∗)-NSM. Finally, by Lemma 7, we +conclude that {Mt} is a (� D)-NSM.1 +We now argue that � D is too rich to admit nontrivial NSMs. The argument is by contradiction—we +assume that Mt > Mt−1 for some t with nontrivial probability, and use this to construct a law in � D that +violates the NSM property. The argument repeatedly exploits the topological equivalence of (Rd)t and Rdt +under the product and metric topologies respectively. We shall denote the Lebesgue measure in m dimensions +as Lebm, and we note that the product Lebesgue measure on (Rd)t is identical to Lebdt, and use the latter to +denote the former. +Let us proceed with the argument. For a natural number t, define the event At := {Mt > Mt−1, Mt−1 < ∞}, +i.e. that {Mt} increases at time t. It suffices to argue that no matter the t, the mass of At is zero, since +Γ(Mt−1 = ∞) must be zero due to integrability of Mt−1. For the sake of contractiction, assume Γ(At) > 0. For +n ∈ N, define the approximations An +t := {Mt ≥ Mt−1 + 1/n, Mt−1 ≤ n}. The An +t form an increasing sequence +of sets, and converge to {Mt > Mt−1, Mt−1 < ∞} = At. +Now, since Γ(At) > 0 and At ∈ Ft, we conclude that Lebdt(At) > 0 due to the mutual absolute continutity of +Gaussians and Lebesgue measures on Euclidean spaces. Without loss of generality, we may assume Lebdt(At) < +∞ (since otherwise we may pass to a subset of At such that of positive and finite mass, using sigma-finiteness +of the Lebesgue measure, and run the argument on this subset). Since An +t ր At, we have by regularity of +measure that Lebdt(An +t ) → Lebdt(At), and in particular there exists an n such that Lebdt(An +t ) ∈ (0, ∞). Fix +such an n for the remainder of the argument. +Recall that an open rectangle in Rm is a Cartesian product of open intervals, i.e. +a set of the form +× +m +i=1(ai, bi) for ai < bi. Similarly, we say that R is an open rectangle in (Rd)t if there exist open Rd-rectangles +S1 . . . St such that R =× +t +s=1 Ss. The following statement is a consequence of basic topological and measure +theoretic properties of Euclidean spaces, which we prove in §A.3. +Lemma 8. Let E ⊂ (Rd)t be such that Lebdt(E) > 0. For every natural m ∈ N, there exists an open rectangle +R in (Rd)t such that +Lebdt(R) > 0 +and +Lebdt(R ∩ E) ≥ +m +m + 1Lebdt(R). +Exploiting the above result, we may construct a sequence of rectangles in (Rd)t, {Rm}m∈N each of positive +mass such that +Lebdt(An +t ∩ Rm) +Lebdt(Rm) +≥ +m +m + 1. +1We can also argue this more directly: +observe that taking closed fork-convex hull is an idempotent operation, i.e. +f-conv(f-conv(P)) = f-conv(P) (which follows from the facts that closed fork-convex hulls are fork-convex, and that closures +of closed sets are invariant). Therefore, using the chain of Lemmata of §3.2, f-conv(G∞) ⊃ � G∗, and so {Mt} is a (� D)-NSM. +11 + +Now, since each Rm is a rectangle, there exists a law Dm ∈ � D such that the prefix restriction Dm|t = +Unif(Rm). Indeed, if Rm =× +t +s=1 Sm +s , then Dm = �{Dm +s }, where Dm +s = Unif(Sm +s ) for s ≤ t, and Dm +s = Γ for +s > t. We claim that for large m, Dm witness a violation of the NSM property for {Mt}. We demonstrate this +using the process {Nt} := {min(Mt, n + 1)}. +Notice that if {Mt} is a P-NSM, then so is {Nt}, since +E[Nt|Ft−1] ≤ min(E[Mt|Ft−1], E[n + 1|Ft−1]) = min(Mt−1, n + 1) = Nt−1, +and the nonnegativity follows since both Mt and n + 1 are nonnegative. Further, since Mt−1 ≤ n on An +t , it +follows that Nt ≥ Nt−1 + 1/n on An +t as well, since n + 1/n ≤ n + 1. +Consequently, we have +EDm[Nt] ≥ EDm +� +(Nt−1 + 1/n)1{Xt +1 ∈ An +t } +� ++ 0 += EDm +� +Nt−11{Xt +1 ∈ An +t } +� ++ Dm(An +t ) +n +≥ EDm +� +Nt−11{Xt +1 ∈ An +t } +� ++ +m +n(m + 1), +where the final inequality exploits the fact that at least a m/(m + 1) fraction of the mass of Rm lies in An +t , +and we have used the nonnegativity of Nt. +However, since Nt−1 is upper bounded by n + 1, we observe that +0 ≤ EDm[Nt−11{Xt +1 ∈ (An +t )c}] ≤ (n + 1)Dm((An +t )c) ≤ n + 1 +m + 1, +and so +EDm[Nt−11{Xt +1 ∈ An +t }] = EDm[Nt−1] − EDm[Nt−11{Xt +1 ∈ (An +t )c] ≥ EDm[Nt−1] − +n + 1 +(m + 1). +But now, we conclude that +EDm[Nt] ≥ EDm[Nt−1] + (m/n) − (n + 1) +m + 1 +. +Choosing m > 3n2, and exploiting n ≥ 1, this implies that EDm[Nt] > EDm[Nt−1], thus contradicting the su- +permartingale property of {Nt} under Dm (since supermartingales must have non-increasing mean sequences). +We conclude that it cannot hold that Γ(At) > 0, i.e., Mt ≤ Mt−1 Γ-almost surely. +But, since t is arbitrary, we immediately conclude that +Γ(∃t ≥ 2 : Mt > Mt−1) ≤ +� +t≥2 +Γ(Mt > Mt−1) = 0. +The argument for NMs follows from this as well. If {Mt} is a � D-NM, then it is also an NSM, and thus +almost surely does not increase. But this means that M1 − Mt ≥ 0 is also a nonnegative supermartingale, and +therefore does not increase, which implies that Mt also does not decrease almost surely. +Remark. It may be possible to develop a different argument that does not explicitly need to pass through +the notion of fork-convex hulls. Perhaps one could directly work with the An +s above, and replace Dm a by +sufficiently skinny Gaussian Gm such that Gm(An +s ∩ R) ≈ Gm(R) ≈ 1. However, there would still be sufficiently +many technical details to iron out, so such an approach is not necessarily shorter or cleaner. More importantly +however, our chosen path of development above leads to a richer characterisation of fork-convex hulls of +i.i.d. processes with densities, and further directly illustrates the utility of such a characterisation. It thus +deepens our understanding of the important geometric concept of fork-convexity. +4 +The Sequential Universal Likelihood Ratio E-Process +We begin by recalling the definition of e-processes from the introduction. +12 + +Definition An {Ft}-adapted process {Et} is said to be an e-process for a set of sequential laws P if +sup +P∈P +sup +τ E[Eτ] ≤ 1, +where the second supremum is over all stopping times. Further, if for some n ≥ 1 it holds a.s. with respect to +all P ∈ P that E1 = E2 = · · · = En−1 = 1, then we say that Et is an e-process for P started at time n. +Next, we define the universal likelihood ratio (ULR) process [WRB20], which forms the main object of +interest for this section. +Definition Let E denote a sequence of estimators {Et}t≥0 such that each Et : (Rd)t → D. At any t, denote ˆqt = +Et(X1, . . . , Xt). Finally, let ˆpt denote the log-concave maximum likelihood estimate over the data X1, . . . , Xt +(which exists if t > d). The ULR process is the statistic +Rt(Xt +1; E ) := 1{t ≤ d} + 1{t > d} +� +d+1≤s≤t +ˆqs−1(Xs) +ˆpt(Xs) . +We shall often suppress the dependence of Rt on Xt +1 and E . The initial setting of Rt = 1 for t ≤ d is to +account for the fact that log-concave MLEs are known to exist only if at least d + 1 samples are available. +As discussed in the introduction, {Rt} constitutes an e-process due to the predictability of ˆqt−1 and the +fact that they are probability densities. We formally state the validity of Rt as a proposition. +Proposition 9. For any E , the process {Rt} is an e-process for L∞ started at time d + 1. Consequently, +rejecting the null hypothesis when Rt ≥ 1/α results in an α-valid test for log-concavity. +On exact MLEs. The measures ˆpt need not exactly maximise the likelihood ratio in the above. Indeed, if +instead of the exact log-concave MLE ˆpt we instead an estimate ˜pt such that +� +s≤t +log ˜pt(Xs) ≥ − log(1/ε) + +� +s≤t +log ˆpt(Xs), +then εRt ·� ˆpt(Xs) +˜pt(Xs) is an e-process, and this can be thresholded at 1/α as before. This observation is pertinent +since practical procedures for computing the log-concave MLE of a dataset are inexact, and only approximate +the solution up to a (user-specified) additive gap in the log-likelihood objective, and require computation that +scales polynomially with the inverse of this additive gap. +For the remainder of this section, we shall equate laws P ∈ D1 with their density, denoted p. +4.1 +Consistency of the ULR E-Process for Testing Log-Concavity +Consistency of the ULR e-process depends strongly on the underlying estimator E . Indeed, as an extreme +example, consider the case of ˆqt(Xt) = 1{Xt = X1}, for which the resulting Rt is a.s. 0 for any time t ≥ d + 1 +so long as the law P is continuous, and the test is thus powerless against such laws. +It thus follows that the ULR e-process can only yield power against a set of laws determined by the +estimator E . Concretely, we shall argue the same against the following set of ‘well estimable’ laws. Below, dH +below denotes the Hellinger distance. +Definition For a sequential estimator E , and a density p ∈ D1, define the prediction regret for a sequence +{Xt} as +ρt(E ; p) := +� +s≤t +log p(Xs) − log ˆqs−1(Xs). +Further, let Lp denote the log-concave M-projection of p. We define the class of distributions that are well +estimable by E with respect to log-concavity as +Q(E ; c) := +� +p ∈ D1 : P ∞ +� +lim sup +t→∞ +ρt(E ; p) +td2 +H(p, Lp) ≤ c +� += 1 +� +. +13 + +The main result of this section is that the ULR-based test is powerful against the above well-estimable +laws, which is shown later in this section. +Theorem 10. There exists a constant c > 1/25 such that if p ∈ Q(E ; c) \ L, then P ∞(Rt → ∞) = 1. +Consequently, the ULR e-process yields a consistent test against i.i.d. draws from any distribution in Q(E ; c). +The well-estimability condition above essentially requires that the distribution can be estimated well in a +log-loss sense. For i.i.d. distributions, one expects that for reasonable E , the estimates ˆqt converge to some ˆq, +and thus the regret grows for large t as ρt ≈ tKL(p∥ˆq) (which could grow sublinearly in t if KL(p∥ˆqt) → 0, but +the latter convergence is not required). The class Q thus roughly consists of distributions can be estimated well +in KL divergence. Such estimation can be a challenging task in complete generality, since the KL divergence is +quite sensitive to mismatch in the tails of distributions. However, under mild restrictions such as compactness +of support and smoothness, such estimability is quite forthcoming. Indeed, we give the following statement to +illustrate this point. This is proved in §B.3. +Corollary 11. Let DBox,Lip,B denote the set of 1-Lipschitz densities supported on the unit box [−1, 1]d and +bounded between [1/B, B]. There exists a sequence of sieve maximum likelihood estimators E such that for +every c > 0, DBox,Lip,B ⊂ Q(E ; c), i.e., the ULR e-process yields a consistent test against i.i.d. draws from +such distributions. +It is further interesting that the consistency of the test does not require that the regret ρt/t → 0, and +only that it gets small enough relative to the Hellinger distance between p and its log-concave M-projection +Lp. This signals that deviations from log-concavity may be detected far before the underlying law can be +estimated, which is quite favourable theoretically, although its practical effects depend significantly on how +large a c can be taken in Theorem 10. +Proof of Theorem 10. We begin by defining σt(p) = � +s≤t log p(Xs) − log ˆpt(Xs). Observe that +log Rt = σt(p) − ρt(E ; P). +Further, by assumption, we have that p ∈ Q(E , c) for some c, and thus for any ζ > 0, we have that +ρt ≤ (1 + ζ)ctd2 +H(p, Lp), +for large enough t. Consequently, to show that Rt → ∞, it suffices to show that P ∞ almost surely, +lim inf +t→∞ +σt +td2 +H(p, Lp) ≥ (1 + 2ζ)c. +(3) +It is at this point that the following lemma is useful, the proof of which is left to §B.1. +Lemma 12. For any p ∈ D1, it holds that +P ∞ +� +lim inf +t→∞ +σt(P) +td2 +H(P, LP ) ≥ 1 +25 +� += 1. +The claim (3) thus follows so long as (1 + 2ζ)c ≤ 1/25, and since ζ > 0 can be taken arbitrarily small, +this allows us to take any c < 1/25. We note that the constants in this argument are loose, and informal +calculations suggest that it may be possible to improve c up to about 1/6. +The proof of Lemma 12 relies on strong convergence properties of the log-concave MLE ˆpt to the log-concave +M-projection Lp. Recall that for a pair of functions u ≤ v, a bracket [u, v] is the set of all functions that lie +between u and v everywhere. By exploiting a characterisation of the convergence properties of log-concave +MLEs due to Cule and Samworth [CS10], Dunn et al. [DGWR21, Lem. 1] show that there is a small bracket +that is well separated from P such that ˆpt eventually lies in this bracket. Conditioning on this event, we +then exploit a classical result of Wong and Shen [WS95] which show linear growth of σt(p) with condiitonal +probability at least 1−exp (−Ω(t)) , at which point the lemma follows by Borel-Cantelli. As mentioned before, +see §B.1 for the full proof of Lemma 12. +14 + +4.2 +Power of the ULR E-Process for Testing Log-Concavity +The argument underlying Theorem 10 is also amenable to deriving rates, under further restrictions on the +underlying law P. As in the previous section, we argue this using the decomposition log Rt = σt(p) − ρt(E ; p). +4.2.1 +Challenges, and Context from the Theory of Log-Concave MLEs +With the above approach, the argument breaks into two parts. Frstly, we assume that we use a good enough +estimator E so that ρt is not too large with high probability. Such an assumption is necessary for the approach +we take, although in principle the test can be analysed using a different decomposition, in which case this +assumption may perhaps be weakened. In any case, we observe that for concrete alternate hypotheses such as +laws with Lipschitz densities supported on the unit hypercube, ρt can indeed be appropriately controlled. It +is worth noting, however, that the resulting rate bounds are strongly driven by the behaviour of ρt, and thus +the estimator being considered, which limits the power of the results to follow. +The second part of the argument requires us to show that σt is large, i.e., to argue that the log-concave +MLE cannot represent the underlying law very well when it is not log-concave. While a natural statement, +arguing this is challenging because this requires us to understand the behaviour of log-concave MLEs ‘off-the- +model,’ i.e., when the data is not drawn from a log-concave distribution itself. With the notable exception of +Barber and Samworth [BS21], this task has not been undertaken in the literature, with most works focusing +on on-the-model minimax rate bounds [KS16; KDR19; Han21; CDSS18]. Let us consider this in some detail. +Tight analysis of the on-the-model log-concave estimation problem fundamentally relies on a subtle re- +duction of the rates of log-concave MLEs to the problem of controlling deviations of empirical processes over +convex sets, i.e., to that of controlling supC |P(C) − Pt(C)| under data drawn from P, where Pt is the empir- +ical law, and the supremum is over convex sets in a bounded domain [CDSS18]. Using this observation and a +refined study of these deviations, Kur et al. [KDR19] recently showed tight on-the-model estimation rates of +the form dH(ˆpt, p) = O(n−1/(d+1)) when p ∈ L and d ≥ 3 (Han showed similar results, along with extensions +to s-concave densities [Han21]). While significant elements of this study can be extended to analysing off- +the-model behavior, the analysis ultimately cannot be applied to our situation. The gap arises because their +argument only upper bounds the quantity +θt := EX∼P [log Lp(X)] − EX∼P [log ˜pt(X)], +where for a small constant c, ˜pt ∝ max(c, pt) is a slight modification the log-concave MLE. When p = Lp, this +object is a KL-divergence, and so is lower bounded. However, when p ̸= Lp (that is, P is not log-concave), +this quantity is may well be negative. +Notice that this is a problem for us precisely because E[σt]/t ≈ +θt + EX∼p[log p(X) − log Lp(X)]. When p ̸∈ L, the second term can indeed be shown to be large, but the lack +of a lower bound on the first term limits the applicability of such results. We also note that other aspects +of the argument, which are relatively simple in on-the-model analysis (for instance, arguing that the mass p +places on sets of the form {Lp(x) < γ} is small), are also rendered inoperative in off-the-model analysis. +Of course, we can in principle exploit the results of Barber and Samworth instead. However, these re- +sults give quite poor rates. Roughly speaking, Theorem 5 of their paper [BS21] shows that off-the-model, +dH(ˆpt, Lp) ≲ t−1/4d, and thus any analysis that exploits this result cannot hope to show that σt is large for +t ≪ dH(p, L)−4d. This power of 4d arises since the analysis of [BS21] passes through a reduction to convergence +of empirical laws in Wasserstein distance (which gives the relatively benign factor of d), and further suffers a +1/4th power slowdown relative to this convergence (which is both unavoidable, and leads to a 4d exponent). +Our analysis sidesteps these issues by controlling the growth of σt on the basis of bracketing entropy (see +§B.1) bounds for the class of bounded log-concave laws on compact supports. Our bound below holds for all +d but appears to be new for d ≥ 4. Indeed, we show the following statement. +Lemma 13. Let Ld,B denote the set of laws with log-concave densities that supported on [−1, 1]d and uniformly +upper bounded by a constant B. There exists a constant Cd dependending only on the dimension such that +H[](Ld,B, ζ) = Cd �Θ +� +(B/ζ)max(d/2,d−1)� +, +where the �Θ hides terms that scale polylogarithmically with ζ or B. +15 + +We note that the lower bounds on the bracketing entropy implicit in the statement of Lemma 13 were +already shown by Kim & Samworth [KS16, Thm. 8], who further also showed the corresponding upper bounds +for d ≤ 3. While not the central point of the paper, we develop upper bounds for the same when d ≥ 4. +There are two salient technical points regarding the bound above. Firstly, observe that for d ≥ 4, the entropic +bounds lie in the non-Donsker regime, i.e., when the Dudley integral +� ε +0 +� +H[](Ld,B, ζdζ does not converge due +to a blow-up near ζ = 0, which typically (but not always) represents a slowdown in the convergence rates that +can be shown via entropy integrals. Secondly, for d ≥ 3, the bound grows as ζ−(d−1) rather than as ζ−d/2. +The latter quantity is pertinent it is close to the growth rate of the bracketing entropy of convex sets which +is ζ−(d−1)/2; see §B.2.3. This fact underlies the power of the previously discussed reduction of the analysis +of log-concave MLE rates to control on the deviations of empirical processes over convex sets, which admit a +slower entropy growth. +Lemma 13 is proved in §B.2.3. The growth bounds of this result ensure that for t ≳ dH(p, L)2(d−1), σt is +linearly large in t, even when the underlying law is not log-concave. This 2(d−1) exponent should be compared +to the aforementioned 4dth power scaling that one expects to emerge from using the Wasserstein continuity +based approach discussed above. Of course, the dependence on d could potentially be improved even further. +For instance, if the on-the-model analysis can indeed be extended to off-the-model, it is plausible to expect +dependence of the form d + 1 instead of 2d − 2. However, this remains a challenging problem for future work. +It is worth noting that while the techniques for the bounds in Lemma 13 exist in the literature, the +bounds themselves appear to not have explicitly been observed. We believe that this might be because it was +previously observed that due to existing lower bounds on this entropy, the resulting growth rate bounds that +emerged from entropic considerations could not be optimal for the rate analysis of the log-concave MLE, at +least in on-the-model settings. It should also be noted that the bounds above are explicitly for compactly +supported log-concave laws (which is a restriction, but a relatively mild one, due to the exponentially decaying +tail enjoyed by all log-concave densities). Further note that that the brackets we construct for this setting +are ‘improper’, i.e. the bracketing functions are themselves not log-concave, which may limit utility in direct +analysis of the difference between ˆpt and Lp, but is good enough when studying the behaviour of σt. +4.2.2 +Bounding Typical Rejection Times for the ULR Test +As discussed previously, our analysis of σt passes through a bracketing entropy bound for bounded, compactly +supported log-concave laws. For such bounds to be effective, we need to ensure that the log-concave MLE ˆpt +itself is bounded. This is enabled by the quantity ∆P , defined for a law P as +∆P := +min +v:∥v∥=1 EP [|⟨v, X − EP [X]⟩|], +which was identified by Barber and Samworth [BS21] as a means to lower bound the covariance of the log- +concave projection of P, which in turn can be exploited to upper bound the supremum of Lp and (indirectly) +ˆpt. Observe that ∆P roughly corresponds to the minimum eigenvalue of the covariance matrix of P — indeed, +it is best seen as a robust version of the same. +With this in hand, we are ready to state our main result, the proof of which is the subject of §B.2.2. Recall +that dH(p, L) = infq∈L dH(p, q) and τα := inf{t : Rt ≥ 1/α} is the rejection time. +Theorem 14. Suppose p is supported on [−1, 1]d, and let πt → 0 be a sequence such that for every t, +P ∞ +� ρt(E , p) +td2 +H(p, L) ≥ 1 +25 +� +≤ πt. +Then there exists a constant c ∈ [1/600, 1] and a natural number T0 such that for any t ≥ T0 + log(1/α) +cd2 +H(p,L), +P ∞ (τα > t) ≤ πt + 1 +c exp +� +−ctd2 +H(p, L) +� ++ 1 +c exp +� +−ct∆2 +P /d2� +, +and +T0 = Cd · �O +� +∆− max(d2/2,d2−d) +P +dH(p, L)− max((d+4)/2,2(d−1)) + d2∆−2 +P +� +, +where Cd depends only on d, and the �O hides terms depending polylogarithmically on dH(p, L), ∆P . +16 + +Observe that from the statement above we may conclude that the average rejection time is bounded as +E[τα] = +� +t +P ∞(τα > t) ≤ +� +t +πt + O(T0). +Here, the first term is driven by the predictability of p using the estimators E , while the second term is driven +by our analysis of the noise scale of log-concave density estimation in off-the-model scenarios. +In typical situations, the former of these terms will dominate the resulting bounds, since typical alternate +classes will be much larger than the class of log-concave distributions. For instance, using results on the +estimation of uniformly lower-bounded Lipschitz densities [WS95, e.g.], we show the following result about +the set DBox,Lip,B introduced in Corollary 11. +Corollary 15. For any constant B > 0, there exists a sequence of sieve maximum likelihood estimators E such +that if p ∈ DBox,Lip,B, the ULRT rejection time τα is bounded in expectation as E[τα] = �O(dH(p, L)−2(d+3)). +The proof is in §B.3. We remark that the above rates adapt to the extent to which the underlying law +p violates log-concavity in the sense that the time-scales of rejection are driven by dH(p, L). Indeed, this +represents an important advantage of sequential tests as opposed to batched tests, in that validity is retained, +and detection is guaranteed at an adapted time-scale. +On tightness. We note that the exponents of Theorem 14, and in particular, Corollary 15 are likely loose +for the problem of testing log-concavity. This is an artefact of the analysis; for instance, the slow rate in +Corollary 15 is largely determined by the rate requirements for estimating Lipschitz laws on the unit box, +which arises due to the πt terms present in Theorem 14. It is possible that this aspect can be improved, since +nothing neccessitates that we use an estimator that captures the underlying density p well. +Indeed, instead of analysing the prediction regret with respect to p itself, we could decompose R = σt(q) − +ρt(E ; q) for some other law q, perhaps lying in a smaller class of densities Q than those possible for p. As long +as (i) E does as good a job at prediction under the log-loss as any law in Q, and (ii) no matter what p ̸∈ L +is, there is a law in Q that is ‘closer to’ p than any law in L, a similar analysis should be possible, although +this requires possibly subtle off-model control on the behaviour of E , as well as a careful choice of Q itself +to control the relative values of distances such as dH(p, Q) and dH(p, L). One such approach which appears +promising for log-concave laws is to exploit s-concave densities to play the role of Q, which are particularly +attractive since they form a rich extension of the class of log-concave laws, but nevertheless enjoy identical +minimax MLE convergence rates as them [HW16; Han21]. +5 +Algorithmic Proposal, and Simulation Study +We now proceed to algorithmically describe the ULR e-process based test for log-concavity, and investigate +the behaviour of a concrete implementation of the same on a simple parametric family. +5.1 +Computational Aspects, Batching, and a Concrete Testing Algorithm +Under specification of the sequential estimators E , and a method for fitting the log-concave MLE, the statistic +Rt is explicitly computable, and thus naturally leads to implementations. While the e-process is powerful +against wide classes of alternatives, its implementation suffers from a fundamental computational issue, that +arises due to the recomputation of ˆqt and ˆpt in each round. This cost grows superlinearly with t since since the +entire denominator � ˆpt(Xs) must be evaluated on the entirety of the stream, and the cost of estimating this +ˆpt is itself superlinear in the number of samples t. A second issue arises upon increasing the data dimensions d, +since computational costs of estimating ˆpt grow quite fast with this. Even though polynomial in d algorithms +exist for computing the log-concave MLE [Axe+19], the fastest available method for this is typically hundreds +of times slower when processing ∼ 100 points in even the modest d = 5 when compared to the time needed +to process the same sized dataset for d = 1 [RS19]. We address this issue by exploiting batching to reduce +the computational load, which makes computations viable in the moderate d ≤ 4. The idea is to wait to +17 + +accumulate I > 1 fresh samples before recomputing Rt, rather than updating it at every round.2 +Let us point out that such batched updates still retain the e-process property, and thus validity, as long +as the ˆqt−1 remain nonanticipating over the entire batch. Concretely, we may set a schedule, captured by an +increasing sequence of times T = {tk}, and evaluate the statistic +Rt(T ) = Rt−1(T )1{t ̸∈ T } + +� +1{t ∈ T } +� +j≤k(t) +tj +� +s=tj−1 +ˆqtj−1(Xs) +ˆptk(Xs) , +where k(t) = max{k : tk ≤ t}. In words, the schedule divides streams into a sequence of batches of size +tk − tk−1, and each time a new batch is accumulated, we evaluate a new estimate ˆq on the previous batches, +and re-evaluate the log-concave MLE on the entirety of the data seen. This process continues to be dominated +by a batched version of Ft(P), which retains the martingale property under P ∞, thus yielding validity. The +simplest viable schedule is to set tk = kI for a constant ‘batching interval’ I. This effectively boils down to +testing the log-concavity of p⊗I, which is valid since tensor products of log-concave laws remain log-concave. +Notice that such batching may result in a reduction in power. For instance, rejection can only occur at the +time tk, and further the statistic may be deflated because data points with a large signal may be ‘washed-out’ +due to milder behaviour across the remainder of the batch. Nevertheless, we find in simulation studies that +this drop in power is nominal, and comes at the cost of a significant improvement in runtime. +With this in hand, we can provide an explicit algorithmic description of our test below. +Algorithm 1 Log-Concave Universal Likelihood Ratio Test +1: Input: Batching schedule {tk}∞ +k=1 with t1 ≥ d + 1, estimator E , level α. +2: Initialise: Rt ← 1 for t ≤ t1, K ← 1, N1 = 1, t ← 1. +3: while Rt < 1/α do +4: +if t = tK then +5: +ˆq = E (Xt−tK−1 +1 +). +6: +Nt ← Nt−1 · �tK +s=tK−1+1 ˆq(Xs). +7: +ˆp ← L (XtK +1 ). +8: +Rt ← Nt · +��tK +s=1 ˆp(Xs) +�−1 +. +9: +K ← K + 1. +10: +else +11: +Rt ← Rt−1. +12: +Nt ← Nt−1. +13: +t ← t + 1. +5.2 +Evaluating the ULR E-Process Test +We investigate the behaviour of the test of Algorithm 1 on the following simple test-bed family of laws, where +ed = 1d/ +√ +d is the unit vector along the all-ones direction in Rd,. +p(x; µ, d) := +1 +2(2π)d/2 +� +exp +� +−∥x − µ +2 ed∥2/2 +� ++ exp +� +−∥x + µ +2 ed∥2/2 +�� +, +i.e., balanced two component Gaussian mixture laws with means ± µ +2 ed and identity covariance. The norm +of the mean-difference is precisely µ, which we assume without loss of generality to be nonnegative. A small +modification of this family of laws was also used as a test-bed for the non-sequential test proposed by Dunn +et al. [DGWR21]. +These laws are extremely convenient for proof-of-concept investigation of tests of log-concavity. Indeed, +observe that up to a rotation, the d-dimensional law is a tensorisation of the one-dimensional law with a +2Note of course, that for our simulation study, the repetition of simulations required to study power and size mean that we +only implement our fully nonparametric test for up to d = 4. Nevertheless, even this is reasonable to run for d = 6, wherein a +single run over a horizon of 100 steps takes about 20s. +18 + +log-concave law (specifically a standard Gaussian in d − 1-dimensions). Since log-concavity properties are +invariant under rotations, and since the log-concave M-projection of product laws is a product of the marginal +log-concave M-projections [SW14], this gives a very simple characterisation of the log-concavity properties of +this law. Concretely, the distance from log-concavity is purely a function of the norm of the mean-difference +µ, and p(x; µ, d) is log-concave if and only if µ > 2. These laws thus give us a simple way to check both the +size and power of the test statistics, as well as study the effect of increase in dimension on the power. +Finally, we give details of the simulations. All data reported is a mean over 100 runs of each experiment. +All simulations are run up to 100 time steps, which is mainly for computational practicality. Note thus that +our size estimates are systematically lower than the true size (with infinte horizon). We run the case of d = 1, +which is computationally the cheapest, over the longer horizon of 500 time steps to illustrate that not much +changes in this case, at least as regards the empirical validity of our test. For d = 1, 2, 3, the tests are batched +at an interval of I = 20, while for computational practicality the test is batched at I = 25 for d = 4. These +are significant fractions of the time horizon studied, but do not significantly lower power, at least for d = 1, +as demonstrated by explicit simulation. +All code is implemented in R. The nonparametric estimator used for E is the kernel density estimate +as implemented in the ks package [CD18], and the log-concave MLE used is either from the logConDens +package [DR11] for d = 1 or the fmlcd package (d = 2, 3, 4) [RS19]. We note that the latter is not guaranteed +to return the log-concave MLE since it optimises a non-convex approximation to the program defining the +same. However, we find that compared to alternatives like the logConcDEAD package [CSS10], the fmlcd +implementation retains similar validity and power, but runs significantly faster. We also investigate using +parametric Gaussian mixture model fits to illustrate the effect of inefficiency in E on power, for which we use +the EM algorithm as implemented in the mclust package [SFMR16]. All simulations were executed on an +AMD Ryzen 5650U processor, a medium range CPU for a laptop computer. +5.2.1 +Fully Nonparametric tests +Figure 2 shows the behaviour of our instantiation of the algorithm with the fully nonparametric approach of +using kernel density estimators as E over p(·; µ, d) for d = 1, 2 as µ is varied, run at the size α = 0.1, with +I = 20 for d ∈ {1, 2, 3} and I = 25 for d = 4. We plot five traces which record the fraction of runs out of 100 +independent runs that the test rejected the null hypothesis at times smaller than 20, 40, 60, 80, and 100, where +100 was the horizon over which the test was run. +There are three major observations. Firstly, we observe that the test shows excellent validity. Indeed, the +null hypothesis holds true for µ ≤ 2, and the test does not reject more than a 0.02 fraction of the time in +either case in this scenario. Secondly, we observe that at least for sufficiently large µ, all of the tests do reject +within 100 steps. Finally, we notice that the power sharply drops as d increases. To concretely discuss this, +let µ∗(d) be defined as the smallest value of µ for which PX∼p(·;µ,d)(τ0.1 ≤ 100) = 0.9. The plots in figure 2 +give us estimates of µ∗(d), which increase sharply with d—from about 6 in d = 1 to over 1000 in d = 4.3 +This reduction in power is perhaps expected, given the considerable deterioration in the nonparametric +estimation rates with d. Nevertheless, we may question how much of the above decay in power is driven by +the inefficiencies in fitting log-concave MLEs, and how much accrues due to the inefficiency of kernel density +estimation. We investigate this effect in §5.2.2 by studying Oracle tests. +Longer Run for d = 1. +To show that the validity persists over longer time horizons, we implement the +fully nonparametric method over 500 time steps for d = 1, using I = 50. Observe in Figure 3 that rejection +under the null µ ≤ 2 is well controlled even at this increased timescale, while rejection rates steadily improve +as the horizon grows, although the improvement is somewhat marginal over the horizon of 500 versus 200. +3With pilot simulations in d = 5, we observe that µ∗(5) ≈ 1500. We note that these simulations were already too costly, in +terms of time, to implement completely for d = 5, due both to the increased costs of fitting MLEs in higher dimensions, and due +to the fact that as rejection rates decrease with dimension, more runs need to be executed over the whole horizon, which extends +the total cost of the experiment. We hope to implement the method on larger computational resources for such moderate ds. +19 + +Figure 2: Performance of the fully nonparametric test. Empirical rejection rates (over 100 simulations) +at α = 0.1 versus the mean difference µ for fully nonparametric test implementations over four cases: d = +1, I = 20 (top left); d = 2, I = 20 (top right); d = 3, I = 20 (bottom left) ; d = 4, I = 25 (bottom right). The +thin horizontal line plots the level α = 0.1, and the vertical line marks µ = 2 since p(·; µ, d) ∈ L ⇐⇒ µ ≤ 2. +Observe the strong validity properties in all plots, as well as the deterioration of power in higher dimensions, +as signalled by the sharp increases in the scales of the X-axis. +Figure 3: Performance of the fully nonparamet- +ric test over long horizons. Empirical rejection +rates (over 100 simulations) in the setting of Figure 2 +for d = 1, ran over a horizon of length 500 with +I = 50. Observe that the validity persists over this +longer horizon, and that power improves for µ > 2. +Figure 4: Effect of I on the fully nonparametric +test. Empirical rejection rates (over 100 simulations) +in the setting of Figure 2 for d = 1 and with varying +I ∈ {1, 10, 20, 50}. Observe that the rejection rates +for I = 10, 20 are roughly the same as for I = 1, +while I = 50 suffers large losses. +20 + +Figure 5: Performance of the partial oracle test. Empirical rejection rates (over 100 simulations) at +α = 0.1 versus the mean difference µ for the partial oracle test implementations over four cases: d = 1, I = 20 +(top left); d = 2, I = 20 (top right); d = 3, I = 20 (bottom left) ; d = 4, I = 25 (bottom right). Observe that +in each plot, the power improves starkly relative to the fully nonparametric test (Figure 2), as indicated by a +strong contraction of the scale of the X-axis, especially in higher dimensions. +Effect of Batching Interval. +As seen in Figure 4, batch sizes of I = 10 and 20 have a mild effect on the +rejection rates under alternate setting (µ ≥ 2) when compared to the direct I = 1. Interestingly, note that +I = 20 does somewhat better than I = 10 in the setting of moderate µ (the range 4 − 6), and slightly loses +power for larger intermediate µ (the range 6 − 8). In turn, the no batching setting, i.e., I = 1, is observed to +suffer deterioration in its size (µ < 2), although this remains at an acceptable level. +The large batch size I = 50 suffers the same validity issues as I = 1, but does even better than it for small +but non-null values of µ (2-5). Power considerably deteriorates for larger µ (5-10). While it is unclear how +much of this is an artefact of the fact that the length of the horizon is only 100, and how much is directly +due to the larger batching interval, the fact that I = 10, 20 perform well suggests that so long as the batching +interval is a relatively small fraction of the horizon length, the loss in power is not too bad. +5.2.2 +Oracle Tests, and the Effect of the Quality of E +Oracle Tests. +To probe the effect of the lossiness of the kernel density estimate on the power of the fully +nonparametric test, we run ‘partial-oracle’ and ‘full-oracle’ oracle tests, which adjust E to exploit concrete +information about the underlying laws p(·; µ, d). In the partial-oracle, we adjust E to estimate a two-component +Gaussian mixture model instead of a kernel density estimate, and in the full-oracle case, we directly set +ˆqt−1(·) = p(·; µ, d), i.e., we exactly evaluate the density. +We expect that under data drawn from p(·; µ, d), these tests are more powerful than the fully nonparametric +tests discussed above, since the regret ρt(E ; p) would reduce in the case of the partial oracle due to a reduced +21 + +Figure 6: Perforamance of the full oracle test. Empirical rejection rates (over 100 simulations) at α = 0.1 +versus the mean difference µ for the full oracle test implementations over four cases: d = 1, I = 20 (top left); +d = 2, I = 20 (top right); d = 3, I = 20 (bottom left) ; d = 4, I = 25 (bottom right). Observe the sharp +improvement in power compared to Figure 2, especially in high dimensions, as indicated by a strong contraction +in the scale of the X-axis. Observe also the improvement in power compared to Figure 5, in that the curves +reach high power at about half the µ that is needed for the partial oracle test. +complexity of the estimation class; and, of course, would reduce exactly to 0 in the case of the full oracle. In +either case, this effectively serves to increase Rt. These oracle tests thus let us probe the extent of the loss in +power at a fixed µ (and thus a fixed distance from log-concavity) that arise purely due to the decay in rate of +convergence of the log-concave MLE. In particular, the full oracle test captures exactly this effect, while the +partial oracle test approaches this in a soft way. Figure 5 shows the performance of the partial oracle tests, +and Figure 6 shows the same for the full oracle test for d ∈ {1, 2, 3, 4}. +Comparing Figures 2 and 5, we see that for using the partial oracle yields a marked increase in power, +at least for d > 1. This is evident in d = 2 by observing that the purple lines (overall rejection rate within +100 times steps) rises higher and is nonzero at smaller values of µ, as well as observing that the typical +rejection time decreases substantially (for instance, rejection never happened below time step 60 in the fully +nonparametric case, but is quite prevalent at higher µs under the partial oracle). In d = 3, 4 the effect is +much starker - notice that the scale of the plot completely changes, from order of hundreds to tens in d = 3. +This suggest that using the parametric mixture of Gaussians estimate offers strong improvements over the +nonparametric KDE estimate due to the reduced variance scale of this estimator. +The above effect is seen even more starkly in the case of the fully oracle test, where each of the rejection +rate curves is further improved (Figure 6). For instance, our estimate of µ∗(d) (the smallest µ such that +Pp(·;µ,d)(τ0.1 ≤ 100) = 0.9)) is about halved for the full oracle case when compared to the partial oracle (and +improved manifold relative to the fully nonparametric test). +22 + +The Quality of E has a Strong Effect. +These observations from the oracle tests indicate that the quality +of the estimate offered by E is very important in driving the overall power of the test. In these oracle examples, +the quality improved by reducing the variance scale of the estimator, whilst keeping the bias at 0 (since the +law p(·; µ, d) is representible by each of the estimator outputs). +Of course, in practice we cannot always hope to reduce the variance scale of our estimates whilst keeping +the bias zero. Nevertheless, there is a tradeoff between the two implicit here. Indeed, as we discussed briefly in +§4, it is possible to use a biased E in the test, i.e. one that does not strictly estimate p, so long as the output of +E does a better job of representing p than the log-concave MLE. The strong dependence of the testing power +on E indicates the critical need to investigate this design freedom, and to study how the trade-off between +the variance, in terms of the convergence rates of ˆqt, and the bias, i.e., the distance of lim ˆqt from p, should +be balanced to optimise the testing power. +6 +Discussion +Our work has shown that the sequential testing of log-concavity throws interesting challenges, in that the +prevalent paradigm of test martingales cannot be fruitfully applied to this practically relevant setting. In the +process of doing this, we developed a characterisation of the closed fork-convex hulls of independent sequential +laws on a continuous space, thus contributing to the theory of this new tool that characterises the nonnegative +supermartingale property. We then showed that the universal likelihood e-process instead does yield powerful +tests for log-concavity. In particular, we demonstrated that these tests are consistent against large classes of +nonparametric alternate laws, and further admit nontrivial rates, and made contributions to the off-the-model +analysis of the convergence of log-concave MLEs, as well as the general theory of the power analysis of universal +tests in order to do so. These properties are validated by running the test over a simple parametric family +of laws, which further demonstrates the critical role of the sequential estimator E in the power of the test. +Taking a broad view, the above can also be seen as a contribution to the emerging literature on e-processes, +and in particular as additional evidence for the case that the study of sequential testing at large must exploit +this powerful yet simple tool. +A number of directions, both theoretical and methodological remain open in this interesting subject, a few +of which we discuss below. +Regarding fork-convexity, our characterisation in §3 and §A of the closed fork-convex hulls of i.i.d. Gaus- +sians can possibly be further enriched, and it would be very interesting to understand precisely which laws lie +in this set. Additionally, notice that sequentially testing the Gaussiantiy of an i.i.d. process itself is a basic +problem that again cannot be tested using martingales (at least with respect to the natural filtration of the +data). Construction and analysis of such sequential Gaussianity tests is a natural and interesting direction. +Of course universal inference is again a natural approach for this class, but it may be possible to take ad- +vantage of translation and rotation invariance of the null hypothesis (all Gaussians) using methods developed +in [PLHG22]. +Regarding the ULR e-process based test for log-concavity, §5 shows that the power of the fully nonpara- +metric test can be quite limited particularly as the data dimension increases. This observation was also made +in the non-sequential setting by [DGWR21], who proposed using random one-dimensional projections as an +interesting method to ameliorate this. In this test, rather than computing the full d-dimensional kernel and +log-concave estimates, one projects the data onto many one-dimensional subspaces, and averages the e-values +(nonnegative test statistics with expectation at most one under the null) that result from a one-dimensional +test carried on each of these projected datasets. This approach not only has computational benefits due to +the speed of one-dimensional density estimation methods, but also shows statistical benefits in the scenario +of §5, in that the decay of power is considerably limited with dimension. Such projected tests are of course +possible in the sequential setting as well, and are a natural next step to investigate, both methodologically +and in terms of their theoretical properties. +On a broader scale, both the theoretical bounds and the simulations illustrate the critical role that the +quality of the estimator E plays, both specifically in the power of the test for log-concavity, but also more +generally in the use of the universal likelihood ratio e-process. 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In: The Annals of Statistics (1995), pp. 339–362 (cit. on pp. 14, 17, +33, 37). +26 + +A +Proof of Triviality and Properties of Fork-Convex Hulls +This appendix is devoted to showing the structural lemmata regarding fork-convex hulls, and discussing +technical aspects of our arguments. +A.1 +Details on the Local L1(Γ) Closure +Let us begin by explicitly detailing the notion of convergence implicit in closed fork-convex combinations. +Recall that the f-conv(P) is the closure of of f-conv(P) with respect to L1(Γ)-convergence of likelihood +ratio processes at every fixed time t. Let us unpack this statement in simple terms. Let Pn be some sequence +in f-conv(P) of density ratio Zn +t := ZPn +t . We say that Pn → P if for every t, it holds that Zn +t → Zt in L1(Γ). +Since Zt and Zn +t are Ft measurable objects, this convergence is simply in L1(Γ|t). Stating that the convergence +needs to happen at every fixed time t means that this convergence need not be uniform in t: it is fine for Zn +100 +to converge more slowly than Zn +1 , for instance. This notion of convergence may be metrised by +∆(P, Q) := +� +t∈N +2−t∥ZP +t − ZQ +t ∥L1(Γ). +We note that ∆ is bounded, since +∥ZP +t − ZQ +t ∥L1(Γ) = +� ���P|t(dxt +1) − Q|t(dxt +1) +��� ≤ +� +P|t(dxt +1) + +� +Q|t(dxt +1) = 2. +With this in hand, we first show the following auxiliary claim that is repeatedly used. +Lemma 16. Let P be a set of sequential laws, and let R be any sequential law. Suppose there exists a sequence +of sequential laws {RT} such that each RT ∈ f-conv(P), and for all t ≤ T, ZRT +t += ZR +t . Then R ∈ f-conv(P). +Proof. We claim that RT → R. Indeed, since ZRT +t += ZR +t for all t ≤ T, +∆(RT , R) ≤ +� +t>T +2−t · 2 = 2−(T −1). +Thus, limT →∞ ∆(RT , R) = 0, meaning RT → R. Since the closed fork-convex hull of P includes such limits by +definition, the claim is proved. +The above lends significant convenience to our arguments, since it allows us to only construct processes +matching some claimed member of the fork-convex hull up to finite times, which is typically easy to do in our +arguments below using just finite fork-convex combinations. +A.2 +Proofs about the Fork-Convex Hull of Independent Sequential Laws +We may now proceed with the proofs of the Lemmata omitted from §3. +Proof of Lemma 4. As detailed in the main text, by taking repeated fork-convex combinations, it follows that +RT ∈ f-conv(P∞), where +R1 := P ∞ +1 , RT := +� +RT −1 T −1,0 +−→ P ∞ +T +� +, +where validity of the mixture weight 0 exploits the mutual absolute continuity of laws in P. We conclude by +Lemma 16. +Proof of Lemma 5. It suffices to show that for all finite k, P∞ +k +⊂ f-conv(P∞), since P∗ = � +k Pk, and Pk ⊂ +Pk+1 for all k. +For T ∈ N and two laws P, Q on Rd, define the sequential law RP,Q,T as the law of an +independent sequence {Xt} such that Xt ∼ P for t ≤ T and Xt ∼ Q for t > T , i.e. RP,Q,T = +� +P ∞ T,0 +−→ Q∞� +. +For T ∈ N, define Pk,T as the set of sequential laws of the form RP,Q,T with P ∈ Pk and Q ∈ P. +27 + +We first claim that Pk,T ⊂ f-conv(P ∞). We show this inductively in k. Fix any T , and observe that +trivially P1,T lies in this fork-convex hull. For k ≥ 2, we may represent each P ∈ Pk as P = αP 1 + (1 − α)P 2 +for some α ∈ [0, 1], P 1 ∈ Pk−1 and P 2 ∈ P. We need to show that for any such P, and any Q ∈ P, RP,Q,T lies +in the fork-convex hull of P∞. By the induction hypothesis, RP 1,Q,T ∈ f-conv(P∞), and RP 2,Q,T ∈ f-conv(P∞). +But then define the laws +S0 := RP 1,Q,T , �Sτ := +� +Sτ−1 τ−1,α +−→ RP 2,Q,T +� +, Sτ := +� +�Sτ +τ,0 +−→ RP 1,Q,T +� +. +We note that every fork-convex combination above has valid weights since P is m.a.c., and so no density +process is ever 0. We claim that ST = RP,Q,T . +Indeed, let p1, p2, q respectively denote the densities (with respect to the standard Gaussian) of P 1, P 2, +and Q, and let Z1 +t and Z2 +t be the density processes of RP 1,Q,T and RP 2,Q,T respectively. These can be explicitly +evaluated as +Zi +t = +� +s≤min(t,T ) +pi(Xs) · +t� +s=min(t,T +1) +q(Xs), +where i ∈ {1, 2}, and we note that for u < v, �u +s=v · = 1. Observe that for each i, and any t1 < T, and t > t1, +we have +Zi +t +Zi +t1 += +min(t,T ) +� +s=min(t1,T )+1 +pi(Xs) · +t� +s=min(t,T +1) +q(Xs). +We shall inductively claim that for each τ, the density process of Sτ satisfies +ZSτ +t += +� +s≤min(t,τ) +(αp1(Xs) + (1 − α)p2(Xs)) · +min(t,T ) +� +s=min(t,τ+1) +p1(Xs) · +t� +s=min(t,T +1) +q(Xs). +Indeed, the base claim is trivial since for τ = 0 since S0 = RP 1,Q,T . Assuming the induction hypothesis for +τ, we observe that since ˜Sτ+1 is a fork-convex combination of Sτ and RP 2,Q,T at time τ, it shares the density +process of Sτ up to time τ, while after that time the density is a mixture of the two density processes, giving +Z~Sτ+1 +t += +� +s≤min(t,τ) +(αp1(Xs) + (1 − α)p2(Xs)) +× + +α + + + +min(t,T ) +� +s=min(t,τ+1) +p1(Xs) · +t� +s=min(t,T +1) +q(Xs) + + + + (1 − α) + + + +min(t,T ) +� +s=min(t,τ+1) +p2(Xs) · +t� +s=min(t,T +1) +q(Xs) + + + + + , +where we have used the behaviour of Zi +t/Zi +τ above for t ≥ τ + 1. +Finally, Sτ+1 mixes the above with RP 1,Q,T at time τ + 1 with a mixture weight of 0. This means that the +suffix law of Sτ+1 beyond the time τ + 2 is exactly equal to the law of RP 1,Q,T , while the prefix up to time +τ + 1 is left alone. In other words, +ZSτ +t += +� +s≤min(t,τ) +(αp1(Xs) + (1 − α)p2(Xs)) · +τ+1 +� +s=min(t,τ+1) +(αp1(Xs) + (1 − α)p2(Xs)) +× +min(t,T ) +� +s=min(t,τ+2) +p1(Xs) · +T +� +s=min(t,T +1) +q(Xs). +The claim follows upon noticing that the first two products can be merged into � +s≤min(t,τ+1)(αp1(Xs) + +(1 − α)p2(Xs). +With this in hand, the argument is straightforward. For any element P ∈ P∞ +k , we note that there exists +some member of Pk,T , say PT such that the density process of PT matches that of P up to time T . Applying +Lemma 16 immediately yields the claim. +28 + +Proof of Lemma 6. Let P = �{Pt} for any arbitrary sequence of Pt ∈ P. +We need to show that P ∈ +f-conv(� P). But, since Pt ∈ P for each t, for each t there further exist sequences {P n +t }n∈N, with each +P n +t +∈ P, such that P n +t +→ Pt in L1(Γ). +Let Q := {P n +t +: t, n ∈ N}. +We note that � Q ⊂ � P +=⇒ +f-conv(� Q) ⊂ f-conv(� P). Let Q := f-conv(� Q). We shall argue that P ∈ Q. +Let PT be the sequential law with density process +ZPT +t += +�� +s≤t ps(Xs) +if t ≤ T +� +s≤T ps(Xs) · � +T T . +If we can show that for each T , PT ∈ Q, then the claim will follow, since PT → �{Pt} as in the argument of +Lemma 16, and since Q is closed under the relevant notion of convergence. +We shall show this inductively. Let P1,n be a sequential law with density Z1,n +t +:= pn +1(X1)·� +s>min(1,t) p1 +s(Xs). +Notice that P1,n ∈ � Q ⊂ Q for every n. Further, +∆(P1,n, P1) ≤ ∥Z1,n +1 +− ZP1 +1 ∥L1(Γ) → 0. +Thus P1 ∈ Q. +Now suppose that PT −1 ∈ Q for some T ≥ 2. For T, n ∈ N, define Qn as the sequential law of density ratio +ZQn +t +:= + + + + + +� +s T +. +It trivially follows that Qn ∈ � Q ⊂ Q for all n. Now, define +PT,n = +� +PT −1 T −1,0 +−→ Qn� +, +which is valid since each P n +t and Pt has are mutually absolutely continuous. But ZPT,n +t += ZPT −1 +t += ZPT +t +for +t ≤ T − 1, and for t ≥ T, +ZPT,n +t +− ZPT +t += ZPT +T −1 · (pn +T (XT ) − pT (Xt)) · +t� +s=T +1 +ps(Xs). +It follows that +∥ZPT,n +t +− ZPT +t ∥ = +� +0 +t < T +∥P n +T − PT ∥L1(Γ) +t ≥ T , +and therefore, ∆(PT,n, PT) ≤ ∥P n +T − PT ∥L1(Γ) → 0. By closeness of Q, we conclude that PT ∈ Q. +A.3 +Proof of Lemma 8 +Proof of Lemma 8. Fix an m ∈ N. Since E has positive mass and is measurable, there exists an open set +O ∈ (Rd)t such that O ⊃ E and Lebdt(O) ≤ (1 + 1/m)Lebdt(E). Observe here that ‘most’ of the mass of O +lies within E. +Since O is open, there exists a sequence of disjoint open rectangles Ri in (Rd)t such that � Ri ⊂ O ⊂ � Ri, +and +Lebdt +�� +Ri +� += +� +Lebdt(Ri) = Lebdt(O). +Further, since most of the mass of O lies in E, we conclude that there exists at least one i such that +Lebdt(Ri) > 0 +and +Lebdt(E ∩ Ri) ≥ +m +m + 1Lebdt(Ri). +Indeed, otherwise we would have +Lebdt(E) = Lebdt(E ∩ O) = +� +Lebdt(E ∩ Ri) < +m +m + 1 +� +Lebdt(Ri) ≤ +m +m + 1 · m + 1 +m +Lebdt(E), +which is impossible. +29 + +A.4 +Technical Aspects of Fork-Convex Hulls and Our Triviality Argument +We comment on some technical aspects of the argument underlying the non-existence of nontrivial NSMs. +Specifically, we discuss the necessity of our definition of nontriviality, and the m.a.c. condition repeatedly used +in the argument, how the argument can be extended to consider log-concave laws over bounded sets, and +finally issues that arise when one tries to relax the definition of fork-convex combinations to handle support +mismatch. +Going beyond almost sure triviality. +The main text defines trivial NSMs (and NMs) as those that are +Γ-almost surely non-increasing (respectively, constant). Could one instead show that there are no nontrivial +L∞-NSMs in the stronger sense that such processes must be non-increasing (as opposed to only almost surely +non-increasing)? This turns out to be impossible, as witnessed by the following process +Mt := +1 +1 − 1{∃(t1, t2, t3, t4) ∈ [1 : t]4 : Xt1 = Xt2, Xt3 = Xt4, Xt1 ̸= Xt3}. +Since log-concave measures can have at most one atom (due to unimodality), it follows that {Mt} is an +L∞-martingale (indeed, it is almost surely just a constant 1, as stated by the theorem). However, Mt does +diverge to ∞, and this occurs almost surely against any i.i.d. sequential law which has at least two atoms, +for instance, a coin flip process. This means that while it may not be possible to reject processes with a +Lebesgue density using test martingales, it is possible to reject atomic processes. Structurally, this example +has to do with the fact that one cannot approach point masses in an L1 sense using measures with density. +Therefore, although L∞-NSMs must also be NSMs for independent processes with densities, this does not +extend to sequences drawn from distributions with atoms. In another sense, this issue is the same as the +problem discussed below regarding loss of the NSM property under extensions of fork-convex combinations of +laws with support mismatch, in that two laws with distinct single atoms each have parts that are mutually +singular. +The role of the mutual absolute continuity condition on P. +The definition of fork-convex combinations +of two laws P and Q at time s involves the ratio of density processes ZP +s/ZQ +s. This ratio must indeed appear, as +can be seen from the algorithmic viewpoint of §3 to account for the fact that if R is the fork-convex combination, +then the prefix law R|s = P|s. However, if ZQ +s = 0, i.e. if for {Xt} ∼ R, the prefix Xs +1 lies in a set that is almost +surely impossible under Q, then the above ratio is meaningless. This observation underlies the condition that +if ZQ +s = 0, then the mixture weight h must be exactly 1. +Our argument ultimately asserts that any law of the form �{Pt} lies in f-conv(P∞). +However, our +constructions to demonstrate this fact rely on setting h = 0 in order to generate switches between different +laws in P. Our assumption of mutual absolute continuity is to enable precisely this flexibility without running +into the issue discussed in the previous paragraph. +The role of Gaussians in our argument. +Since we used the density of the Gaussians in order to show +that L∞-NSMs must also be � D-NSMs, it behooves us to ask how essential L ⊃ G is to the main point of the +result.4 In the argument, Gaussians play two roles: firstly, since all Gaussians are supported on the entirety +of the domain, this class is m.a.c., and we can flexibly take fork-convex combinations. Secondly, the triviality +of Gaussian NSMs follows since mixtures of Gaussians are L1-dense in the set of densities on the reals. Any +subset of L that satisfies these two properties will suffice for our purposes. +Extending the argument to log-concave laws on subsets of Rd. +We finally observe that our argument +extends in a straightforward manner to log-concave laws on restricted subsets of the reals: for a bounded +convex set K, define LK to be log-concave densities supported on K. +Then all L∞ +K -NSMs are trivial, in +the sense that they are almost surely nonincreasing with respect to the reference measure (Unif(K))∞. This +follows because truncated Gaussians are again dense and supported on the entirety of the domain K. +4Notwithstanding that the result is interesting in its own right for Gaussians, which tells us that there is a simple, and very +natural, parametric family that cannot be tested via nonnegative supermartingales. +30 + +To see this, first observe that if γ := � αiφi is a mixture of Gaussians, then for any K of nonzero Lebesgue +mass, the truncation γ|K is also a mixture of truncated Gaussians. Indeed, define θi = +� +K φi. Then +γ|K(x) = +� +αi +� αiθi +φi(x) · 1{x ∈ K} = +� +αiθi +� αiθi +φi|K(x). +Now, let p be any density supported on K, and let γn → p be a sequence of mixtures of Gaussians converging +so that dn := +� +|p − γn| → 0. Then, defining πn = +� +Kc γn, we have +� +|p − γn|K| = +� +K +|p(1 − πn) − γn| +1 − πn +≤ +� +K +|p − γn| +1 − πn ++ +� +K +πnp +1 − πn +≤ πn + +� +|p − γn| +1 − πn +. +Further, since p is supported on K, πn = +� +Kc γn ≤ +� +Kc γn + +� +K |p − γn| = dn. Therefore, +TV(p, γn|K) ≤ +2dn +1 − dn +→ 0. +But this means that we can run the entire argument of §3 but with Gaussians truncated over K, and draw +the same conclusion. +Can we extend nontrivial fork-convex combinations to all laws? +As we discussed above, due to the +“ZQ +T = 0 =⇒ h = 1” condition in the definition of fork-convex combinations, it is not possible to take arbitrary +fork-convex combinations between sequential laws. In the extreme case of P = P ∞ and Q = Q∞ for P, Q that +have disjoint support, the only possible fork-convex combinations are mixtures of the form αP ∞ + (1 − α)Q∞. +While this technicality did not pose a serious issue for the current paper, this situation is quite unsatisfying +in general. After all, the algorithmic view of fork-convex combinations is very natural, and extends to such +disjoint support situations easily. +One can formalise this algorithmic picture by exploiting conditional densities. For a sequence of (appro- +priately measurable) maps kP +t : (Rd)t−1 × Rd → R≥0, denoted kP +t (xt|xt−1 +1 +), we say that {kP +t } is the conditional +density process of P if for each xt−1 +1 +, kt(·|xt−1 +1 +) is a density with respect to Γ, and for any t, A ∈ Ft, +P(Xt +1 ∈ A) = +� +A +� +s≤t +ks(xs|xs−1 +1 +)Γ(dxt +1). +More generally, we can define a similar notion via Markov kernels. We observe that, by definition, it holds +that if P has a conditional density process, then for any t and Γ-almost every xt +1 that +ZP +t (xt +1) = +� +s≤t +ks(xs|xs−1 +1 +). +Using the above characterisation, we can give the following natural extended definition of fork-convex +combinations: for two sequential laws P, Q with conditional density processes {kP +t}, {kQ +t} respectively, a law R +is a fork-convex combination of P and Q at time T with FT -measurable weight h if +ZR +t = +�� +s≤t kP +s(xs|xs−1 +1 +) +t ≤ T +� +s≤T kP +s(xs|xs−1 +1 +) · +� +h �t +s=T +1 kP +s(xs|xs−1 +1 +) + (1 − h) �t +s=T +1 kQ +s(xs|xs−1 +1 +) +� +t > T , +(4) +the difference being that we now do not impose the restriction that h = 1 if ZQ +T = 0. Simplistically, this is +possible since we are never dividing by the potentially null ZQ +T , and more formally, this is considering the +formal ratio ZQ +t /ZQ +T , which is interpreted in the natural way as � kQ +s(xs|xs−1 +1 +). The above extended definition +genealizes our previous definition of fork-convex combinations, and we can extend the same to the fork-convex +hull and its closure. +While the density process above is a perfectly sound mathematical object, such an extension is not fruitful +because of a failure to preserve the NSM property under these extended fork-convex combinations in general. +To illustrate why the above extended definition fails to maintain the NSM property (unlike the restricted +one used in the paper), consider the following example. +31 + +Example 1. P = (Unif(0, 1))∞ and Q = (Unif(1, 2))∞, and the process +Mt := +� +2 +∃s1, s2 ≤ t : Xs1 ∈ (0, 1), Xs2 ∈ (1, 2) +1 +otherwise +. +This process is an NSM (indeed, a martingale) under both P, Q. However, under any nontrivial fork-convex +combination of these two laws, this process must start at 1, and with positive probability grow to 2 but never +fall, and thus cannot be a supermartingale. +Under the hood, the issue in the example above arises due to the fact that under the extended definition, +for t ≥ T +1, {ZR +t > 0} = {ZP +t > 0}∪{ZP +T > 0, �t +T +1 ks(Xs|Xs−1 +1 +) > 0}, but the NSM property of {Mt} under +P or Q only controls the conditional expectations of MtZP +t 1{ZP +t > 0} and MtZQ +t 1{ZQ +t > 0} under Γ, which +leaves the conditional behaviour of MtZR +t uncontrolled when R places mass on events that are null under one +of these laws. +It should be noted that in the above example there is a version of the process {Mt}, i.e., a process {� +Mt} +such that P(∀t, Mt = � +Mt) = Q(∀t, Mt = � +Mt) = 1, but such that {� +Mt} is a martingale even under extended +fork-convex combinations. Concretely this process is just � +Mt = 1. One may thus wonder if this phenomenon +holds true in greater in generality: is it the case that if {Mt} is an NSM under P and Q, then there is a version +{� +Mt} of it (under P and Q) such that {� +Mt} is an NSM against any extended fork-convex combination of P +and Q, without the restriction “ZQ +T = 0 +=⇒ +h = 1”? This turns out also to be impossible in general, as +demonstrated by the following example. +Example 2. Let P = Unif(0, 1)∞ and Q = Unif(0, 1/2)∞. Define ρt = 1{Xt ∈ (0, 1/2)} for t ≥ 1, and ρ0 = 0. +Let {Nt} be an adapted process such that +Nt = + + + + + +1 +ρt−1 = 1 +3/2 +ρt−1 = 0, ρt = 1 +1/2 +ρt−1 = 0, ρt = 0 +. +Finally define Mt = � +s≤t Nt. It is easy to check that Mt is an NM under both P and Q. +Now suppose R is an extended fork-convex combination of P, Q at time T ≥ 1 with mixture weight h < 1. +This means that with probability 1 − h, it holds that Xt ∈ (0, 1/2) with certainty for all t ≥ T + 1. As as result, +we can explicitly compute that +E[NT +1|FT ] = ρT + (1 − ρT ) ((1 − h) · 3/2 + h(1/2 · 3/2 + 1/2 · 3/2)) = +� +1 +ρT = 1 +1 + (1 − h)/2 +ρT = 0 , +and so as long as h < 1, E[NT +1|FT ] > 1 if ρT = 0, and therefore {Mt} violates the NSM property under R +at the time T + 1. Note here that it is hard to construct any nontrivially different version of the above process +since the law of P dominates that of Q. +In light of the above discussion, generalised definitions of fork-convex combinations are at loggerheads with +maintaining the NSM property these combinations. Of course, since our purpose in using fork-convexity is to +assert the triviality of NSMs over large classes of sequential laws, this latter property is essential to maintain for +such statistical applications. At the same time, while the restricted original definition does maintain the NSM +property, the included restriction is unsatisfying, and in conflict with the algorithmic intuition underlying +the idea of these combinations. Finding an appropriate generalised definition of fork-convex combinations +that abstains from imposing these support conditions, but nevertheless retains NSM closure under the NSM +property is an interesting, and challenging, question left for future work. +B +Proofs of Consistency and Power Analysis +Recall the notation σt(P) := � +s≤t log p(Xs) − log ˆpt(Xs). The main arguments of this section control the +behavious of σt(P), in particular arguing that if the Hellinger distance of P from log-concavity is large, then +32 + +σt(P) must eventually grow linearly. We show this in asymptotic and nonasymptotic regimes in §B.1 and §B.2 +respectively. +Corollary 11 and Corollary 15 each relies on further control on the behaviour of ρt(E ; p) = � +s≤t log p(Xs)− +log ˆqs−1(Xs) when p is a bounded Lipschitz law on the unit box. This argument is left to §B.3. +B.1 +Proof of Consistency +Our arguments rely on the following bracketing tail estimate, developed by Wong and Shen to analyse the +behaviour of sieve-based maximum likelihood estimates [WS95, Thm. 1]. The estimate involves the bracketing +entropy of a class of laws Q under the Hellinger metric. We refer the reader to the text of Van der Vaart and +Wellner [VW96] for a thorough introduction, and give a brief account. +A bracket [u, v] is defined by two functions u(x), v(x) such that u(x) ≤ v(x) for all x, and consists of the +set of all functions f such that u(x) ≤ f(x) ≤ v(x) for all x. Since we shall only be interested in functions +that are densities, we may restrict attention to nonnegative functions. The Hellinger size of such a bracket +[u, v] is defined as |[u, v]| = ∥√u − √v∥2/2. We say that a class of distributions Q is bracketed by {[ui, vi]}N +i=1 +if for all Q ∈ Q, there exists an i such that q ∈ [ui, vi], where recall that for a distribution Q, we denote its +density by q. Note that this bracketing is typically “improper”, i.e., ui, vi will generally not lie in Q (because +q integrates to one, and so its lower bracket u will integrate to less than one, and its upper bracket v will +integrate to more than one). The Hellinger bracketing entropy of Q at scale ζ is the logarithm of the most +parsimonious bracketing of Q by brackets of size at most ζ, i.e. +H[](Q, ζ) := inf{log N : Q has an N-sized Hellinger bracketing at scale ζ} +Note, of course, that bracketing entropies are nonincreasing in ζ. +Lemma 17. (Simplification of [WS95, Thm. 1]) For a class of distributions Q and a natural number t, define +εt as the smaller number ε such that +� √ +2ε +ε2/28 +� +H[](Q, ζ/10)dζ ≤ 2−11√ +tε2. +For every t and ε ≥ εt, it holds that for any law P such that dH(P, Q) ≥ ε, we have +P ⊗t + + inf +q∈Q +� +s≤t +log p(Xs) − log q(Xs) ≤ tε2/24 + + ≤ 4 exp +� +−Ctε2� +, +where C > 2−14 is a constant. +Informally, if P is far enough from Q in the Hellinger metric (where far enough is determined by the +bracketing entropy of Q), then it is exponentially unlikely (in the sample size) for the maximum log-likelihood +under Q to be linearly close to the log-likelihood under P. Exploiting this observation in our context requires +us to argue that eventually, the log-concave MLE ˆpt must lie in a set with small entropy. To this end, we +appeal to the following result due to Dunn et al., which extends the convergence analysis of Cule & Samworth +[CS10]. +Lemma 18. [DGWR21, Lem. 1] Consider any distribution P ∈ D1, not necessarily log-concave. For any +η > 0, there exists a bracket [uη, vη] of size at most η that contains the log-concave projection LP , and +eventually also contains the log-concave MLE ˆpt: P ∞(∃t0 : ∀t ≥ t0, ˆpt ∈ [uη, vη]) = 1. +In words, the lemma states that for large enough t, the log-concave MLE ˆpt is certain to lie in a very +small bracket around the log-concave projection LP of the true distribution P. With this in hand, we are in +a position to show Lemma 12, the main statement underlying the proof of Theorem 10. +Proof of Lemma 12. Let ε := dH(P, LP ) > 0. Define ηε = ε2/211. Using Lemma 18, we know that there exists +a bracket [u∗, v∗] such that |[u∗, v∗]| ≤ ηε and, almost surely, ˆpt ∈ [u∗, v∗] for all large enough t. But observe +33 + +that H[]([u∗, v∗], ε2/211) = 0, since the size of [u∗, v∗] is already ηε. Further, since LP ∈ [u∗, v∗], by the triangle +inequality, +dH(P, [u∗, v∗]) = +inf +Q∈[uη,vη] dH(P, Q) ≥ dH(P, LP ) − |[u∗, v∗]| ≥ ε(1 − 2−11) ≥ ε · +� +24/25. +Let us define +�σt(P) := +inf +q∈[u∗,v∗] +� +s≤t +log p(Xs) − log q(Xs). +By exploiting the above observations, Lemma 17 yields that for every t, +P ∞ � +�σt(P) ≤ tε2/25 +� +≤ 4 exp +� +−Ctε2� +. +Note further that if ˆpt ∈ [u∗, v∗], then since ˆpt is a maximum likelhood estimate, it must hold that +σt(P) = �σt(P). Let Es := {∀t ≥ s, ˆpt ∈ [u∗, v∗]} be the event that ˆpt lies in the small bracket after time s, +and At := {σt(P)/tε2 ≥ 1/25} be the event that σt(P) is larger than tε2/25. +By Lemma 17, for every fixed time s and t ≥ s, +P ∞(Ac +t ∩ Es) ≤ 4 exp +� +−Ctε2� +, +and since this upper bound is summable, by the Borel-Cantelli Lemma +0 = P ∞ +� +lim sup +t +(Ac +t ∩ Es) +� += P ∞ +� +(lim sup +t +Ac +t) ∩ Es +� +, +and so for any time s, +P ∞(lim sup +t +Ac +t) ≤ P ∞(lim sup +t +Ac +t ∩ Es) + P ∞(Ec +s) = P ∞(Ec +s). +By Lemma 18, ˆpt must eventually almost surely fall in [u∗, v∗], lims→∞ P ∞(Ec +s) → 0. Further notice that +lim sup +t +Ac +t = {σt(P)/tε2 < 1/25 infinitely often} = {lim inf σt(P)/tε2 < 1/25}. +Putting the observations together, we conclude upon sending s → ∞ that +P ∞(lim inf σt(P)/tε2 < 1/25) ≤ lim +s→∞ P ∞(Ec +s) = 0. +B.2 +Proofs Underlying the Power Analysis +We shall begin by stating the key lemmata underlying our argument, which exploit our bracketing entropy +control from Lemma 13 along with results in the literature that bound the maximum value attained by a +log-concave density in order to make the same effective. We then prove the main result, and conclude by +proving Lemma 13. +B.2.1 +Controlling the Maximum Value Attained by the Log-Concave MLE +The rate analysis quantitatively exploits Lemma 17. To do so, we first need bracketing entropy bounds for +log-concave laws, which is precisely the subject of Lemma 13. We recall that this controls the bracketing +entropy of the class Ld,B of log-concave laws with densities supported on [−1, 1]d that are bounded from above +by B, showing that +H[](Ld,B, ζ) = �O((B/ζ)max(d/2,(d−1))). +The role of B in the above is quantitatively unimportant as long as this constant does not scale with +relevant parameters. This fact is assured for log-concave laws with near identity covariance. Intuitively, since +the covariance is lower bounded in all directions, the laws cannot concentrate too much, and thus the value of +the density at the mode cannot be too large. This observation is encapsulated in the following result, which +follows trivially from the work of Kim & Samworth. +34 + +Lemma 19. [KS16, Cor. 6] Let Lγ +d denote the set of log-concave laws distributed on [−1, 1]d with covariances +lower bounded in the positive semidefinite order by γI. Then there exists a dimension dependent constant Cd +such that for any f ∈ Lγ +d, +max +x∈[−1,1]d f(x) ≤ γ−d/2Cd. +Of course, our bounds in Theorem 14 depend on ∆P , which roughly speaking only controls that the +covariance of the underlying law P. The relevance of this quantity arises from the following observation, due +to Barber and Samworth. +Lemma 20. [BS21, Cor. 8] Let P ∈ D1 be a law supported on [−1, 1]d such that +∆P := +min +v:∥v∥=1 Ep[|⟨v, X − Ep[X]⟩|] > 0. +Then there exists a dimension dependent constant cd such that Cov(LP ) ⪰ cd∆2 +P I. Further, there exists a +dimension-independent constant C such that for any t ≥ 2Cd3/∆2 +P , it holds with probability at least 1 − +2 exp +� +−Ct∆2 +P /d2� +that Cov(ˆpt) ⪰ cd∆2 +P +4 +I for the log-concave MLE ˆpt. +Proof of Lemma 20. The first observation is a direct restatement of Lemma 7 of Barber and Samworth. The +second statement follows from the fact that over v : ∥v∥ = 1, v �→ ⟨v, X − EP [X]⟩ is bounded by 2 +√ +d, and +is clearly continuous in v. Thus exploiting standard subGaussian concentration results over the unit ball, it +follows that with probability at least 1−2 exp +� +−Ct∆2 +P /d2� +, it holds that for the empirical law pt = 1 +t +� +s≤t δXs, +min +v:∥v∥=1 Ept[|⟨v, X − Ept[X]⟩|] ≥ ∆P /2. +But notice that ˆpt = Lpt, from which the claim follows by the first part. +Merging Lemmas 19 and 20 immediately yields the following observation, which serves as a concrete bound +for the scale of B we need to employ in Lemma 13. +Lemma 21. There exists a constant Cd depending only on d such that for any t ≥ 2Cdd3/∆2 +P , it holds with +probability at least 1 − 2 exp +� +−Cdt∆2 +P /d2� +that +max +x∈[−1,1]d ˆpt(x) ≤ Cd∆−d +P . +Proof. Employing Lemma 19, we observe that {max ˆpt ≤ (cd∆2 +P /4)−d/2} ⊂ {Cov(ˆpt) ⪯ cd∆2 +P /4I}, and the +latter has probability at least 1 − 2 exp +� +−t(C∆2 +P /d2) +� +for t ≥ 2Cd3/∆2 +P . Take Cd = max(C, c−d/2 +d +). +B.2.2 +Proof of Bounds on Rejection Times +With the above in hand, we may proceed with the main argument. +Proof of Theorem 14. Recall the definition σt := � +s≤t log p(Xs) − log ˆpt(Xs). We shall first lower bound σt +with high probability. +Let B be a quantity that we will choose later. Let εt denote the solution to the fixed point equation from +Lemma 17, instantiated with the bracketing entropy of Ld,B. Further, let define the event +Et := {ˆpt ∈ Ld,B}. +For any t, provided that such that εt ≤ dH(p, L) and ˆpt ∈ Ld,B, Lemma 17 yields that +σt ≥ +inf +q∈Ld,B:dH(p,q)≥dH(p,L) log +� +s≤t +p(Xs) +q(Xs) ≥ td2 +H(p, L) +24 +(5) +with probability at least 1 − exp +� +−Ctd2 +H(p, L) +� +− P ∞(Ec +t). +35 + +In the rest of the proof, we will determine the range of t that leads to a small enough value for εt to +ensure that the condition εt ≤ dH(p, L) is met and, at the same time, control P ∞(Ec +t). To this end, we deploy +Lemma 13. First, observe that for d ≥ 3 and for any positive constants c and C +� Cε +cε2 +� +˜O(Bd−1ζ−(d−1))dζ = �O(B(d−1)/2ε−(d−3)). +Note that polylogarithmic terms do not affect the main growth of the integral.5 Therefore, solving the fixed +point equation +�O(B(d−1)/2ε−(d−3)) = ε2t1/2, +we obtain that for d ≥ 3 +εt(B) = �O(B1/2t−1/2(d−1)), +where we highlight the dependence on the as yet undetermined quantity B. +A similar argument using the entropy bound ζ−d/2 yields εt(B) = �O(Bd/(d+4)t−2/(d+4)) for d ∈ {1, 2}. +Now define +T1(B) = inf{t : εt(B) ≤ dH(p, L)} +and observe that +T1(B) = +� �O(B(d−1)(dH(p, L))−2(d−1)) +d ≥ 3 +�O(Bd/2(dH(p, L)−(4+d)/2)) +d ∈ {1, 2} . +Finally, by Lemma 21, for B ≥ Cd∆−d +P +and t ≥ T2 := C∆2 +P /d2 the probability of the event Et is at least +1 − 2 exp +� +−tC∆2 +P /d2� +. Let us set B∗ = Cd∆−d +P +and let T0 := max(T1(B∗), T2). We obtain that the lower +bound +σt ≥ td2 +H(p, L) +24 +holds with probability at least 1−C exp +� +−tcd2 +H(p, L) +� +−C exp +� +−tc∆2 +P /d2� +for t ≥ T0. Now, observe that at any +time t ≥ max(T0, 600 log(1/α) +d2 +H(p,L) +), it holds with probability at least 1−πt−C exp +� +−td2 +H(p, L) +� +−C exp +� +−Ct∆2 +P /d2� +that +log Rt = σt − ρt ≥ td2 +H(p, L) +600 +≥ log(1/α), +and thus the probability that the rejection time τα := inf{t : Rt ≥ 1/α} exceeds the above bound is bounded +by πt + C exp +� +−td2 +H(p, L) +� ++ C exp +� +−Ct∆2 +P /d2� +. +B.2.3 +Proof of Bracketing Entropy Bound on Log-Concave Laws +We proceed to show Lemma 13. We note that the upper bound for d ≤ 3 was shown by Kim and Samworth +[KS16]. Below we focus on d ≥ 4. We shall exploit two existing results in the literature regarding convex sets +and functions. The first is essentially due to Bronshtein (also see [KDR19, Lem. 3]). +Lemma 22. [Bro76] Let Kd denote the collection of convex sets in [−1, 1]d. For any ζ > 0, there exists a +collection of pairs of convex sets Kd,ζ ⊂ Kd × Kd with log |Kd,ζ| = O(ζ−(d−1)/2) such that +• Every (K, K) ∈ Kd,ζ satisfies Lebd(K \ K) ≤ ζ. +• For every K ∈ Kd, exists (K, K) ∈ Kd,ζ satisfying K ⊂ K ⊂ K. +In other words, the bracketing entropy of convex sets under the set difference metric is controlled at rate +(d − 1)/2. Importantly, the bracketing demonstrated above is proper. This result may be extended to the +following bracketing entropy bound on convex functions as by Gao and Wellner. +Lemma 23. [GW17, Thm. 1.5] Let K be a convex set in [−1, 1]d, and let CK,B be the set of convex functions +upper bounded by B over K. Then the L2(K) bracketing entropy of CK,B at scale ζ is bounded as O((B/ζ)(d−1)). +5This can be seen by iterating the relation +� +xn logm x = xn+1 logm x +n+1 +− +m +n+1 +� +xn logm−1(x). +36 + +Above, the L2(K) metric is the usual L2 distance ∥f − g∥L2(K) = ( +� +K(f − g)2dx)1/2, and the L2(K) +bracketing entropy is the bracketing entropy when the size of a bracket [u, v] is |[u, v]| = ∥u − v∥L2(K). +With the above in hand, we may proceed with the proof. +Proof of Lemma 13. For any log-concave law f, let S := {x ∈ Rd : f(x) ≥ ζ3} = {x ∈ Rd : log f(x) ≥ 3 log ζ}. +Since f is log-concave, the set S is convex. As a result, by Lemma 22, there exists some convex set ˜S ∈ Kd,ζ2/B +such that Leb(S \ ˜S) ≤ ζ2/B and ˜S ⊂ S. Let ˜C ˜S,ζ,B denote a ζ-bracketing of convex functions bounded by +B on ˜S. Since, on ˜S, the function − log f is convex and is upper bounded by − log B, by Lemma 23 there +exists a bracket [−u, −l] ∈ ˜C ˜S,ζ/B,log B/ζ3 such that, on ˜S, l ≤ log f ≤ u, and +� +˜S(u(x) − l(x))2dx ≤ ζ2/B2. +Note that, on ˜S, f is lower bounded by −3 log ζ and that, without loss of generality, we may assume that +supx∈ ˜S u(x) ≤ log B, since this is already a pointwise upper bound on f. +Next, we construct the functions +x ∈ [−1, 1]d �→ U(x) := + + + + + +eu(x) +x ∈ ˜S +B +x ∈ S \ ˜S +ζ3 +x ∈ [−1, 1]d \ S +, +x ∈ [−1, 1]d �→ L(x) := + + + + + +el(x) +x ∈ ˜S +ζ3 +x ∈ S \ ˜S +0 +x ∈ [−1, 1]d \ S +. +Observe that U ≥ f ≥ L on [−1, 1]d. Furthermore, for ζ < 2−d, +� +( +√ +U − +√ +L)2dx = +� +˜S +(eu(x)/2 − el(x)/2)2dx + +� +S\ ˜S +Bdx + +� +[−1,1]d\S +ζ3dx +≤ +� +˜S +eu(x)(1 − eu(x)−l(x)/2)2dx + B · ζ2/B + 2dζ3 +≤ +� +˜S +B2(u(x) − l(x))2/4 dx + 2ζ2 +≤ B2 · ζ2/B2 + 2ζ2 = 3ζ2, +where we have exploited the fact that z �→ ez/2 is Lipschitz on [−∞, log C], with derivative bounded by +e(log B)/2/2 = +√ +B/2 to argue that e(u(x)−l(x))/2 − e0 ≤ +√ +B|u(x) − l(x) − 0|/2. +Since this construction can be carried out for any f, we conclude that we can construct a bracketing cover +of Ld,B at scale O(ζ) as the union of the bracketing covers of convex functions on each of the smaller sets in +Kd,ζ2/B. By Lemmas 22 and 23, the size of this cover is +exp +� +O((B/ζ)d−1) +� +· exp +� +O((log B/ζ3)(B)/ζ)d−1� += exp +� +�O((B/ζ)d−1) +� +, +and the claim now follows. Let us again observe that the resulting cover is improper, in that the maps U(x) +and L(x) are not log-concave. +B.3 +Regret Control for Bounded Lipschitz Laws on the Unit Box +As this subsection demonstrates, both Corollaries 15 and 11 rely on arguing that laws in DBox,Lip,B can be +estimated in a low-regret manner online. We argue this by exploiting the following result, which follows as a +simplification of the results of Wong and Shen on sieve estimators. +Lemma 24. (Adaptation of [WS95, Cor. 1 & Thm. 6])For every P ∈ DBox,Lip,B and t ≥ 1, there exists a +sieve MLE ˆq(·) = ˆq(·; Xt +1) and a constant A > 1 depending only on B such that for every ζ ≥ ζt, +P ∞ +� +KL(p∥ˆq) > 1 +Aζ2 log(1/ζ) +� +≤ A exp +� +−t +ζ2 +A log(1/ζ) +� +, +where ζt = �O(t−1/2(d+2)). +37 + +Proof. The cited results of Wong and Shen apply because densities of laws in DBox,Lip,B are uniformly upper +bounded. This directly yields the entirety of the statement, barring the scale bound on ζt. This scale is +determined by the same entropy integral fixed point equation that appears in Lemma 17, and for this instance, +the bound can be derived by using the standard fact that the Hellinger bracketing entropy of Lipschitz functions +on a box at scale η are controlled as O(η−(d+1)) [Vaa94]. +The sieve estimators in this result can be taken with a fair bit of lassitude. In particular, one explicit choice +is to construct for each ζ > 0 a bracket of the class DBox,Lip,B at scale ζ, and choose a representative density +within each bracket of the class. The sieve MLE then involves choosing a ζ at each time, and estimating the +law as the maximiser of likelihood amongst the aforementioned representative densities. Importantly for us, +the lower brackets in these bracketings can be taken to be uniformly larger than 1/B, and the upper brackets +smaller than B, since p ∈ [1/B, B], and as a result the sieve estimates are uniformly bounded between 1/B +and B. +Below we first show Corollary 15 using the above results, and then show Corollary 11 follows as a simple +consequence of this argument. +Proof of Corollary 15. As argued in the main text, the expected rejection time is bounded as E[τ] ≤ � πt + +O(T0), where T0 = o(dH(p, L)−2(d+3)). We thus only need to show that a sequence πt exists such that � πt is +appropriately small, and that for any t, +P ∞ +� ρt(E ; p) +td2 +H(p, L) ≥ 1 +25 +� +≤ πt, +where p ∈ DBox,Lip,B and E are sieve estimators. We proceed to do so below. +For succinctness, we shall define ε = dH(p, L). Let A be the constant from Lemma 24, and set +T1 := min{t : ζ2 +t log(1/ζt)/A < ε2/200, ζt < 1/√e}. +Further let +ζ(ε) := max{ζ ∈ [0, 1/√e] : ζ2 log(1/ζ) ≤ Aε2/200}. +In the subsequent proof, we shall use Lemma 24 with ζ = ζ(ε) ≥ ζT1. To this end, we note that if ζ(ε) < +1/√e ⇐⇒ Aε2/200 < 1/2e, and in this case the equality ζ(ε)2 log(1/ζ(ε)) = Aε2/200 holds. From this, we +may derive6 that ζ(ε)2 > aε2/ log(1/ε) for some small enough constant a, and so that the exponent of the +upper bound of Lemma 24 is +ζ(ε)2 +A log(1/ζ(ε)) = 400ζ4 +A2ε2 ≥ +ε2 +A′ log(1/ε) +for some large enough constant A′. We shall also assume that A′ ≥ max(1, A). +Let E be a choice of sieve estimators such that for every t, x ∈ [−1, 1]d, ˆqt−1(x) ∈ [1/B, B], which can be +ensured due to the discussion above. Notice, by the independence of the data {Xt}, that for any t, +E[log p(Xt)/ˆqt−1(Xt)|Ft−1] = KL(p∥ˆqt−1). +Let θ ∈ (0, 1) and M ≥ 0 be two parameters of argument that we shall set later. Let us consider the case +of t = T1 + τ for some τ ≥ MT1. +Since for each τ > 0, ζT1+θτ ≤ ζT1 ≤ ζ(ε), the bound of Lemma 24 is effective at each time s ∈ [T1 + θτ : +T1 + τ] with ζ = ζ(ε). As a result, applying Lemma 24 to each s in this range, and exploiting the behaviour +of ζ(ε)2 established above, +P ∞(KL(p∥ˆqs−1) > ε2/200) ≤ A′ exp +� +−s +ε2 +A′ log(1/ε) +� +. +6This equation is equivalent to x log x = y for x = ζ(ε)2, y = Aε2/100 in the range 0 < x < 1/e. +The claim follows +by noting that the map x �→ x log(1/x) is monotonically increasing on [0,1/e], and verifying that for y ∈ [0, 1/2e], +y +2 log(1/y) · +log(2 log(1/y)/y) < y. Indeed, this inequality is equivalent to arguing that log(2 log(1/y)) < log(1/y) +⇐⇒ +y log(1/y) < 1/2, +which holds since the maximum value of y �→ y log(1/y) is 1/e < 1/2. +38 + +Next, by applying the union bound over s ∈ [T1 + θτ : T1 + τ] in the above result, we conclude that +P ∞ +� +∃s ∈ [T1 + θτ : T1 + τ] : KL(p∥ˆqs−1) > ε2 +200 +� +≤ +T1+τ +� +s=T1+θτ +A′ exp +� +−s +ε2 +A′ log(1/ε) +� += A′ exp +� +−(T1 + θτ) +ε2 +A′ log(1/ε) +� +· +1 +1 − exp (−ε2/(A′ log(1/ε)), +≤ A′2 log(1/ε) +ε2 +exp +� +−θτ +ε2 +A′ log(1/ε) +� +. +where the equality sums over the geometric series, and the final inequality uses that T1 ≥ 0 and that for +u < 1, 1/(1 − e−u) ≤ 2 +u. +Next, observe that since for any x ∈ [−1, 1]d, +1 +B2 ≤ +p(x) +ˆqt−1(x) ≤ B2, we have the bound | log(p(Xt)/ˆqt−1(Xt))| ≤ +2 log B. Therefore, the Azuma-Hoeffding inequality is applicable, and yields that for every τ ≥ 1, δ > 0 +P ∞ +� +T1+τ +� +s=T1+θτ +log +p(Xs) +ˆqs−1(Xs) > +T1+τ +� +s=T1+θτ +KL(p∥ˆqs−1) + (τ − θτ)δ +� +≤ exp +� +−(τ − θτ)δ2/8 log2 B +� +. +We proceed by setting δ = ε2/200 in the above, and applying the union bound, to conclude that there +exists a constant C such that +P ∞ +� +T1+τ +� +s=T1+θτ +log +p(Xs) +ˆqs−1(Xs) > (1 − θ)τε2 +100 +� +≤ exp +� +−(1 − θ)τε4 +C log2 B +� ++ C log(1/ε) +ε2 +exp +� +−θτ +ε2 +C log(1/ε) +� +. +(6) +Let us call the right hand side of (6) π(τ, θ). By the definition of ρt, and the boundedness of log +p(x) +ˆqs−1(x) for +every s, it follows that with probability at least 1 − π(τ, θ), +ρT1+τ(E ; p) = +T1+τ +� +s=1 +log +p(Xs) +ˆqs−1(Xs) +≤ 2(T1 + θτ) log B + (1 − θ)τε2 +100 +. +So long as we can choose θ, M such that the upper bound above is smaller than (τ +T1)ε2/25, the inequality +(6) will limit the probability that ρT1+τ > ε2(T1 + τ)/25, which is precisely our goal. But observe that this +indeed occurs if (θ + 1/M) ≤ 3ε2/(200 log B), since in such a case +2(T1 + θτ) log B + (1 − θ)τε2 +100 +≤ τ +� +2(1/M + θ) log B + ε2 +100 +� +≤ τ +� +3ε2 +200 log B · 2 log B + ε2 +100 +� += τε2 +25 ≤ (T1 + τ)ε2 +25 +. +So, we may set θ = min(1/2, ε2/(100 log B)) and M = max(1, (200 log B)/ε2), and conclude that for any +τ ≥ MT1, it holds that +P ∞(ρT1+τ(E ; p)/tε2 > 1/25) ≤ π(τ), +where, for a constant C′, +π(τ) = exp +� +− +τε4 +C′ log2 B +� ++ C′ log(1/ε) +ε2 +exp +� +− +τε2 +C′ log(1/ε) · log B +� +. +39 + +Note that in the terminology of Theorem 14, πt = π(t − T1) for t ≥ (M + 1)T1. Of course we can always +provide the trivial bound πt ≤ 1 for t < (M + 1)T1. It remains to compute the resulting bound on expected +rejection time. To this end, observe by summing the appropriate geometric series that +� +t≥1 +πt ≤ (M + 1)T1 + +∞ +� +τ=MT1 +π(τ) +≤ (M + 1)T1 + +1 +1 − exp +� +−ε4/C′ log2 B +� + +C′ log(1/ε) +ε2(1 − exp (−ε2/C′ log(1/ε) · log B) +≤ O +� 1 +ε2 +� +T1 + �O +� 1 +ε4 +� +, +where the O bounds are as ε → 0, and we have hidden the dependence on B and log(1/ε). But, since in +Lemma 24, ζt = �O(t−1/2(d+2)), and since T1 is the first time that ζ2 +t log2(1/ζt) ≤ Aε2/200, we may conclude +that T1 = �O(ε−2(d+2)). The claim follows upon noticing that O(ε−2) · T1 = �O(ε−2(d+3)), and recalling that +ε = dH(p, L). +We conclude with a brief proof of Corollary 11 that exploits the bounds developed in the argument above. +Proof of Corollary 11. It suffices to argue that using the estimators in the proof of Corollary 15, for any +P ∈ DBox,Lip,B, +P ∞ +� +lim sup ρt(E ; p) +td2 +H(p, L) ≤ 1 +25 +� += 1. +This follows since for each t, +P ∞ +� ρt(E ; p) +td2 +H(p, L) > 1 +25 +� +≤ πt, +and � πt < ∞, which yields precisely the above relation by the Borel-Cantelli Lemma. +40 + diff --git a/99E1T4oBgHgl3EQf8QUL/content/tmp_files/load_file.txt b/99E1T4oBgHgl3EQf8QUL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e996bcbad08d3ff2148688b2c71506896702f9b9 --- /dev/null +++ b/99E1T4oBgHgl3EQf8QUL/content/tmp_files/load_file.txt @@ -0,0 +1,1630 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf,len=1629 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='03542v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='ST] 9 Jan 2023 A Sequential Test for Log-Concavity Aditya Gangrade1, Alessandro Rinaldo1 and Aaditya Ramdas12 agangra2@andrew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='edu, arinaldo@cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='edu, aramdas@cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='edu 1Department of Statistics and Data Science, Carnegie Mellon University 2Machine Learning Department, Carnegie Mellon University Abstract On observing a sequence of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' data with distribution P on Rd, we ask the question of how one can test the null hypothesis that P has a log-concave density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This paper proves one interesting negative and positive result: the non-existence of test (super)martingales, and the consistency of universal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' To elaborate, the set of log-concave distributions L is a nonparametric class, which contains the set G of all possible Gaussians with any mean and covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Developing further the recent geometric concept of fork-convexity, we first prove that there do no exist any nontrivial test martingales or test supermartingales for G (a process that is simultaneously a nonnegative supermartingale for every distribution in G), and hence also for its superset L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Due to this negative result, we turn our attention to constructing an e- process — a process whose expectation at any stopping time is at most one, under any distribution in L — which yields a level-α test by simply thresholding at 1/α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We take the approach of universal inference, which avoids intractable likelihood asymptotics by taking the ratio of a nonanticipating likelihood over alternatives against the maximum likelihood under the null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Despite its conservatism, we show that the resulting test is consistent (power one), and derive its power against Hellinger alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' To the best of our knowledge, there is no other e-process or sequential test for L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 1 Introduction Log-concavity is an important and prevalent modelling assumption in the modern study of shape-constrained nonparametrics [Sam18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Log-concave distributions include many common families of densities, including normal, exponential, extreme-value, and logistic distributions, and further are frequently justified in diverse application domains including economics, reliability theory and filtering in engineering, and survival analysis in medicine [BB06].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' At the same time, the family is technically amenable, and admits a unique maximum likelihood estimate with a well developed minimax theory and computationally efficient estimators [CS10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' CDSS18;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' KDR19;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Axe+19;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' DR11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' RS19;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' CSS10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' As a result, log-concave densities offer practitioners a broadly applicable and usable structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Given the attractive properties of estimation within the log-concave family, tests for membership in the same are an important and necessary line of investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We note that along with the applications mentioned above, such tests also have theoretical interest;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' for instance, in much of computational learning theory, efficient learning algorithms are only known when covariates are sampled according to a log-concave distribution [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' KKMS08].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' While the estimation of log-concave densities has seen significant advances over the past decade or two (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=', the citations above, and the survey by Samworth [Sam18]), testing for log-concavity has been relatively poorly developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Indeed, prior to 2021, there were no valid and powerful tests for the same— both theoretically and practically—outside of certain restricted one-dimensional settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In a significant development, recent work of Dunn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' [DGWR21] has developed such a test, based on the Universal Inference strategy of Wasserman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' [WRB20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Our work is concerned with testing log-concavity in a sequential setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Concretely, we assume that we are given streaming access to a sequence {Xt} that are drawn independently and identically from some d- dimensional density p, and we wish to test the membership of p within the family of log-concave densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Such a sequential test can be identified with a stopping time τ, where stoppage indicates rejection of the null 1 hypothesis, and the test is α-valid if under the null, the probability that τ < ∞ is bounded by α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The principal attractiveness of such sequential tests arises from their adaptivity: rather than fixing a number of samples a priori, the test may adapt to the difficulty of the underlying instance, rejecting earlier in easier settings, and allowing for a greater number of samples to detect subtle deviations from the null hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Below, we first set up some notation, and then proceed to contextualise our study, and give a brief overview of the contributions of our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='1 Problem setup and background We begin by describing the notation needed for our discussion, the testing problem under consideration, and the fundamental notions of test martingales and e-processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We shall give further definitions and details in §2, as well as later in the text as the context arises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Spaces and measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Let {Xt} = (X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' ) denote a sequence of d-dimensional random vectors with entries indexed by t, which are measurable maps from Ω := (Rd)N to Rd, endowed with the cylindrical Borel sigma- algebra B(Rd)N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We use typewriter style fonts, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' P, to denote laws of random processes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' probability measures on (Ω, B(Rd)N)), and standard fonts, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' P to denote laws on (Rd, B(Rd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We use F = {Ft} to denote the natural filtration of the process {Xt}, where Ft := σ(X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' , Xt), for each t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For a Borel probability measure P on Rd, we use P ∞ to denote the law of an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' process drawn according to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We use D to denote the set of probability measures on Rd with Lebesgue densities, and D∞ = {P ∞ : P ∈ D}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For P ∈ D, we use p to denote its Lebesgue density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For technical convenience we define D1 := {P ∈ D : E[max(0, log p(X))] < ∞, E[∥X∥] < ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' A set of laws P is said to be mutually absolutely continuous (m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=') if for all P, Q ∈ P, P ≪ Q ≪ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Finally, we frequently use Xt 1 := (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' , Xt) to denote finite prefixes of {Xt}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Log-concave measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' A function f : Rd → R is said to be log-concave if there exists a concave function g such that f = eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' If f is further a density with respect to the Lebesgue measure, then it is said to be a log-concave density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We denote the set of measures with log-concave densties as L, and use L∞ to denote the set of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' log-concave measures on euclidean sequences, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' L∞ := {P ∞ : P ∈ L}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Sequential test for log-concavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The testing problem of interest is formulated as follows: Let Xt i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' ∼ P for some unknown P ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We wish to test the null hypothesis H0 : P ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' A sequential test corresponds to {Ft}-adapted stopping time, representing the (possibly infinite) time at which the test stops and rejects the null hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We shall refer to this stopping time as the rejection time of the sequential test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' A test is said to be α-valid if its rejection time, τ satisfies that sup P ∈L P ∞(τ < ∞) ≤ α, meaning that, under the null, the probability of ever rejecting, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' of incurring a Type I error, is at most α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Similarly, a test is said to be asymptotically (1 − β)-powerful against Q ⊂ D \\ L if the probability of failing to reject the null under any distribution in the alternative Q (also know as a type II error) is uniformly bounded by β: inf Q∈Q Q∞(τ = ∞) ≤ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' A test is said to be consistent against Q if it is asymptotically 1-powerful against the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Note that, when consistent, these tests are typically called ‘power-one tests’ (following Robbins) to differentiate them from the traditional Waldian sequential testing paradigm for which stopping does not imply rejection of the null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Test martingales, test supermartingales and e-processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We briefly survey key notions underlying our discussion, namely test martingales, and e-processes, leaving details to §3 and §4 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Definition A process {Mt} is a nonnegative supermartingale (NSM) with respect to a filtration {Ft} and a law P if it is adapted, nonnegative, and EP[Mt|Ft−1] ≤ Mt−1 for each t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' If the inequality is further an equality at each t, then {Mt} is a nonnegative martingale (NM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We shall succinctly say that such a process is a P-NSM or P-NM respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 2 Obviously, every P-NM is also a P-NSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' An important basic inequality of Ville [Vil39] controls the tail behaviour of NSMs: if {Mt} is a P-NSM such that M0 = 1, then for every α ∈ (0, 1], P(∃t ≥ 1 : Mt ≥ 1/α) ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The result above is a sequential (time-uniform) analogue of Markov’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Equivalently, one can make claims at arbitrary stopping times: for all stopping times τ, P(Mτ ≥ 1/α) ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This can be seen by applying the optional stopping theorem for NSMs [Mey66, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' V, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 28] and Markov’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We now extend the above notions to composite families of sequential laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Throughout this paper we shall take the filtration to be the natural filtration of the data, and will leave it implicit in our definitions below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Definition For a set of sequential laws P, we say that a process {Mt} is a P-NSM if {Mt} is a P-NSM for every P ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Similarly, {Mt} is a P-NM if it is a P-NM for every P ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' A P-NSM such that M0 = 1 is called a test supermartingale for P, and a P-NM such that M0 = 1 is called a test martingale for P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Observe that test supermartingales satisfy Ville’s inequality for each P ∈ P, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=', if {Mt} is a test super- martingale for P, then for every α ∈ (0, 1], ∀P ∈ P, P(∃t ≥ 1 : Mt ≥ 1/α) ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' (1) Test supermartingales are so named because they form the canonical path to sequentially testing composite hypotheses, which is encapsulated entirely by the above relation, in that valid tests can be derived by rejecting only when a test supermartingale crosses a threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' They are particularly interesting in nonparametric settings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' for example one can use them to sequentially test the mean of a bounded random variable [WR23], for testing symmetry [RRLK20], for two-sample testing [SR21], independence testing [PBKR22], and testing calibration [AHZ21], to mention only a few interesting sequential nonparametric problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We shall discuss test supermartingales extensively in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' E-Processes [RGVS22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' RRLK20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' GHK19;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' HRMS20] are a recently defined class of processes that will also play a central role in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Definition A process {Et} is called an e-process with respect to a sequential law P if it is non-negative, and for every stopping time τ, we have EP[Eτ] ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Similarly, {Et} is an e-process for a class of sequential laws P if it is an e-process with respect to every P ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' E-Processes have a variety of equivalent definitions [RRLK20, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In particular it is sufficient for the process to satisfy EP[Eτ] ≤ 1 for only bounded stopping times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' By the optional stopping theorem (which holds without restriction on stopping times for nonnegative supermartingales), notice that every test supermartingale for a class P is also an e-process for this class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Thus, e-processes generalise the notion of test supermartingales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We observe that a Ville-type relation also holds for e-processes, simply due to Markov’s inequality: if Et is an e-process for P, then for every α ∈ (0, 1], ∀P ∈ P, stopping times τ, P(Eτ ≥ 1/α) ≤ αE[Eτ] ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' (2) Much as Ville’s inequality over the class (1) captures the relevance of test supermartingales to sequential testing, the above inequality captures the relevance of e-processes to the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The notion of e-processes, along with the non-sequential analogue of e-values, is gaining vogue in recent work in statistics due to this key property, along with the fact that e-processes exist for many composite and nonparametric testing problems for which test supermartingales do not exist (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=', the recent survey by Ramdas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' [RGVS22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We will also encounter this situation in the current paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' It is important to note that test supermartingales or e-processes can directly be interpreted as evidence against the null hypothesis: since we expect them to be less than one under the null, the larger their realized value, the more evidence we have that the null hypothesis is wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Thus, there is no explicit need to threshold them at 1/α for some prespecified α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' one can alternatively simply report the final value at the final stopping time of the experiment (which can itself be arbitrarily chosen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Nevertheless, we present this paper in the language of level-α tests because that is far more popular, and we refer the interested reader to the aforementioned references for further discussion on e-processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='2 Inadequacy of Test (Super)Martingales, and the Power of E-Processes One dominant (but sometimes hidden) principle behind sequential testing of composite hypotheses is the use of nonnegative martingales (NMs), or nonnegative supermartingales (NSMs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Concretely, to test a composite hypothesis P ∈ P, one attempts to construct a P-test supermartingale {Mt}, which was defined earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' By Ville’s inequality (1), the chance that Mt ever exceeds 1/α under any null law is bounded by α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Thus, these test supermartingales immediately yield a valid test: reject the null when Mt ≥ 1/α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The associated rejection time, of course, is the Mt-hitting time of 1/α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Such tests have game-theoretic interpretations, through the fact that nonnegative (super)martingales represent wealth processes in betting games [RGVS22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For example, a P-test martingale is the wealth process of a gambler who bets against the hypothesis that the sequence {Xt} is drawn according some law in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The game is designed so that the gambler cannot hope to reliably (in expectation) make money if the null hypothesis is true;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' this is imposed by a restriction that under any law in P, the expected wealth multiplier in each round should be at most unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' However, for sufficiently rich classes P, such a game leaves the gambler powerless;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' the gambler is so constrained by the aforementioned restriction that the only option is to not bet at all (or throw away money).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This phenomenon was first observed in work on testing exchangability in discrete time binary processes by Ramdas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' [RRLK22], who demonstrated that any process {Mt} that is an NSM for all exchangable binary laws is, almost surely, a strictly decreasing process (the wealth starts at one and can only possibly go down).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' As a result, any test based on thresholding such processes must be powerless against any alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Our first technical contribution demonstrates an anlogous phenomenon in the setting of log-concave distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Specifically, we show that the smaller class of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Gaussian processes is not testable using NMs (or NSMs), since all such processes are trivial in the sense of being almost surely constant (or decreasing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The claim is summarised below, where G∞ denotes the set of all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Gaussian laws (of any mean and variance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' (Informal) There are no nontrivial G∞-NSMs or G∞-NMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' A fortiori, there are also no nontrivial L∞-NSMs or L∞-NMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Thus, log-concave densities represent a natural class of distributions that cannot be tested via martingales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Testing via E-Processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Given that one cannot test for log-concavity (or indeed, Gaussianity) using nonegative (super)martingales, we are left in a situation where the prevalent design paradigm for sequential testing is neutralised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' There are two contrasting lines of attack that can be employed instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The first of these involves designing a restricted filtration Gt, distinct from the natural filtration, under which there might exist nontrivial test supermartingales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Ramdas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' [RRLK22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' RGVS22] highlight the remarkable fact that shrinking a filtration could introduce new nontrivial (composite) test martingales when none existed in the original filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Such a strategy was notably used by Vovk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' [VNG03;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' FGNV12] to develop a sequential test for exchangeability, where as mentioned above, no nontrivial test supermartingales exist in the data filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' There are two main disadvantages to such an approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' First, such test martingales only yield an e-process for a restricted set of stopping times (those under the restricted filtration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' From an applied point of view, the use of such an e-process demands discipline from a practitioner—they cannot look at the raw data to decide when to adaptively stop (a predefined stopping rule, like the hitting time of 1/α is okay, but it may never be reached, in which case we may still wish to present the obtained evidence at the stopping time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Second, from a design point of view, the construction of appropriate filtrations is itself a subtle task that is heavily problem-dependent, and thus designing such tests is more of an art than a science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In particular, no such construction is known or obvious for sequential log-concavity testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In contrast, we follow the alternative strategy of testing via an e-process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Recall that a process {Et} is an e-process for a set of sequential laws P if, for every stopping time τ and every P ∈ P, EP[Eτ] ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Such processes bear a deep relationship to the aforementioned test martingales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Indeed, it has been argued that (admissible) e-processes must take the form infP∈P M P t , where each {M P t } is a P-NM [RRLK20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The same observation lends e-processes a gambling interpretation as the wealth process of a gambler against a ‘family of games’, wherein the gambler simultaneously plays a game against each P ∈ P, and their wealth is taken as the smallest wealth amongst these games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The gambler can then make money only if each of these games makes money, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=', if ∀P ∈ P, M P t grows without bound, which would then indicate that every P ∈ P can be rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' E-Processes offer a similar testing approach as the previously discussed test supermartingales, as elucidated 4 by the inequality (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Indeed, given an e-process {Et} for P, we can construct an α-valid test of membership in P by rejecting only if Et ≥ 1/α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Indeed, in this case, the rejection time is τα := inf{t ≥ 1 : Et ≥ 1/α}, and using the inequality (2), we may conclude that ∀P ∈ P, P(τα < ∞) ≤ α, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' this test is valid for the composite null P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Note further that the validity extends beyond this: let σ be any other stopping time with respect to the natural filtration of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We further have that P(Eσ ≥ 1/α) ≤ α, and thus no extraneous stopping criterion can affect the validity of the test, as long as rejection occurs only if Eσ ≥ 1/α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The theory and applications of e-processes have seen considerable development in the recent literature on sequential analysis (along with the more basic notion of e-variables in batched settings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The concept is attractive thanks to its flexibility and simplicity (despite generalizing nonnegative martingales), but construct- ing powerful e-processes is partly science and partly art [RGVS22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In composite testing, e-processes are of central importance since they do not encounter the same pitfalls as NSMs and NMs, and there do indeed exist nontrivial e-processes even on classes where no such NSMs exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Indeed, in some sense, e-processes can be shown to lie at the very core of sequential composite testing [RLKR22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='3 Test Using Universal Likelihood Ratios: A simple E-Process The universal inference strategy [WRB20] gives a simple and generic construction of e-processes when a maximum likelihood estimate can be easily computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' To contextualise this approach, we first consider the case of a point null and alternative P ∞ and Q∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In this case, classical sequential testing theory posits that the sequential likelihood ratio Lt = t� s=1 q(Xs) p(Xs) yields a valid and powerful test upon thresholding at 1/α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Indeed, under the null, {Lt} is an e-process, since it is an NM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Against simple nulls but composite alternatives, likelihood ratios such as the above are typically adjusted to account for the variety of possible alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' One way to do this is to replace the above numerator with an estimate ˆqs(Xs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Importantly, as long as this ˆqs is nonanticipating, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=', is Ft−1-measurable (depending only on the first t − 1 datapoints), the martingale property continues to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' To highlight this nonanticipation, we shall denote these estimators as ˆqs−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' A second option is to mix over alternatives, perhaps using some non-informative “prior”, but we will go with the first option in this paper because we are dealing with a highly nonparametric alternative (essentially the complement of all log-concave laws, or the unspecified subset of those against which one may hope to have power) — it is easy to use kernel density estimates for ˆqs, but not so easy to mix over such a loosely specified nonparametric alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The sequential universal likelihood ratio statistic (ULR) extends the above to composite nulls when a maximum likelihood estimator (MLE) is computable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Concretely, the statistic is as follows: let ˆqt−1 be any predictable probability density, that is ˆqt−1 may be expressed as a function of only {X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' , Xt−1} and additional independent randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' As before, we should think of ˆq as trying to estimate the underlying law p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Let ˆpt be the MLE over the null class L with the data Xt 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=', ˆpt = arg max ˆp∈L � s≤t log ˆp(Xs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Notice that, unlike ˆqt−1, the MLE ˆpt makes use of Xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The sequential ULR statistic is the process Rt := � s≤t ˆqs−1(Xs) ˆpt(Xs) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 5 (Of course, if the numerator was simply � s≤t ˆqt(Xs), where ˆqt is an MLE over a larger class calculated using {X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' , Xt}, then we would get the usual generalized likelihood ratio process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' However, we will handle very rich nonparametric alternatives over which computing the MLE is for all practical purposes impossible, and further, for irregular models like log-concave distributions, such generalized likelihood ratios are very ill-behaved and not well understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=') The principal factor underlying the utility of Rt is that it is an e-process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Indeed, for any P ∈ L, and t, Rt is dominated by Ft(P) = � s≤t ˆqs−1(Xs)/p(Xs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Further, {Ft(P)} is a P ∞-martingale started at 1, due to the predictability of ˆq, and thus for any stopping time τ, EP ∞[Rτ] ≤ EP ∞[Fτ(P)] ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Notice in the argument above that while the e-process is dominated by a P ∞-martingale, it is not itself a martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Indeed, this property is crucial to the existence of nontrivial e-processes even when there are no such test martingales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We note that this property of domination by a P ∞-NM for every P ∈ L (or in general P-NM for P ∈ P) is equivalent to the e-process property itself, and can be taken as an alternate definition of the same [RRLK20, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Due to the above observation, the ULR e-process yields a valid test upon thresholding at 1/α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The power of any such test relies on the two aspects of how well ˆpt and {ˆqs}s≤t estimate the underlying law p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Indeed, we argue in §4 that if p ̸∈ L, then � s≤t p(Xs)/ˆpt(Xs) must grow exponentially with t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Thus, as long as the sequential estimates ˆqt approximate p well in a cumulative regret sense, the procedure above must be consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Concretely, define the regret of prediction using {ˆqt} as ρt(ˆq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' P) := � s≤t (− log ˆqs−1(Xs)) − � s≤t (− log p(Xs)), so that better estimation results in lower regret, and define the ‘well-estimable’ class Q(ˆq) := {P : ρt(ˆq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' P)/t → 0 P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' as t ր ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In Section 4 we show the following: Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' (Informal) Let Rt denote the ULR e-process with the sequential estimator E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Then the test that rejects when Rt ≥ 1/α is α-valid, and consistent against Q(ˆq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In fact, in §4, we demonstrate a more refined version of the above statement, which allows ρt to grow linearly, but at a rate bounded by the distance of p from log-concavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In any case, we comment that the class Q(·) above is quite rich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For instance, using sieve estimators yields low-regret estimation in the above log-loss sense for nonparametric classes such as laws on compact intervals with smooth and bounded densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The ULR e-process thus gives a powerful test for log-concavity against a rich set of alternates, even though no test martingale can deliver such properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Our work thus offers further insight into the sequential testing of rich composite nulls, and the primacy that e-processes must take in the modern study of the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Along with the above asymptotic consistency result, we further derive finite rejection rate bounds by controlling the typical rejection time of the ULR e-process in terms of the Hellinger distance of the alternaive law from log-concavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In particular, we show explicit bounds on typical rejection times against Lipschitz and bounded laws on the unit box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The above theoretical exploration is augmented with simulation studies on a simple parametric family comprising a mixture of two Gaussians to empirically evaluate the validity and power of the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We find that in small dimensions d ≤ 3, the tests show excellent validity, as well as reasonable power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We further use this simulation study to highlight the role of the quality of the estimators ˆqt in the power of the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Summary of Contributions To summarise, this paper is concerned with the theoretical and methodolog- ical aspects of sequential testing for log-concavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We first show a negative result that demonstrates that the approach of constructing test (super)martingales is powerless for testing this class of laws, and along the way also offering simple characterisations of the fork-convex hull of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' sequential laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In the positive direction, we propose using the Universal Inference based e-process as a way to test log-concavity in the absence of test martingales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We theoretically demonstrate both the consistency of the resulting sequential test, along with concrete adaptive bounds on typical rejection time under a wide class of alternatives, and illustrate the same via simulation studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 6 2 Definitions, and Background on Log-Concave Distributions We begin with basic background on log-concave distributions, and necessary notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We refer the reader to the survey of Samuard and Wellner for further details [SW14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Log-Concave Laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' A distribution P on (Rd, B(Rd)) is called logarithmically concave (henceforth log- concave) if for every pair of compact sets A, B and λ ∈ (0, 1), P(λA + (1 − λ)B) ≥ P(A)λP(B)1−λ, λA + (1 − λ)B is the Minkowski sum {λx + (1 − λ)y : x ∈ A, y ∈ B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' It is well known that a distribution that admits a density with respect to the Lebesgue measure is log-concave if and only if P(dx) = eg(x)dx for a concave function g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Recall that L denotes the class of log-concave distributions with density on Rd, while L∞ denote the set of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' sequential laws P ∞ for P ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Log-Concave M-projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Recall that D denotes the set of laws on (Rd, B(Rd)) that admit densities with respect to the Lebesgue measure, and that D1 := {P ∈ D : E[max(0, log p(X))] < ∞, E[∥X∥] < ∞}, where p(·) is the density of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For every P ∈ D1, there exists a unique law LP := arg min L∈L KL(P∥L), where KL(·∥·) is the KL-divergence, called its log-concave M-projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We shall abuse notation and use Lp to denote the Lebesgue density of LP (one is admitted as long as P ∈ D1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For a set of points {xt 1}, t ≥ d + 1, the log-concave maximum likelihood estimator (MLE) is the log-concave M-projection of the empirical law Pt = � s≤t δxs/t, denoted ˆPt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Most commonly, we shall refer to its Lebesgue density, ˆpt, which may equivalently be defined as ˆpt := arg max log f is a concave function f≥0, � f=1 � s≤t log f(xs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The log-concave MLE has extremely favourable theoretical properties when Xt 1 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' ∼ P for some P ∈ D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For instance ˆpt → Lp in the strong sense that ∃a > 0 : � ea∥x∥|ˆpt(x) − Lp(x)| → 0 almost surely [CS10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Locally Absolutely Continuous Sequential Measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Let Γ denote the standard Gaussian law on Rd, γ its density, and let Γ = Γ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Notice that Γ is the law of a white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' A sequential law P is said to be locally absolutely continuous (l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=') with respect to Γ, denoted P ≪loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Γ, if for all t, the law of the finite prefix P|t(·) := P(Xt 1 ∈ ·) is absolutely continuous with respect to Γ|t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Such l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' laws admit a density process, denoted ZP t := dP|t dΓ|t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' As an example, if P = P ∞ for some law P with Lebesgue density p, then P ≪loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Γ, and ZP t = � s≤t p(Xs)/γ(Xs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Of course, we may specify sequential laws (that are ≪loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Γ) by specifying their density processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Note that ZP t is a likelihood ratio process with respect to Γ, and so is a Γ-martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We shall henceforth use Γ as a reference measure for sequential laws, and almost entirely work under laws that are ≪loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Notice that if ZP t−1 > 0 then for {Xt} ∼ P, ZP t /ZP t−1 is the Γ-conditional density of Xt given Xt−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Further, ZP t−1 = 0 =⇒ ZP t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' As a result, we may write for any adapted process {Mt} that ZP t−1EP[Mt(Xt 1)|Ft−1] = ZP t−1EP[Mt(Xt 1)1{ZP t > 0}|Ft−1] = EΓ[MtZP t 1{ZP t > 0}|Ft−1] = EΓ[MtZP t |Ft−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' From this, we observe that a process {Mt} is a P-NSM if and only if {ZP t Mt} is a Γ-NSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Indeed, if the former is true, then we conclude from the above that EΓ[MtZP t ] ≤ ZP t−1Mt−1, while if the latter is true, then we can conclude that ZP t−1EP[Mt|Ft−1] ≤ ZP t−1Mt−1 ⇐⇒ 1{ZP t−1 > 0}EP[Mt|Ft−1] ≤ 1{ZP t−1 > 0}Mt−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Since 1{ZP t−1 > 0} holds P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=', it follows that EP[Mt|Ft−1] ≤ Mt−1 P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=', and so Mt is a P-NSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' By maintaining equalities in the above analysis, the analogous statement also holds for NMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' These facts are quite useful in our later study of fork-convex hulls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 7 3 There Are No Nontrivial Test Supermartingales for Log-concavity We begin with defining a natural notion of triviality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Definition An NSM {Mt} is said to be trivial if Γ(∃t : Mt > Mt−1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' An NM {Mt} is said to be trivial if Γ(∃t : Mt ̸= Mt−1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In words, a NSM (NM) is trivial if, almost surely, it is a non-increasing (constant) process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For the remain- der of this section, we will set {Ft} to be the natural filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We recall the notion of test supermartingales for a class of laws P, which we shall refer to as just nonnegative supermartingales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Definition For a set sequential laws P, we say that a process {Mt} is a P-NSM if {Mt} is a P-NSM for every P ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Similarly, {Mt} is a P-NM is it is a P-NM for every P ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' With these definitions in hand, we state the main result of this section, the proof of which is left to §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' There are no nontrivial G∞-NSMs or G∞-NMs under the natural filtration, and a fortiori, there are no nontrivial L∞-NSMs or L∞-NMs under the natural filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' As discussed by Ramdas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' [RRLK22], the above result implies that any valid level-α sequential test for log-concavity based on thresholding L∞-NSMs or L∞-NMs must be powerless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Indeed, in the former case, such a test against any law that is locally absolutely continuous with respect to Γ will almost surely never exceed its starting value, and thus will almost surely never reject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Intuition behind the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The result arises from a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' To illustrate this, suppose {Mt} is a G∞- NSM, and that for some time t, given Ft−1, it increases on the event {Xt ∈ O} for some open ball O, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=', conditionally on Xt−1 1 = xt−1 1 , {Xt ∈ O} ⊂ {Mt > Mt−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Notice that due to nonnegativity, at worst it could be zero outside the ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Now, consider a Gaussian GO of such a small variance that GO(O) ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' By tuning this variance, we can ensure that Mt > Mt−1 with probability arbitrarily close to 1 given the history, and since the drop in the Mt remains bounded outside of the ball, this ensures that the conditional expectation of Mt strictly increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Since this violates the supermartingale property against G∞ B ∈ G∞, we must conclude that no such ball O exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Of course, the set on which {Mt} increases need not contain any ball, but still be of nontrivial mass, not to mention that this set may vary with the history in a complex way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We address such gaps by exploiting the notion of fork-convexity [RRLK22] which serves as a sequential analogue of convexity especially germane to (super)martingale properties, and is treated in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In particular, it holds that any process {Mt} that is a G∞-NSM (or NM) is also a NSM (or NM) with respect to any sequential law in the ‘fork-convex hull’ of G∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The main argument then demonstrates that the fork-convex hull of G∞ is incredibly rich, and contains the laws of arbitrary independent processes with density (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=', processes of jointly independent {Xt} such that Xt ∼ pt ∈ D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This large set of laws entirely obstructs the NSM (or NM) property from holding in any nontrivial manner, essentially using a robust version of the previous intuitive example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Schematically, we take the following route to establish this result, where the forward direction of each implication exploits fork-convex combinations (and the reverse is trivial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' G∞-NSM G∞ ∗ -NSM ( � G∗)-NSM ( � G∗)-NSM ( � D)-NSM Trivial ⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒ Figure 1: Schematic view of the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' G∗ is the set of all finite mixtures of Gaussians and G∗ denotes its L1 closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For any set P, the class � P consists of independent sequential laws with marginals in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' See §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='2 for definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='1 Fork-convex Combinations In an algorithmic sense, for two laws P, Q, an α-convex combination R = αP +(1−α)Q is the law of the output of the following procedure: independently sample U ∼ P and V ∼ Q, and output X = U or V according to the outcome of an independent α-coin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Fork-convex combinations are the natural sequential extension of such 8 a procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Concretely, we sample two trajectories {Ut} ∼ P and {Vt} ∼ Q, release Xt = Ut for t ≤ s for some time s, and then flip a h-coin (where h can depend on the history) to decide whether the subsequent tail is Xt = Ut or Xt = Vt for t > s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Notice that this is a much richer notion than convex combinations: firstly, the decision to release Ut or Vt only needs to be made for a tail of the output sequence, and secondly, the mixture proportion can depend on the history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Formally, this is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Definition ([RRLK22]) Let P, Q ≪loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Γ be sequential laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Let s ∈ N, and let h ∈ [0, 1] be an Fs-measurable random variable such that Γ(h < 1, ZQ s = 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Then the (s, h)-fork-convex combination of P with Q is the sequential law R with density process ZR t := ZP t 1{t ≤ s} + � hZP t + (1 − h)ZQ t ZP s ZQs � 1{t > s}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We shall denote this succinctly as R = � P s,h −→ Q � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Notice that fork-convex combinations probabilistically allow single data-dependent change-points, or ‘switches’, from a law P to Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The ratio ZP s/ZQ s accounts for the fact that the prefix up to time s was drawn according to P in the case of a switch, and the condition on h ensures that ZQ s ̸= 0 when we switch to Q (informally meaning that the initial segment of data was not impossible under Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The importance of the above definition lies in the fact that fork-convex combinations preserve (super)martingale properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Recall from §2 that {Mt} is a P-NSM if and only if {ZP t Mt} is a Γ-NSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Now suppose {Mt} is both a P-NSM and Q-NSM, let R be a (s, h)-fork-convex combination of P and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For t ≥ s + 1, and we have EΓ[ZR t Mt|Ft−1] = hEΓ[ZP t Mt|Ft−1] + (1 − h)ZP s ZQs EΓ[ZQ t Mt] ≤ hZP t−1Mt−1 + (1 − h)ZP s ZQs ZQ t−1Mt−1 = ZR t−1Mt−1, where we have utilized the fact that h and Z· s are Ft−1-measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The same calculation is trivial for t ≤ s, and follows similarly for the martingale property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The same property extends considerably beyond finite combinations to closed fork-convex hulls, which generalise the standard notion of closed convex hull of sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Definition ([RRLK22]) A set is said to be fork-convex if it contains all fork-convex combinations of its elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Let P be a set of sequential laws that are locally absolutely continuous with respect to Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The fork-convex hull of P, denoted f-conv(P) is the intersection of all fork-convex sets containing P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The closed fork-convex hull of P, denoted f-conv(P) is the closure of its fork-convex hull with respect to L1(Γ) convergence of the likelihood ratio processes at every fixed time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Explicitly, the closure in the definition includes all processes Q such that there exists a sequence Qn with density process {ZQn t } such that ∀t, ZQn t → ZQ t in L1(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We shall refer to this as the local L1(Γ) closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This closure induces considerable flexibility into closed fork-convex hulls, making the notion a powerful concept in light of the following phenomenon, observed by Ramdas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' [RRLK22, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 13] whose argument we reproduce below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For a set of sequential laws P, a process is a P-NSM if and only if it is a f-conv(P)-NSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The result is evident for the fork-convex hull as an extension of the previous two-point calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This extends to closures as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Let {Mt} be the process in question, and suppose Pn → P in the sense above for Pn ∈ f-conv(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Let Zn t := ZPn t and Zt := ZP t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We know that for each t, Zn t → Zt in L1(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We need to show that ZtMt is a Γ-NSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' To this end, fix a t, and, by passing to a subspace, assume that Zn t → Zt and Zn t−1 → Zt−1 pointwise a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Now, since Zn t Mt is a Γ-martingale, using Fatou’s lemma yields EΓ[ZtMt|Ft−1] = EΓ[lim inf Zn t Mt|Ft−1] ≤ lim inf EΓ[Zn t Mt|Ft−1] ≤ lim inf Zn t−1Mt−1 = Zt−1Mt−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' It is worth noting that while the NSM property is preserved under closures above, the same is not necessarily true of the martingale property due to the use of Fatou’s Lemma when handling closures in the above proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Nevertheless, the NM (and indeed the martingale property without appeal to non-negativity) persists under fork-convex hulls, without the closure, giving us the following characterisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For a set of sequential laws P, a process is a P-NM if and only if it is a f-conv(P)-NM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='2 The Fork-Convex Hull of Independent Sequential Laws Proposition 2 gives us a concrete attack to showing the triviality of G∞-NSMs: we shall show that the fork- convex hull of this set is far too rich to allow the existence of nontrivial NSMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The bulk of our argument develops simple structural characterisations of fork-convex hulls of independent sequential laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This section describes this characterisation using three properties, whose proof we leave to §A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We begin with a key definition that sets notation for ‘independent sequential laws’ from a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Definition Let P be a set of distribution on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For a sequence of distributions {Pt}t∈N, we define �{Pt} as the sequential distribution of a stochastic process {Xt}t∈N such that all Xt are jointly independent, and for each t ∈ N, Xt ∼ Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We further define � P := {�{Pt} : Pt ∈ P ∀t}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' the set of laws of independent stochastic processes with laws at each time lying in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Note that � P is a much richer set than the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' sequential laws, which we denote P∞ := {P ∞ : P ∈ P}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In light of this, the following result demonstrates the richness of fork-convex hulls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Recall that a set of laws is mutually absolutely continuous (m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=') if every pair of laws contained in it is mutually absolutely continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Let P ⊂ D be a m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' set of laws with density on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Then, f-conv(P∞) ⊃ � P ⊃ P∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' To sketch the argument underlying the above, fix any P = �{Pt}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' It suffices to demonstrate a sequence of laws {RT}T ∈N, each generated by finite fork-convex combinations of P∞-laws (and their fork-convex combina- tions) such that for t ≤ T, the density process of RT and P agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The conclusion then follows under closure, since RT → P in the appropriate sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The concrete witness for the above Lemma is the following sequence R1 := P ∞ 1 , RT = � RT −1 T −1,0 −→ P ∞ T � , where each fork-convex combination is valid since P is m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In essence, this exploits the fact that fork-convex combinations let one switch between laws after a time of our choosing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' See A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='2 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Next, we exploit the convex combination properties of fork-convex combinations to demonstrate that fork- convex combinations of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' laws includes i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' products over mixtures as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' To this end, let us define the mixture classes as below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Definition Let P be a set of distributions on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For k ∈ N, we let Pk be the class of laws formed by k-fold mixtures of laws in P, and denote P∗ = � k∈N Pk as the class of laws formed by finite mixtures of laws in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Note that P∗ is well defined since Pk form an increasing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The second key result shown in §A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='2 is Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Let P ⊂ D be a m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' set of laws on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Then f-conv(P∞) ⊃ P∞ ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The key observation underlying the above is already demonstrated in showing that f-conv(P∞) ⊃ P∞ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' To see this, fix any P, Q ∈ P, and α ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We need to demonstrate a sequence of laws RT constructed via repeated fork-convex combinations that match the density process of R := (αP + (1 − α)Q)∞ for times up to T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This is realised as follows: R0 := P ∞, ST := � RT −1 T −1,α −→ Q∞� , RT := � ST T,0 −→ P ∞� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In the above, ST matches the density process of R up to time T by mixing between RT −1 (whose tail behaves as P ∞) and Q∞ appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' RT then switches the tail of ST to behave as P ∞ to enable the recursion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This argument extends to P∞ k for any arbitrary k by inducting over k (which is possible since a member of Pk is a mixture of a Pk−1 law and a P law).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Since k is arbitrary, this immediately extends to P∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Finally, we exploit the closure properties of fork-convex hulls under L1(Γ) to extend fork-convex hulls from product measures over a set to product measures over closures of that set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Let P be a set of distributions on Rd that have densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Then f-conv(� P) ⊃ � P, where P is the L1(Γ)-closure of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The above lemma is a straightforward consequence of the closure properties as detailed in §A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='3 Proof of the Absence of Nontrivial Test Martingales The previous section demonstrates that taking closed fork-convex hulls can yield significant expansion to product laws over sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This section exploits these properties to demonstrate the triviality of G∞-NSMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The key observation underlying this is the following standard fact about the richness of Gaussian mixtures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Recall that P denotes the L1(Γ)-closure of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' G∗ is L1(Γ)-dense in the set of all distributions with densities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=', G∗ = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The L1(Leb)-denseness of mixtures of Gaussians in D is a classical fact;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' for instance see the work of Alspach and Sorenson [AS72] or Lo [Lo72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' More recently, a considerably more robust result was presented by Bacharoglu [Bac10], who shows that Gaussian mixtures are dense in nonnegative simple functions in both an L1 and an L∞ sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This also suffices for our purposes since nonnegative simple functions are themselves L1- dense in nonnegative integrable functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The L1(Γ)-denseness follows since Γ admits a uniformly bounded density with respect to the Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We note that our argument extends to any such set, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=', to any P such that P = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The Gaussians serve as a convenient witness within L for which this property holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' With this in hand, we proceed as below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Let {Mt} be a G∞-NSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' First observe by Lemma 5 and Proposition 2 that as a conse- quence, {Mt} is also a G∞ ∗ -NSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Next, by Lemma 4 and Proposition 2, it is further a (� G∗)-NSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Similarly, by Lemma 6 and Proposition 2, we conclude that {Mt} is also a (� G∗)-NSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Finally, by Lemma 7, we conclude that {Mt} is a (� D)-NSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='1 We now argue that � D is too rich to admit nontrivial NSMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The argument is by contradiction—we assume that Mt > Mt−1 for some t with nontrivial probability, and use this to construct a law in � D that violates the NSM property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The argument repeatedly exploits the topological equivalence of (Rd)t and Rdt under the product and metric topologies respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We shall denote the Lebesgue measure in m dimensions as Lebm, and we note that the product Lebesgue measure on (Rd)t is identical to Lebdt, and use the latter to denote the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Let us proceed with the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For a natural number t, define the event At := {Mt > Mt−1, Mt−1 < ∞}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' that {Mt} increases at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' It suffices to argue that no matter the t, the mass of At is zero, since Γ(Mt−1 = ∞) must be zero due to integrability of Mt−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For the sake of contractiction, assume Γ(At) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For n ∈ N, define the approximations An t := {Mt ≥ Mt−1 + 1/n, Mt−1 ≤ n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The An t form an increasing sequence of sets, and converge to {Mt > Mt−1, Mt−1 < ∞} = At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Now, since Γ(At) > 0 and At ∈ Ft, we conclude that Lebdt(At) > 0 due to the mutual absolute continutity of Gaussians and Lebesgue measures on Euclidean spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Without loss of generality, we may assume Lebdt(At) < ∞ (since otherwise we may pass to a subset of At such that of positive and finite mass, using sigma-finiteness of the Lebesgue measure, and run the argument on this subset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Since An t ր At, we have by regularity of measure that Lebdt(An t ) → Lebdt(At), and in particular there exists an n such that Lebdt(An t ) ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Fix such an n for the remainder of the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Recall that an open rectangle in Rm is a Cartesian product of open intervals, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' a set of the form × m i=1(ai, bi) for ai < bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Similarly, we say that R is an open rectangle in (Rd)t if there exist open Rd-rectangles S1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' St such that R =× t s=1 Ss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The following statement is a consequence of basic topological and measure theoretic properties of Euclidean spaces, which we prove in §A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Let E ⊂ (Rd)t be such that Lebdt(E) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For every natural m ∈ N, there exists an open rectangle R in (Rd)t such that Lebdt(R) > 0 and Lebdt(R ∩ E) ≥ m m + 1Lebdt(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Exploiting the above result, we may construct a sequence of rectangles in (Rd)t, {Rm}m∈N each of positive mass such that Lebdt(An t ∩ Rm) Lebdt(Rm) ≥ m m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 1We can also argue this more directly: observe that taking closed fork-convex hull is an idempotent operation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' f-conv(f-conv(P)) = f-conv(P) (which follows from the facts that closed fork-convex hulls are fork-convex, and that closures of closed sets are invariant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Therefore, using the chain of Lemmata of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='2, f-conv(G∞) ⊃ � G∗, and so {Mt} is a (� D)-NSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 11 Now, since each Rm is a rectangle, there exists a law Dm ∈ � D such that the prefix restriction Dm|t = Unif(Rm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Indeed, if Rm =× t s=1 Sm s , then Dm = �{Dm s }, where Dm s = Unif(Sm s ) for s ≤ t, and Dm s = Γ for s > t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We claim that for large m, Dm witness a violation of the NSM property for {Mt}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We demonstrate this using the process {Nt} := {min(Mt, n + 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Notice that if {Mt} is a P-NSM, then so is {Nt}, since E[Nt|Ft−1] ≤ min(E[Mt|Ft−1], E[n + 1|Ft−1]) = min(Mt−1, n + 1) = Nt−1, and the nonnegativity follows since both Mt and n + 1 are nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Further, since Mt−1 ≤ n on An t , it follows that Nt ≥ Nt−1 + 1/n on An t as well, since n + 1/n ≤ n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Consequently, we have EDm[Nt] ≥ EDm � (Nt−1 + 1/n)1{Xt 1 ∈ An t } � + 0 = EDm � Nt−11{Xt 1 ∈ An t } � + Dm(An t ) n ≥ EDm � Nt−11{Xt 1 ∈ An t } � + m n(m + 1), where the final inequality exploits the fact that at least a m/(m + 1) fraction of the mass of Rm lies in An t , and we have used the nonnegativity of Nt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' However, since Nt−1 is upper bounded by n + 1, we observe that 0 ≤ EDm[Nt−11{Xt 1 ∈ (An t )c}] ≤ (n + 1)Dm((An t )c) ≤ n + 1 m + 1, and so EDm[Nt−11{Xt 1 ∈ An t }] = EDm[Nt−1] − EDm[Nt−11{Xt 1 ∈ (An t )c] ≥ EDm[Nt−1] − n + 1 (m + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' But now, we conclude that EDm[Nt] ≥ EDm[Nt−1] + (m/n) − (n + 1) m + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Choosing m > 3n2, and exploiting n ≥ 1, this implies that EDm[Nt] > EDm[Nt−1], thus contradicting the su- permartingale property of {Nt} under Dm (since supermartingales must have non-increasing mean sequences).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We conclude that it cannot hold that Γ(At) > 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=', Mt ≤ Mt−1 Γ-almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' But, since t is arbitrary, we immediately conclude that Γ(∃t ≥ 2 : Mt > Mt−1) ≤ � t≥2 Γ(Mt > Mt−1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The argument for NMs follows from this as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' If {Mt} is a � D-NM, then it is also an NSM, and thus almost surely does not increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' But this means that M1 − Mt ≥ 0 is also a nonnegative supermartingale, and therefore does not increase, which implies that Mt also does not decrease almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' It may be possible to develop a different argument that does not explicitly need to pass through the notion of fork-convex hulls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Perhaps one could directly work with the An s above, and replace Dm a by sufficiently skinny Gaussian Gm such that Gm(An s ∩ R) ≈ Gm(R) ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' However, there would still be sufficiently many technical details to iron out, so such an approach is not necessarily shorter or cleaner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' More importantly however, our chosen path of development above leads to a richer characterisation of fork-convex hulls of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' processes with densities, and further directly illustrates the utility of such a characterisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' It thus deepens our understanding of the important geometric concept of fork-convexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 4 The Sequential Universal Likelihood Ratio E-Process We begin by recalling the definition of e-processes from the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 12 Definition An {Ft}-adapted process {Et} is said to be an e-process for a set of sequential laws P if sup P∈P sup τ E[Eτ] ≤ 1, where the second supremum is over all stopping times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Further, if for some n ≥ 1 it holds a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' with respect to all P ∈ P that E1 = E2 = · · · = En−1 = 1, then we say that Et is an e-process for P started at time n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Next, we define the universal likelihood ratio (ULR) process [WRB20], which forms the main object of interest for this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Definition Let E denote a sequence of estimators {Et}t≥0 such that each Et : (Rd)t → D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' At any t, denote ˆqt = Et(X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' , Xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Finally, let ˆpt denote the log-concave maximum likelihood estimate over the data X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' , Xt (which exists if t > d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The ULR process is the statistic Rt(Xt 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' E ) := 1{t ≤ d} + 1{t > d} � d+1≤s≤t ˆqs−1(Xs) ˆpt(Xs) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We shall often suppress the dependence of Rt on Xt 1 and E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The initial setting of Rt = 1 for t ≤ d is to account for the fact that log-concave MLEs are known to exist only if at least d + 1 samples are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' As discussed in the introduction, {Rt} constitutes an e-process due to the predictability of ˆqt−1 and the fact that they are probability densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We formally state the validity of Rt as a proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For any E , the process {Rt} is an e-process for L∞ started at time d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Consequently, rejecting the null hypothesis when Rt ≥ 1/α results in an α-valid test for log-concavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' On exact MLEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The measures ˆpt need not exactly maximise the likelihood ratio in the above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Indeed, if instead of the exact log-concave MLE ˆpt we instead an estimate ˜pt such that � s≤t log ˜pt(Xs) ≥ − log(1/ε) + � s≤t log ˆpt(Xs), then εRt ·� ˆpt(Xs) ˜pt(Xs) is an e-process, and this can be thresholded at 1/α as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This observation is pertinent since practical procedures for computing the log-concave MLE of a dataset are inexact, and only approximate the solution up to a (user-specified) additive gap in the log-likelihood objective, and require computation that scales polynomially with the inverse of this additive gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For the remainder of this section, we shall equate laws P ∈ D1 with their density, denoted p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='1 Consistency of the ULR E-Process for Testing Log-Concavity Consistency of the ULR e-process depends strongly on the underlying estimator E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Indeed, as an extreme example, consider the case of ˆqt(Xt) = 1{Xt = X1}, for which the resulting Rt is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 0 for any time t ≥ d + 1 so long as the law P is continuous, and the test is thus powerless against such laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' It thus follows that the ULR e-process can only yield power against a set of laws determined by the estimator E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Concretely, we shall argue the same against the following set of ‘well estimable’ laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Below, dH below denotes the Hellinger distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Definition For a sequential estimator E , and a density p ∈ D1, define the prediction regret for a sequence {Xt} as ρt(E ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' p) := � s≤t log p(Xs) − log ˆqs−1(Xs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Further, let Lp denote the log-concave M-projection of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We define the class of distributions that are well estimable by E with respect to log-concavity as Q(E ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' c) := � p ∈ D1 : P ∞ � lim sup t→∞ ρt(E ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' p) td2 H(p, Lp) ≤ c � = 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 13 The main result of this section is that the ULR-based test is powerful against the above well-estimable laws, which is shown later in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' There exists a constant c > 1/25 such that if p ∈ Q(E ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' c) \\ L, then P ∞(Rt → ∞) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Consequently, the ULR e-process yields a consistent test against i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' draws from any distribution in Q(E ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The well-estimability condition above essentially requires that the distribution can be estimated well in a log-loss sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' distributions, one expects that for reasonable E , the estimates ˆqt converge to some ˆq, and thus the regret grows for large t as ρt ≈ tKL(p∥ˆq) (which could grow sublinearly in t if KL(p∥ˆqt) → 0, but the latter convergence is not required).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The class Q thus roughly consists of distributions can be estimated well in KL divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Such estimation can be a challenging task in complete generality, since the KL divergence is quite sensitive to mismatch in the tails of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' However, under mild restrictions such as compactness of support and smoothness, such estimability is quite forthcoming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Indeed, we give the following statement to illustrate this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This is proved in §B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Corollary 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Let DBox,Lip,B denote the set of 1-Lipschitz densities supported on the unit box [−1, 1]d and bounded between [1/B, B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' There exists a sequence of sieve maximum likelihood estimators E such that for every c > 0, DBox,Lip,B ⊂ Q(E ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' c), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=', the ULR e-process yields a consistent test against i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' draws from such distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' It is further interesting that the consistency of the test does not require that the regret ρt/t → 0, and only that it gets small enough relative to the Hellinger distance between p and its log-concave M-projection Lp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This signals that deviations from log-concavity may be detected far before the underlying law can be estimated, which is quite favourable theoretically, although its practical effects depend significantly on how large a c can be taken in Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Proof of Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We begin by defining σt(p) = � s≤t log p(Xs) − log ˆpt(Xs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Observe that log Rt = σt(p) − ρt(E ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Further, by assumption, we have that p ∈ Q(E , c) for some c, and thus for any ζ > 0, we have that ρt ≤ (1 + ζ)ctd2 H(p, Lp), for large enough t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Consequently, to show that Rt → ∞, it suffices to show that P ∞ almost surely, lim inf t→∞ σt td2 H(p, Lp) ≥ (1 + 2ζ)c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' (3) It is at this point that the following lemma is useful, the proof of which is left to §B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For any p ∈ D1, it holds that P ∞ � lim inf t→∞ σt(P) td2 H(P, LP ) ≥ 1 25 � = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The claim (3) thus follows so long as (1 + 2ζ)c ≤ 1/25, and since ζ > 0 can be taken arbitrarily small, this allows us to take any c < 1/25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We note that the constants in this argument are loose, and informal calculations suggest that it may be possible to improve c up to about 1/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The proof of Lemma 12 relies on strong convergence properties of the log-concave MLE ˆpt to the log-concave M-projection Lp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Recall that for a pair of functions u ≤ v, a bracket [u, v] is the set of all functions that lie between u and v everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' By exploiting a characterisation of the convergence properties of log-concave MLEs due to Cule and Samworth [CS10], Dunn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' [DGWR21, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 1] show that there is a small bracket that is well separated from P such that ˆpt eventually lies in this bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Conditioning on this event, we then exploit a classical result of Wong and Shen [WS95] which show linear growth of σt(p) with condiitonal probability at least 1−exp (−Ω(t)) , at which point the lemma follows by Borel-Cantelli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' As mentioned before, see §B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='1 for the full proof of Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 14 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='2 Power of the ULR E-Process for Testing Log-Concavity The argument underlying Theorem 10 is also amenable to deriving rates, under further restrictions on the underlying law P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' As in the previous section, we argue this using the decomposition log Rt = σt(p) − ρt(E ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='1 Challenges, and Context from the Theory of Log-Concave MLEs With the above approach, the argument breaks into two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Frstly, we assume that we use a good enough estimator E so that ρt is not too large with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Such an assumption is necessary for the approach we take, although in principle the test can be analysed using a different decomposition, in which case this assumption may perhaps be weakened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In any case, we observe that for concrete alternate hypotheses such as laws with Lipschitz densities supported on the unit hypercube, ρt can indeed be appropriately controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' It is worth noting, however, that the resulting rate bounds are strongly driven by the behaviour of ρt, and thus the estimator being considered, which limits the power of the results to follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The second part of the argument requires us to show that σt is large, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=', to argue that the log-concave MLE cannot represent the underlying law very well when it is not log-concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' While a natural statement, arguing this is challenging because this requires us to understand the behaviour of log-concave MLEs ‘off-the- model,’ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=', when the data is not drawn from a log-concave distribution itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' With the notable exception of Barber and Samworth [BS21], this task has not been undertaken in the literature, with most works focusing on on-the-model minimax rate bounds [KS16;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' KDR19;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Han21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' CDSS18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Let us consider this in some detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Tight analysis of the on-the-model log-concave estimation problem fundamentally relies on a subtle re- duction of the rates of log-concave MLEs to the problem of controlling deviations of empirical processes over convex sets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=', to that of controlling supC |P(C) − Pt(C)| under data drawn from P, where Pt is the empir- ical law, and the supremum is over convex sets in a bounded domain [CDSS18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Using this observation and a refined study of these deviations, Kur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' [KDR19] recently showed tight on-the-model estimation rates of the form dH(ˆpt, p) = O(n−1/(d+1)) when p ∈ L and d ≥ 3 (Han showed similar results, along with extensions to s-concave densities [Han21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' While significant elements of this study can be extended to analysing off- the-model behavior, the analysis ultimately cannot be applied to our situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The gap arises because their argument only upper bounds the quantity θt := EX∼P [log Lp(X)] − EX∼P [log ˜pt(X)], where for a small constant c, ˜pt ∝ max(c, pt) is a slight modification the log-concave MLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' When p = Lp, this object is a KL-divergence, and so is lower bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' However, when p ̸= Lp (that is, P is not log-concave), this quantity is may well be negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Notice that this is a problem for us precisely because E[σt]/t ≈ θt + EX∼p[log p(X) − log Lp(X)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' When p ̸∈ L, the second term can indeed be shown to be large, but the lack of a lower bound on the first term limits the applicability of such results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We also note that other aspects of the argument, which are relatively simple in on-the-model analysis (for instance, arguing that the mass p places on sets of the form {Lp(x) < γ} is small), are also rendered inoperative in off-the-model analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Of course, we can in principle exploit the results of Barber and Samworth instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' However, these re- sults give quite poor rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Roughly speaking, Theorem 5 of their paper [BS21] shows that off-the-model, dH(ˆpt, Lp) ≲ t−1/4d, and thus any analysis that exploits this result cannot hope to show that σt is large for t ≪ dH(p, L)−4d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This power of 4d arises since the analysis of [BS21] passes through a reduction to convergence of empirical laws in Wasserstein distance (which gives the relatively benign factor of d), and further suffers a 1/4th power slowdown relative to this convergence (which is both unavoidable, and leads to a 4d exponent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Our analysis sidesteps these issues by controlling the growth of σt on the basis of bracketing entropy (see §B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='1) bounds for the class of bounded log-concave laws on compact supports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Our bound below holds for all d but appears to be new for d ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Indeed, we show the following statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Let Ld,B denote the set of laws with log-concave densities that supported on [−1, 1]d and uniformly upper bounded by a constant B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' There exists a constant Cd dependending only on the dimension such that H[](Ld,B, ζ) = Cd �Θ � (B/ζ)max(d/2,d−1)� , where the �Θ hides terms that scale polylogarithmically with ζ or B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 15 We note that the lower bounds on the bracketing entropy implicit in the statement of Lemma 13 were already shown by Kim & Samworth [KS16, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 8], who further also showed the corresponding upper bounds for d ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' While not the central point of the paper, we develop upper bounds for the same when d ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' There are two salient technical points regarding the bound above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Firstly, observe that for d ≥ 4, the entropic bounds lie in the non-Donsker regime, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=', when the Dudley integral � ε 0 � H[](Ld,B, ζdζ does not converge due to a blow-up near ζ = 0, which typically (but not always) represents a slowdown in the convergence rates that can be shown via entropy integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Secondly, for d ≥ 3, the bound grows as ζ−(d−1) rather than as ζ−d/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The latter quantity is pertinent it is close to the growth rate of the bracketing entropy of convex sets which is ζ−(d−1)/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' see §B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This fact underlies the power of the previously discussed reduction of the analysis of log-concave MLE rates to control on the deviations of empirical processes over convex sets, which admit a slower entropy growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Lemma 13 is proved in §B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The growth bounds of this result ensure that for t ≳ dH(p, L)2(d−1), σt is linearly large in t, even when the underlying law is not log-concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This 2(d−1) exponent should be compared to the aforementioned 4dth power scaling that one expects to emerge from using the Wasserstein continuity based approach discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Of course, the dependence on d could potentially be improved even further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For instance, if the on-the-model analysis can indeed be extended to off-the-model, it is plausible to expect dependence of the form d + 1 instead of 2d − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' However, this remains a challenging problem for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' It is worth noting that while the techniques for the bounds in Lemma 13 exist in the literature, the bounds themselves appear to not have explicitly been observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We believe that this might be because it was previously observed that due to existing lower bounds on this entropy, the resulting growth rate bounds that emerged from entropic considerations could not be optimal for the rate analysis of the log-concave MLE, at least in on-the-model settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' It should also be noted that the bounds above are explicitly for compactly supported log-concave laws (which is a restriction, but a relatively mild one, due to the exponentially decaying tail enjoyed by all log-concave densities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Further note that that the brackets we construct for this setting are ‘improper’, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' the bracketing functions are themselves not log-concave, which may limit utility in direct analysis of the difference between ˆpt and Lp, but is good enough when studying the behaviour of σt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='2 Bounding Typical Rejection Times for the ULR Test As discussed previously, our analysis of σt passes through a bracketing entropy bound for bounded, compactly supported log-concave laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For such bounds to be effective, we need to ensure that the log-concave MLE ˆpt itself is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This is enabled by the quantity ∆P , defined for a law P as ∆P := min v:∥v∥=1 EP [|⟨v, X − EP [X]⟩|], which was identified by Barber and Samworth [BS21] as a means to lower bound the covariance of the log- concave projection of P, which in turn can be exploited to upper bound the supremum of Lp and (indirectly) ˆpt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Observe that ∆P roughly corresponds to the minimum eigenvalue of the covariance matrix of P — indeed, it is best seen as a robust version of the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' With this in hand, we are ready to state our main result, the proof of which is the subject of §B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Recall that dH(p, L) = infq∈L dH(p, q) and τα := inf{t : Rt ≥ 1/α} is the rejection time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Suppose p is supported on [−1, 1]d, and let πt → 0 be a sequence such that for every t, P ∞ � ρt(E , p) td2 H(p, L) ≥ 1 25 � ≤ πt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Then there exists a constant c ∈ [1/600, 1] and a natural number T0 such that for any t ≥ T0 + log(1/α) cd2 H(p,L), P ∞ (τα > t) ≤ πt + 1 c exp � −ctd2 H(p, L) � + 1 c exp � −ct∆2 P /d2� , and T0 = Cd · �O � ∆− max(d2/2,d2−d) P dH(p, L)− max((d+4)/2,2(d−1)) + d2∆−2 P � , where Cd depends only on d, and the �O hides terms depending polylogarithmically on dH(p, L), ∆P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 16 Observe that from the statement above we may conclude that the average rejection time is bounded as E[τα] = � t P ∞(τα > t) ≤ � t πt + O(T0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Here, the first term is driven by the predictability of p using the estimators E , while the second term is driven by our analysis of the noise scale of log-concave density estimation in off-the-model scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In typical situations, the former of these terms will dominate the resulting bounds, since typical alternate classes will be much larger than the class of log-concave distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For instance, using results on the estimation of uniformly lower-bounded Lipschitz densities [WS95, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' ], we show the following result about the set DBox,Lip,B introduced in Corollary 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Corollary 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For any constant B > 0, there exists a sequence of sieve maximum likelihood estimators E such that if p ∈ DBox,Lip,B, the ULRT rejection time τα is bounded in expectation as E[τα] = �O(dH(p, L)−2(d+3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The proof is in §B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We remark that the above rates adapt to the extent to which the underlying law p violates log-concavity in the sense that the time-scales of rejection are driven by dH(p, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Indeed, this represents an important advantage of sequential tests as opposed to batched tests, in that validity is retained, and detection is guaranteed at an adapted time-scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' On tightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We note that the exponents of Theorem 14, and in particular, Corollary 15 are likely loose for the problem of testing log-concavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This is an artefact of the analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' for instance, the slow rate in Corollary 15 is largely determined by the rate requirements for estimating Lipschitz laws on the unit box, which arises due to the πt terms present in Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' It is possible that this aspect can be improved, since nothing neccessitates that we use an estimator that captures the underlying density p well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Indeed, instead of analysing the prediction regret with respect to p itself, we could decompose R = σt(q) − ρt(E ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' q) for some other law q, perhaps lying in a smaller class of densities Q than those possible for p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' As long as (i) E does as good a job at prediction under the log-loss as any law in Q, and (ii) no matter what p ̸∈ L is, there is a law in Q that is ‘closer to’ p than any law in L, a similar analysis should be possible, although this requires possibly subtle off-model control on the behaviour of E , as well as a careful choice of Q itself to control the relative values of distances such as dH(p, Q) and dH(p, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' One such approach which appears promising for log-concave laws is to exploit s-concave densities to play the role of Q, which are particularly attractive since they form a rich extension of the class of log-concave laws, but nevertheless enjoy identical minimax MLE convergence rates as them [HW16;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Han21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 5 Algorithmic Proposal, and Simulation Study We now proceed to algorithmically describe the ULR e-process based test for log-concavity, and investigate the behaviour of a concrete implementation of the same on a simple parametric family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='1 Computational Aspects, Batching, and a Concrete Testing Algorithm Under specification of the sequential estimators E , and a method for fitting the log-concave MLE, the statistic Rt is explicitly computable, and thus naturally leads to implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' While the e-process is powerful against wide classes of alternatives, its implementation suffers from a fundamental computational issue, that arises due to the recomputation of ˆqt and ˆpt in each round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This cost grows superlinearly with t since since the entire denominator � ˆpt(Xs) must be evaluated on the entirety of the stream, and the cost of estimating this ˆpt is itself superlinear in the number of samples t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' A second issue arises upon increasing the data dimensions d, since computational costs of estimating ˆpt grow quite fast with this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Even though polynomial in d algorithms exist for computing the log-concave MLE [Axe+19], the fastest available method for this is typically hundreds of times slower when processing ∼ 100 points in even the modest d = 5 when compared to the time needed to process the same sized dataset for d = 1 [RS19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We address this issue by exploiting batching to reduce the computational load, which makes computations viable in the moderate d ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The idea is to wait to 17 accumulate I > 1 fresh samples before recomputing Rt, rather than updating it at every round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='2 Let us point out that such batched updates still retain the e-process property, and thus validity, as long as the ˆqt−1 remain nonanticipating over the entire batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Concretely, we may set a schedule, captured by an increasing sequence of times T = {tk}, and evaluate the statistic Rt(T ) = Rt−1(T )1{t ̸∈ T } + � 1{t ∈ T } � j≤k(t) tj � s=tj−1 ˆqtj−1(Xs) ˆptk(Xs) , where k(t) = max{k : tk ≤ t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In words, the schedule divides streams into a sequence of batches of size tk − tk−1, and each time a new batch is accumulated, we evaluate a new estimate ˆq on the previous batches, and re-evaluate the log-concave MLE on the entirety of the data seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This process continues to be dominated by a batched version of Ft(P), which retains the martingale property under P ∞, thus yielding validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The simplest viable schedule is to set tk = kI for a constant ‘batching interval’ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This effectively boils down to testing the log-concavity of p⊗I, which is valid since tensor products of log-concave laws remain log-concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Notice that such batching may result in a reduction in power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For instance, rejection can only occur at the time tk, and further the statistic may be deflated because data points with a large signal may be ‘washed-out’ due to milder behaviour across the remainder of the batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Nevertheless, we find in simulation studies that this drop in power is nominal, and comes at the cost of a significant improvement in runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' With this in hand, we can provide an explicit algorithmic description of our test below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Algorithm 1 Log-Concave Universal Likelihood Ratio Test 1: Input: Batching schedule {tk}∞ k=1 with t1 ≥ d + 1, estimator E , level α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 2: Initialise: Rt ← 1 for t ≤ t1, K ← 1, N1 = 1, t ← 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 3: while Rt < 1/α do 4: if t = tK then 5: ˆq = E (Xt−tK−1 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 6: Nt ← Nt−1 · �tK s=tK−1+1 ˆq(Xs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 7: ˆp ← L (XtK 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 8: Rt ← Nt · ��tK s=1 ˆp(Xs) �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 9: K ← K + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 10: else 11: Rt ← Rt−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 12: Nt ← Nt−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 13: t ← t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='2 Evaluating the ULR E-Process Test We investigate the behaviour of the test of Algorithm 1 on the following simple test-bed family of laws, where ed = 1d/ √ d is the unit vector along the all-ones direction in Rd,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' p(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' µ, d) := 1 2(2π)d/2 � exp � −∥x − µ 2 ed∥2/2 � + exp � −∥x + µ 2 ed∥2/2 �� , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=', balanced two component Gaussian mixture laws with means ± µ 2 ed and identity covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The norm of the mean-difference is precisely µ, which we assume without loss of generality to be nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' A small modification of this family of laws was also used as a test-bed for the non-sequential test proposed by Dunn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' [DGWR21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' These laws are extremely convenient for proof-of-concept investigation of tests of log-concavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Indeed, observe that up to a rotation, the d-dimensional law is a tensorisation of the one-dimensional law with a 2Note of course, that for our simulation study, the repetition of simulations required to study power and size mean that we only implement our fully nonparametric test for up to d = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Nevertheless, even this is reasonable to run for d = 6, wherein a single run over a horizon of 100 steps takes about 20s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 18 log-concave law (specifically a standard Gaussian in d − 1-dimensions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Since log-concavity properties are invariant under rotations, and since the log-concave M-projection of product laws is a product of the marginal log-concave M-projections [SW14], this gives a very simple characterisation of the log-concavity properties of this law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Concretely, the distance from log-concavity is purely a function of the norm of the mean-difference µ, and p(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' µ, d) is log-concave if and only if µ > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' These laws thus give us a simple way to check both the size and power of the test statistics, as well as study the effect of increase in dimension on the power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Finally, we give details of the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' All data reported is a mean over 100 runs of each experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' All simulations are run up to 100 time steps, which is mainly for computational practicality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Note thus that our size estimates are systematically lower than the true size (with infinte horizon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We run the case of d = 1, which is computationally the cheapest, over the longer horizon of 500 time steps to illustrate that not much changes in this case, at least as regards the empirical validity of our test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For d = 1, 2, 3, the tests are batched at an interval of I = 20, while for computational practicality the test is batched at I = 25 for d = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' These are significant fractions of the time horizon studied, but do not significantly lower power, at least for d = 1, as demonstrated by explicit simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' All code is implemented in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The nonparametric estimator used for E is the kernel density estimate as implemented in the ks package [CD18], and the log-concave MLE used is either from the logConDens package [DR11] for d = 1 or the fmlcd package (d = 2, 3, 4) [RS19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We note that the latter is not guaranteed to return the log-concave MLE since it optimises a non-convex approximation to the program defining the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' However, we find that compared to alternatives like the logConcDEAD package [CSS10], the fmlcd implementation retains similar validity and power, but runs significantly faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We also investigate using parametric Gaussian mixture model fits to illustrate the effect of inefficiency in E on power, for which we use the EM algorithm as implemented in the mclust package [SFMR16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' All simulations were executed on an AMD Ryzen 5650U processor, a medium range CPU for a laptop computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='1 Fully Nonparametric tests Figure 2 shows the behaviour of our instantiation of the algorithm with the fully nonparametric approach of using kernel density estimators as E over p(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' µ, d) for d = 1, 2 as µ is varied, run at the size α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='1, with I = 20 for d ∈ {1, 2, 3} and I = 25 for d = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We plot five traces which record the fraction of runs out of 100 independent runs that the test rejected the null hypothesis at times smaller than 20, 40, 60, 80, and 100, where 100 was the horizon over which the test was run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' There are three major observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Firstly, we observe that the test shows excellent validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Indeed, the null hypothesis holds true for µ ≤ 2, and the test does not reject more than a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='02 fraction of the time in either case in this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Secondly, we observe that at least for sufficiently large µ, all of the tests do reject within 100 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Finally, we notice that the power sharply drops as d increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' To concretely discuss this, let µ∗(d) be defined as the smallest value of µ for which PX∼p(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='µ,d)(τ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='1 ≤ 100) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The plots in figure 2 give us estimates of µ∗(d), which increase sharply with d—from about 6 in d = 1 to over 1000 in d = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='3 This reduction in power is perhaps expected, given the considerable deterioration in the nonparametric estimation rates with d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Nevertheless, we may question how much of the above decay in power is driven by the inefficiencies in fitting log-concave MLEs, and how much accrues due to the inefficiency of kernel density estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We investigate this effect in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='2 by studying Oracle tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Longer Run for d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' To show that the validity persists over longer time horizons, we implement the fully nonparametric method over 500 time steps for d = 1, using I = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Observe in Figure 3 that rejection under the null µ ≤ 2 is well controlled even at this increased timescale, while rejection rates steadily improve as the horizon grows, although the improvement is somewhat marginal over the horizon of 500 versus 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 3With pilot simulations in d = 5, we observe that µ∗(5) ≈ 1500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We note that these simulations were already too costly, in terms of time, to implement completely for d = 5, due both to the increased costs of fitting MLEs in higher dimensions, and due to the fact that as rejection rates decrease with dimension, more runs need to be executed over the whole horizon, which extends the total cost of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We hope to implement the method on larger computational resources for such moderate ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 19 Figure 2: Performance of the fully nonparametric test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Empirical rejection rates (over 100 simulations) at α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='1 versus the mean difference µ for fully nonparametric test implementations over four cases: d = 1, I = 20 (top left);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' d = 2, I = 20 (top right);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' d = 3, I = 20 (bottom left) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' d = 4, I = 25 (bottom right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The thin horizontal line plots the level α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='1, and the vertical line marks µ = 2 since p(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' µ, d) ∈ L ⇐⇒ µ ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Observe the strong validity properties in all plots, as well as the deterioration of power in higher dimensions, as signalled by the sharp increases in the scales of the X-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Figure 3: Performance of the fully nonparamet- ric test over long horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Empirical rejection rates (over 100 simulations) in the setting of Figure 2 for d = 1, ran over a horizon of length 500 with I = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Observe that the validity persists over this longer horizon, and that power improves for µ > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Figure 4: Effect of I on the fully nonparametric test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Empirical rejection rates (over 100 simulations) in the setting of Figure 2 for d = 1 and with varying I ∈ {1, 10, 20, 50}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Observe that the rejection rates for I = 10, 20 are roughly the same as for I = 1, while I = 50 suffers large losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 20 Figure 5: Performance of the partial oracle test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Empirical rejection rates (over 100 simulations) at α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='1 versus the mean difference µ for the partial oracle test implementations over four cases: d = 1, I = 20 (top left);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' d = 2, I = 20 (top right);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' d = 3, I = 20 (bottom left) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' d = 4, I = 25 (bottom right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Observe that in each plot, the power improves starkly relative to the fully nonparametric test (Figure 2), as indicated by a strong contraction of the scale of the X-axis, especially in higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Effect of Batching Interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' As seen in Figure 4, batch sizes of I = 10 and 20 have a mild effect on the rejection rates under alternate setting (µ ≥ 2) when compared to the direct I = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Interestingly, note that I = 20 does somewhat better than I = 10 in the setting of moderate µ (the range 4 − 6), and slightly loses power for larger intermediate µ (the range 6 − 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In turn, the no batching setting, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=', I = 1, is observed to suffer deterioration in its size (µ < 2), although this remains at an acceptable level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The large batch size I = 50 suffers the same validity issues as I = 1, but does even better than it for small but non-null values of µ (2-5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Power considerably deteriorates for larger µ (5-10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' While it is unclear how much of this is an artefact of the fact that the length of the horizon is only 100, and how much is directly due to the larger batching interval, the fact that I = 10, 20 perform well suggests that so long as the batching interval is a relatively small fraction of the horizon length, the loss in power is not too bad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='2 Oracle Tests, and the Effect of the Quality of E Oracle Tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' To probe the effect of the lossiness of the kernel density estimate on the power of the fully nonparametric test, we run ‘partial-oracle’ and ‘full-oracle’ oracle tests, which adjust E to exploit concrete information about the underlying laws p(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' µ, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In the partial-oracle, we adjust E to estimate a two-component Gaussian mixture model instead of a kernel density estimate, and in the full-oracle case, we directly set ˆqt−1(·) = p(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' µ, d), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=', we exactly evaluate the density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We expect that under data drawn from p(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' µ, d), these tests are more powerful than the fully nonparametric tests discussed above, since the regret ρt(E ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' p) would reduce in the case of the partial oracle due to a reduced 21 Figure 6: Perforamance of the full oracle test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Empirical rejection rates (over 100 simulations) at α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='1 versus the mean difference µ for the full oracle test implementations over four cases: d = 1, I = 20 (top left);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' d = 2, I = 20 (top right);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' d = 3, I = 20 (bottom left) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' d = 4, I = 25 (bottom right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Observe the sharp improvement in power compared to Figure 2, especially in high dimensions, as indicated by a strong contraction in the scale of the X-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Observe also the improvement in power compared to Figure 5, in that the curves reach high power at about half the µ that is needed for the partial oracle test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' complexity of the estimation class;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' and, of course, would reduce exactly to 0 in the case of the full oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In either case, this effectively serves to increase Rt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' These oracle tests thus let us probe the extent of the loss in power at a fixed µ (and thus a fixed distance from log-concavity) that arise purely due to the decay in rate of convergence of the log-concave MLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In particular, the full oracle test captures exactly this effect, while the partial oracle test approaches this in a soft way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Figure 5 shows the performance of the partial oracle tests, and Figure 6 shows the same for the full oracle test for d ∈ {1, 2, 3, 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Comparing Figures 2 and 5, we see that for using the partial oracle yields a marked increase in power, at least for d > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This is evident in d = 2 by observing that the purple lines (overall rejection rate within 100 times steps) rises higher and is nonzero at smaller values of µ, as well as observing that the typical rejection time decreases substantially (for instance, rejection never happened below time step 60 in the fully nonparametric case, but is quite prevalent at higher µs under the partial oracle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In d = 3, 4 the effect is much starker - notice that the scale of the plot completely changes, from order of hundreds to tens in d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This suggest that using the parametric mixture of Gaussians estimate offers strong improvements over the nonparametric KDE estimate due to the reduced variance scale of this estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The above effect is seen even more starkly in the case of the fully oracle test, where each of the rejection rate curves is further improved (Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For instance, our estimate of µ∗(d) (the smallest µ such that Pp(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='µ,d)(τ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='1 ≤ 100) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='9)) is about halved for the full oracle case when compared to the partial oracle (and improved manifold relative to the fully nonparametric test).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 22 The Quality of E has a Strong Effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' These observations from the oracle tests indicate that the quality of the estimate offered by E is very important in driving the overall power of the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In these oracle examples, the quality improved by reducing the variance scale of the estimator, whilst keeping the bias at 0 (since the law p(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' µ, d) is representible by each of the estimator outputs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Of course, in practice we cannot always hope to reduce the variance scale of our estimates whilst keeping the bias zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Nevertheless, there is a tradeoff between the two implicit here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Indeed, as we discussed briefly in §4, it is possible to use a biased E in the test, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' one that does not strictly estimate p, so long as the output of E does a better job of representing p than the log-concave MLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The strong dependence of the testing power on E indicates the critical need to investigate this design freedom, and to study how the trade-off between the variance, in terms of the convergence rates of ˆqt, and the bias, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=', the distance of lim ˆqt from p, should be balanced to optimise the testing power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 6 Discussion Our work has shown that the sequential testing of log-concavity throws interesting challenges, in that the prevalent paradigm of test martingales cannot be fruitfully applied to this practically relevant setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In the process of doing this, we developed a characterisation of the closed fork-convex hulls of independent sequential laws on a continuous space, thus contributing to the theory of this new tool that characterises the nonnegative supermartingale property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We then showed that the universal likelihood e-process instead does yield powerful tests for log-concavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In particular, we demonstrated that these tests are consistent against large classes of nonparametric alternate laws, and further admit nontrivial rates, and made contributions to the off-the-model analysis of the convergence of log-concave MLEs, as well as the general theory of the power analysis of universal tests in order to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' These properties are validated by running the test over a simple parametric family of laws, which further demonstrates the critical role of the sequential estimator E in the power of the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Taking a broad view, the above can also be seen as a contribution to the emerging literature on e-processes, and in particular as additional evidence for the case that the study of sequential testing at large must exploit this powerful yet simple tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' A number of directions, both theoretical and methodological remain open in this interesting subject, a few of which we discuss below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Regarding fork-convexity, our characterisation in §3 and §A of the closed fork-convex hulls of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Gaus- sians can possibly be further enriched, and it would be very interesting to understand precisely which laws lie in this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Additionally, notice that sequentially testing the Gaussiantiy of an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' process itself is a basic problem that again cannot be tested using martingales (at least with respect to the natural filtration of the data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Construction and analysis of such sequential Gaussianity tests is a natural and interesting direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Of course universal inference is again a natural approach for this class, but it may be possible to take ad- vantage of translation and rotation invariance of the null hypothesis (all Gaussians) using methods developed in [PLHG22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Regarding the ULR e-process based test for log-concavity, §5 shows that the power of the fully nonpara- metric test can be quite limited particularly as the data dimension increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This observation was also made in the non-sequential setting by [DGWR21], who proposed using random one-dimensional projections as an interesting method to ameliorate this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In this test, rather than computing the full d-dimensional kernel and log-concave estimates, one projects the data onto many one-dimensional subspaces, and averages the e-values (nonnegative test statistics with expectation at most one under the null) that result from a one-dimensional test carried on each of these projected datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This approach not only has computational benefits due to the speed of one-dimensional density estimation methods, but also shows statistical benefits in the scenario of §5, in that the decay of power is considerably limited with dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Such projected tests are of course possible in the sequential setting as well, and are a natural next step to investigate, both methodologically and in terms of their theoretical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' On a broader scale, both the theoretical bounds and the simulations illustrate the critical role that the quality of the estimator E plays, both specifically in the power of the test for log-concavity, but also more generally in the use of the universal likelihood ratio e-process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' With this in mind, and recalling the implicit ‘bias-variance’ tradeoff in E as discussed in §4 and §5, investigating the choice of E relative to the null class 23 is an interesting question both in terms of practical methodological concerns, as well as theoretical concerns studying the power of e-process based tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Acknowledgments The authors thank Martin Larsson for insightful discussions on fork convexity, and Robin Dunn, for an implementation for a batched universal test for log-concavity that formed the backbone of the code underlying our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Rinaldo and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' were supported in part by the NSF grant DMS-EPSRC 2015489.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' References [AHZ21] Sebastian Arnold, Alexander Henzi, and Johanna F Ziegel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' “Sequentially valid tests for forecast calibration”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In: arXiv preprint arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='11761 (2021) (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' [AS72] Daniel Alspach and Harold Sorenson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' “Nonlinear Bayesian estimation using Gaussian sum ap- proximations”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In: IEEE Transactions on Automatic Control 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='4 (1972), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 439–448 (cit.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' [Vil39] Jean Ville.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' “Etude critique de la notion de collectif”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In: Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Soc 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='11 (1939), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 824 (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' [VNG03] Vladimir Vovk, Ilia Nouretdinov, and Alexander Gammerman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' “Testing exchangeability on-line”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In: Proceedings of the 20th International Conference on Machine Learning (ICML-03).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 2003, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 768–775 (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' [VW96] Aad W Vaart and Jon A Wellner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' “Weak convergence”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In: Weak Convergence and Empirical Processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Springer, 1996, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 16–28 (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' [WR23] Ian Waudby-Smith and Aaditya Ramdas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' “Estimating means of bounded random variables by betting”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In: Journal of the Royal Statistical Society (Series B), to appear with discussion (2023) (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' [WRB20] Larry Wasserman, Aaditya Ramdas, and Sivaraman Balakrishnan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' “Universal inference”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In: Pro- ceedings of the National Academy of Sciences 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='29 (2020), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 16880–16890 (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' on pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 1, 5, 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' [WS95] Wing Hung Wong and Xiaotong Shen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' “Probability inequalities for likelihood ratios and conver- gence rates of sieve MLEs”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In: The Annals of Statistics (1995), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 339–362 (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' on pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 14, 17, 33, 37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 26 A Proof of Triviality and Properties of Fork-Convex Hulls This appendix is devoted to showing the structural lemmata regarding fork-convex hulls, and discussing technical aspects of our arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='1 Details on the Local L1(Γ) Closure Let us begin by explicitly detailing the notion of convergence implicit in closed fork-convex combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Recall that the f-conv(P) is the closure of of f-conv(P) with respect to L1(Γ)-convergence of likelihood ratio processes at every fixed time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Let us unpack this statement in simple terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Let Pn be some sequence in f-conv(P) of density ratio Zn t := ZPn t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We say that Pn → P if for every t, it holds that Zn t → Zt in L1(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Since Zt and Zn t are Ft measurable objects, this convergence is simply in L1(Γ|t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Stating that the convergence needs to happen at every fixed time t means that this convergence need not be uniform in t: it is fine for Zn 100 to converge more slowly than Zn 1 , for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This notion of convergence may be metrised by ∆(P, Q) := � t∈N 2−t∥ZP t − ZQ t ∥L1(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We note that ∆ is bounded, since ∥ZP t − ZQ t ∥L1(Γ) = � ���P|t(dxt 1) − Q|t(dxt 1) ��� ≤ � P|t(dxt 1) + � Q|t(dxt 1) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' With this in hand, we first show the following auxiliary claim that is repeatedly used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Lemma 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Let P be a set of sequential laws, and let R be any sequential law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Suppose there exists a sequence of sequential laws {RT} such that each RT ∈ f-conv(P), and for all t ≤ T, ZRT t = ZR t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Then R ∈ f-conv(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We claim that RT → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Indeed, since ZRT t = ZR t for all t ≤ T, ∆(RT , R) ≤ � t>T 2−t · 2 = 2−(T −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Thus, limT →∞ ∆(RT , R) = 0, meaning RT → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Since the closed fork-convex hull of P includes such limits by definition, the claim is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The above lends significant convenience to our arguments, since it allows us to only construct processes matching some claimed member of the fork-convex hull up to finite times, which is typically easy to do in our arguments below using just finite fork-convex combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='2 Proofs about the Fork-Convex Hull of Independent Sequential Laws We may now proceed with the proofs of the Lemmata omitted from §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' As detailed in the main text, by taking repeated fork-convex combinations, it follows that RT ∈ f-conv(P∞), where R1 := P ∞ 1 , RT := � RT −1 T −1,0 −→ P ∞ T � , where validity of the mixture weight 0 exploits the mutual absolute continuity of laws in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We conclude by Lemma 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' It suffices to show that for all finite k, P∞ k ⊂ f-conv(P∞), since P∗ = � k Pk, and Pk ⊂ Pk+1 for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For T ∈ N and two laws P, Q on Rd, define the sequential law RP,Q,T as the law of an independent sequence {Xt} such that Xt ∼ P for t ≤ T and Xt ∼ Q for t > T , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' RP,Q,T = � P ∞ T,0 −→ Q∞� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For T ∈ N, define Pk,T as the set of sequential laws of the form RP,Q,T with P ∈ Pk and Q ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 27 We first claim that Pk,T ⊂ f-conv(P ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We show this inductively in k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Fix any T , and observe that trivially P1,T lies in this fork-convex hull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For k ≥ 2, we may represent each P ∈ Pk as P = αP 1 + (1 − α)P 2 for some α ∈ [0, 1], P 1 ∈ Pk−1 and P 2 ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We need to show that for any such P, and any Q ∈ P, RP,Q,T lies in the fork-convex hull of P∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' By the induction hypothesis, RP 1,Q,T ∈ f-conv(P∞), and RP 2,Q,T ∈ f-conv(P∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' But then define the laws S0 := RP 1,Q,T , �Sτ := � Sτ−1 τ−1,α −→ RP 2,Q,T � , Sτ := � �Sτ τ,0 −→ RP 1,Q,T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We note that every fork-convex combination above has valid weights since P is m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=', and so no density process is ever 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We claim that ST = RP,Q,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Indeed, let p1, p2, q respectively denote the densities (with respect to the standard Gaussian) of P 1, P 2, and Q, and let Z1 t and Z2 t be the density processes of RP 1,Q,T and RP 2,Q,T respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' These can be explicitly evaluated as Zi t = � s≤min(t,T ) pi(Xs) · t� s=min(t,T +1) q(Xs), where i ∈ {1, 2}, and we note that for u < v, �u s=v · = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Observe that for each i, and any t1 < T, and t > t1, we have Zi t Zi t1 = min(t,T ) � s=min(t1,T )+1 pi(Xs) · t� s=min(t,T +1) q(Xs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We shall inductively claim that for each τ, the density process of Sτ satisfies ZSτ t = � s≤min(t,τ) (αp1(Xs) + (1 − α)p2(Xs)) · min(t,T ) � s=min(t,τ+1) p1(Xs) · t� s=min(t,T +1) q(Xs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Indeed, the base claim is trivial since for τ = 0 since S0 = RP 1,Q,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Assuming the induction hypothesis for τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' we observe that since ˜Sτ+1 is a fork-convex combination of Sτ and RP 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='T at time τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' it shares the density process of Sτ up to time τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' while after that time the density is a mixture of the two density processes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' giving Z~Sτ+1 t = � s≤min(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='τ) (αp1(Xs) + (1 − α)p2(Xs)) × \uf8eb \uf8edα \uf8f1 \uf8f2 \uf8f3 min(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='T ) � s=min(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='τ+1) p1(Xs) · t� s=min(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='T +1) q(Xs) \uf8fc \uf8fd \uf8fe + (1 − α) \uf8f1 \uf8f2 \uf8f3 min(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='T ) � s=min(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='τ+1) p2(Xs) · t� s=min(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='T +1) q(Xs) \uf8fc \uf8fd \uf8fe \uf8f6 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' where we have used the behaviour of Zi t/Zi τ above for t ≥ τ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Finally, Sτ+1 mixes the above with RP 1,Q,T at time τ + 1 with a mixture weight of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This means that the suffix law of Sτ+1 beyond the time τ + 2 is exactly equal to the law of RP 1,Q,T , while the prefix up to time τ + 1 is left alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In other words, ZSτ t = � s≤min(t,τ) (αp1(Xs) + (1 − α)p2(Xs)) · τ+1 � s=min(t,τ+1) (αp1(Xs) + (1 − α)p2(Xs)) × min(t,T ) � s=min(t,τ+2) p1(Xs) · T � s=min(t,T +1) q(Xs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The claim follows upon noticing that the first two products can be merged into � s≤min(t,τ+1)(αp1(Xs) + (1 − α)p2(Xs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' With this in hand, the argument is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For any element P ∈ P∞ k , we note that there exists some member of Pk,T , say PT such that the density process of PT matches that of P up to time T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Applying Lemma 16 immediately yields the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 28 Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Let P = �{Pt} for any arbitrary sequence of Pt ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We need to show that P ∈ f-conv(� P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' But, since Pt ∈ P for each t, for each t there further exist sequences {P n t }n∈N, with each P n t ∈ P, such that P n t → Pt in L1(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Let Q := {P n t : t, n ∈ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We note that � Q ⊂ � P =⇒ f-conv(� Q) ⊂ f-conv(� P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Let Q := f-conv(� Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We shall argue that P ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Let PT be the sequential law with density process ZPT t = �� s≤t ps(Xs) if t ≤ T � s≤T ps(Xs) · � T T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' If we can show that for each T , PT ∈ Q, then the claim will follow, since PT → �{Pt} as in the argument of Lemma 16, and since Q is closed under the relevant notion of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We shall show this inductively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Let P1,n be a sequential law with density Z1,n t := pn 1(X1)·� s>min(1,t) p1 s(Xs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Notice that P1,n ∈ � Q ⊂ Q for every n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Further, ∆(P1,n, P1) ≤ ∥Z1,n 1 − ZP1 1 ∥L1(Γ) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Thus P1 ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Now suppose that PT −1 ∈ Q for some T ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For T, n ∈ N, define Qn as the sequential law of density ratio ZQn t := \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 � s T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' It trivially follows that Qn ∈ � Q ⊂ Q for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Now, define PT,n = � PT −1 T −1,0 −→ Qn� , which is valid since each P n t and Pt has are mutually absolutely continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' But ZPT,n t = ZPT −1 t = ZPT t for t ≤ T − 1, and for t ≥ T, ZPT,n t − ZPT t = ZPT T −1 · (pn T (XT ) − pT (Xt)) · t� s=T +1 ps(Xs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' It follows that ∥ZPT,n t − ZPT t ∥ = � 0 t < T ∥P n T − PT ∥L1(Γ) t ≥ T , and therefore, ∆(PT,n, PT) ≤ ∥P n T − PT ∥L1(Γ) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' By closeness of Q, we conclude that PT ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='3 Proof of Lemma 8 Proof of Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Fix an m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Since E has positive mass and is measurable, there exists an open set O ∈ (Rd)t such that O ⊃ E and Lebdt(O) ≤ (1 + 1/m)Lebdt(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Observe here that ‘most’ of the mass of O lies within E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Since O is open, there exists a sequence of disjoint open rectangles Ri in (Rd)t such that � Ri ⊂ O ⊂ � Ri, and Lebdt �� Ri � = � Lebdt(Ri) = Lebdt(O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Further, since most of the mass of O lies in E, we conclude that there exists at least one i such that Lebdt(Ri) > 0 and Lebdt(E ∩ Ri) ≥ m m + 1Lebdt(Ri).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Indeed, otherwise we would have Lebdt(E) = Lebdt(E ∩ O) = � Lebdt(E ∩ Ri) < m m + 1 � Lebdt(Ri) ≤ m m + 1 · m + 1 m Lebdt(E), which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 29 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='4 Technical Aspects of Fork-Convex Hulls and Our Triviality Argument We comment on some technical aspects of the argument underlying the non-existence of nontrivial NSMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Specifically, we discuss the necessity of our definition of nontriviality, and the m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' condition repeatedly used in the argument, how the argument can be extended to consider log-concave laws over bounded sets, and finally issues that arise when one tries to relax the definition of fork-convex combinations to handle support mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Going beyond almost sure triviality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The main text defines trivial NSMs (and NMs) as those that are Γ-almost surely non-increasing (respectively, constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Could one instead show that there are no nontrivial L∞-NSMs in the stronger sense that such processes must be non-increasing (as opposed to only almost surely non-increasing)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This turns out to be impossible, as witnessed by the following process Mt := 1 1 − 1{∃(t1, t2, t3, t4) ∈ [1 : t]4 : Xt1 = Xt2, Xt3 = Xt4, Xt1 ̸= Xt3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Since log-concave measures can have at most one atom (due to unimodality), it follows that {Mt} is an L∞-martingale (indeed, it is almost surely just a constant 1, as stated by the theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' However, Mt does diverge to ∞, and this occurs almost surely against any i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' sequential law which has at least two atoms, for instance, a coin flip process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This means that while it may not be possible to reject processes with a Lebesgue density using test martingales, it is possible to reject atomic processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Structurally, this example has to do with the fact that one cannot approach point masses in an L1 sense using measures with density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Therefore, although L∞-NSMs must also be NSMs for independent processes with densities, this does not extend to sequences drawn from distributions with atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In another sense, this issue is the same as the problem discussed below regarding loss of the NSM property under extensions of fork-convex combinations of laws with support mismatch, in that two laws with distinct single atoms each have parts that are mutually singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The role of the mutual absolute continuity condition on P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The definition of fork-convex combinations of two laws P and Q at time s involves the ratio of density processes ZP s/ZQ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This ratio must indeed appear, as can be seen from the algorithmic viewpoint of §3 to account for the fact that if R is the fork-convex combination, then the prefix law R|s = P|s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' However, if ZQ s = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' if for {Xt} ∼ R, the prefix Xs 1 lies in a set that is almost surely impossible under Q, then the above ratio is meaningless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This observation underlies the condition that if ZQ s = 0, then the mixture weight h must be exactly 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Our argument ultimately asserts that any law of the form �{Pt} lies in f-conv(P∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' However, our constructions to demonstrate this fact rely on setting h = 0 in order to generate switches between different laws in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Our assumption of mutual absolute continuity is to enable precisely this flexibility without running into the issue discussed in the previous paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The role of Gaussians in our argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Since we used the density of the Gaussians in order to show that L∞-NSMs must also be � D-NSMs, it behooves us to ask how essential L ⊃ G is to the main point of the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='4 In the argument, Gaussians play two roles: firstly, since all Gaussians are supported on the entirety of the domain, this class is m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=', and we can flexibly take fork-convex combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Secondly, the triviality of Gaussian NSMs follows since mixtures of Gaussians are L1-dense in the set of densities on the reals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Any subset of L that satisfies these two properties will suffice for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Extending the argument to log-concave laws on subsets of Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We finally observe that our argument extends in a straightforward manner to log-concave laws on restricted subsets of the reals: for a bounded convex set K, define LK to be log-concave densities supported on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Then all L∞ K -NSMs are trivial, in the sense that they are almost surely nonincreasing with respect to the reference measure (Unif(K))∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This follows because truncated Gaussians are again dense and supported on the entirety of the domain K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 4Notwithstanding that the result is interesting in its own right for Gaussians, which tells us that there is a simple, and very natural, parametric family that cannot be tested via nonnegative supermartingales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 30 To see this, first observe that if γ := � αiφi is a mixture of Gaussians, then for any K of nonzero Lebesgue mass, the truncation γ|K is also a mixture of truncated Gaussians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Indeed, define θi = � K φi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Then γ|K(x) = � αi � αiθi φi(x) · 1{x ∈ K} = � αiθi � αiθi φi|K(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Now, let p be any density supported on K, and let γn → p be a sequence of mixtures of Gaussians converging so that dn := � |p − γn| → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Then, defining πn = � Kc γn, we have � |p − γn|K| = � K |p(1 − πn) − γn| 1 − πn ≤ � K |p − γn| 1 − πn + � K πnp 1 − πn ≤ πn + � |p − γn| 1 − πn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Further, since p is supported on K, πn = � Kc γn ≤ � Kc γn + � K |p − γn| = dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Therefore, TV(p, γn|K) ≤ 2dn 1 − dn → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' But this means that we can run the entire argument of §3 but with Gaussians truncated over K, and draw the same conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Can we extend nontrivial fork-convex combinations to all laws?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' As we discussed above, due to the “ZQ T = 0 =⇒ h = 1” condition in the definition of fork-convex combinations, it is not possible to take arbitrary fork-convex combinations between sequential laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In the extreme case of P = P ∞ and Q = Q∞ for P, Q that have disjoint support, the only possible fork-convex combinations are mixtures of the form αP ∞ + (1 − α)Q∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' While this technicality did not pose a serious issue for the current paper, this situation is quite unsatisfying in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' After all, the algorithmic view of fork-convex combinations is very natural, and extends to such disjoint support situations easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' One can formalise this algorithmic picture by exploiting conditional densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For a sequence of (appro- priately measurable) maps kP t : (Rd)t−1 × Rd → R≥0, denoted kP t (xt|xt−1 1 ), we say that {kP t } is the conditional density process of P if for each xt−1 1 , kt(·|xt−1 1 ) is a density with respect to Γ, and for any t, A ∈ Ft, P(Xt 1 ∈ A) = � A � s≤t ks(xs|xs−1 1 )Γ(dxt 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' More generally, we can define a similar notion via Markov kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We observe that, by definition, it holds that if P has a conditional density process, then for any t and Γ-almost every xt 1 that ZP t (xt 1) = � s≤t ks(xs|xs−1 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Using the above characterisation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' we can give the following natural extended definition of fork-convex combinations: for two sequential laws P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Q with conditional density processes {kP t},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' {kQ t} respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' a law R is a fork-convex combination of P and Q at time T with FT -measurable weight h if ZR t = �� s≤t kP s(xs|xs−1 1 ) t ≤ T � s≤T kP s(xs|xs−1 1 ) · � h �t s=T +1 kP s(xs|xs−1 1 ) + (1 − h) �t s=T +1 kQ s(xs|xs−1 1 ) � t > T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' (4) the difference being that we now do not impose the restriction that h = 1 if ZQ T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Simplistically, this is possible since we are never dividing by the potentially null ZQ T , and more formally, this is considering the formal ratio ZQ t /ZQ T , which is interpreted in the natural way as � kQ s(xs|xs−1 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The above extended definition genealizes our previous definition of fork-convex combinations, and we can extend the same to the fork-convex hull and its closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' While the density process above is a perfectly sound mathematical object, such an extension is not fruitful because of a failure to preserve the NSM property under these extended fork-convex combinations in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' To illustrate why the above extended definition fails to maintain the NSM property (unlike the restricted one used in the paper), consider the following example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 31 Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' P = (Unif(0, 1))∞ and Q = (Unif(1, 2))∞, and the process Mt := � 2 ∃s1, s2 ≤ t : Xs1 ∈ (0, 1), Xs2 ∈ (1, 2) 1 otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This process is an NSM (indeed, a martingale) under both P, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' However, under any nontrivial fork-convex combination of these two laws, this process must start at 1, and with positive probability grow to 2 but never fall, and thus cannot be a supermartingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Under the hood, the issue in the example above arises due to the fact that under the extended definition, for t ≥ T +1, {ZR t > 0} = {ZP t > 0}∪{ZP T > 0, �t T +1 ks(Xs|Xs−1 1 ) > 0}, but the NSM property of {Mt} under P or Q only controls the conditional expectations of MtZP t 1{ZP t > 0} and MtZQ t 1{ZQ t > 0} under Γ, which leaves the conditional behaviour of MtZR t uncontrolled when R places mass on events that are null under one of these laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' It should be noted that in the above example there is a version of the process {Mt}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=', a process {� Mt} such that P(∀t, Mt = � Mt) = Q(∀t, Mt = � Mt) = 1, but such that {� Mt} is a martingale even under extended fork-convex combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Concretely this process is just � Mt = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' One may thus wonder if this phenomenon holds true in greater in generality: is it the case that if {Mt} is an NSM under P and Q, then there is a version {� Mt} of it (under P and Q) such that {� Mt} is an NSM against any extended fork-convex combination of P and Q, without the restriction “ZQ T = 0 =⇒ h = 1”?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This turns out also to be impossible in general, as demonstrated by the following example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Let P = Unif(0, 1)∞ and Q = Unif(0, 1/2)∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Define ρt = 1{Xt ∈ (0, 1/2)} for t ≥ 1, and ρ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Let {Nt} be an adapted process such that Nt = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 1 ρt−1 = 1 3/2 ρt−1 = 0, ρt = 1 1/2 ρt−1 = 0, ρt = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Finally define Mt = � s≤t Nt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' It is easy to check that Mt is an NM under both P and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Now suppose R is an extended fork-convex combination of P, Q at time T ≥ 1 with mixture weight h < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This means that with probability 1 − h, it holds that Xt ∈ (0, 1/2) with certainty for all t ≥ T + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' As as result, we can explicitly compute that E[NT +1|FT ] = ρT + (1 − ρT ) ((1 − h) · 3/2 + h(1/2 · 3/2 + 1/2 · 3/2)) = � 1 ρT = 1 1 + (1 − h)/2 ρT = 0 , and so as long as h < 1, E[NT +1|FT ] > 1 if ρT = 0, and therefore {Mt} violates the NSM property under R at the time T + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Note here that it is hard to construct any nontrivially different version of the above process since the law of P dominates that of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In light of the above discussion, generalised definitions of fork-convex combinations are at loggerheads with maintaining the NSM property these combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Of course, since our purpose in using fork-convexity is to assert the triviality of NSMs over large classes of sequential laws, this latter property is essential to maintain for such statistical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' At the same time, while the restricted original definition does maintain the NSM property, the included restriction is unsatisfying, and in conflict with the algorithmic intuition underlying the idea of these combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Finding an appropriate generalised definition of fork-convex combinations that abstains from imposing these support conditions, but nevertheless retains NSM closure under the NSM property is an interesting, and challenging, question left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' B Proofs of Consistency and Power Analysis Recall the notation σt(P) := � s≤t log p(Xs) − log ˆpt(Xs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The main arguments of this section control the behavious of σt(P), in particular arguing that if the Hellinger distance of P from log-concavity is large, then 32 σt(P) must eventually grow linearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We show this in asymptotic and nonasymptotic regimes in §B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='1 and §B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Corollary 11 and Corollary 15 each relies on further control on the behaviour of ρt(E ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' p) = � s≤t log p(Xs)− log ˆqs−1(Xs) when p is a bounded Lipschitz law on the unit box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This argument is left to §B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='1 Proof of Consistency Our arguments rely on the following bracketing tail estimate, developed by Wong and Shen to analyse the behaviour of sieve-based maximum likelihood estimates [WS95, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The estimate involves the bracketing entropy of a class of laws Q under the Hellinger metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We refer the reader to the text of Van der Vaart and Wellner [VW96] for a thorough introduction, and give a brief account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' A bracket [u, v] is defined by two functions u(x), v(x) such that u(x) ≤ v(x) for all x, and consists of the set of all functions f such that u(x) ≤ f(x) ≤ v(x) for all x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Since we shall only be interested in functions that are densities, we may restrict attention to nonnegative functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The Hellinger size of such a bracket [u, v] is defined as |[u, v]| = ∥√u − √v∥2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We say that a class of distributions Q is bracketed by {[ui, vi]}N i=1 if for all Q ∈ Q, there exists an i such that q ∈ [ui, vi], where recall that for a distribution Q, we denote its density by q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Note that this bracketing is typically “improper”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=', ui, vi will generally not lie in Q (because q integrates to one, and so its lower bracket u will integrate to less than one, and its upper bracket v will integrate to more than one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The Hellinger bracketing entropy of Q at scale ζ is the logarithm of the most parsimonious bracketing of Q by brackets of size at most ζ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' H[](Q, ζ) := inf{log N : Q has an N-sized Hellinger bracketing at scale ζ} Note, of course, that bracketing entropies are nonincreasing in ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Lemma 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' (Simplification of [WS95, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 1]) For a class of distributions Q and a natural number t, define εt as the smaller number ε such that � √ 2ε ε2/28 � H[](Q, ζ/10)dζ ≤ 2−11√ tε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For every t and ε ≥ εt, it holds that for any law P such that dH(P, Q) ≥ ε, we have P ⊗t \uf8eb \uf8ed inf q∈Q � s≤t log p(Xs) − log q(Xs) ≤ tε2/24 \uf8f6 \uf8f8 ≤ 4 exp � −Ctε2� , where C > 2−14 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Informally, if P is far enough from Q in the Hellinger metric (where far enough is determined by the bracketing entropy of Q), then it is exponentially unlikely (in the sample size) for the maximum log-likelihood under Q to be linearly close to the log-likelihood under P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Exploiting this observation in our context requires us to argue that eventually, the log-concave MLE ˆpt must lie in a set with small entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' To this end, we appeal to the following result due to Dunn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=', which extends the convergence analysis of Cule & Samworth [CS10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Lemma 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' [DGWR21, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 1] Consider any distribution P ∈ D1, not necessarily log-concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For any η > 0, there exists a bracket [uη, vη] of size at most η that contains the log-concave projection LP , and eventually also contains the log-concave MLE ˆpt: P ∞(∃t0 : ∀t ≥ t0, ˆpt ∈ [uη, vη]) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In words, the lemma states that for large enough t, the log-concave MLE ˆpt is certain to lie in a very small bracket around the log-concave projection LP of the true distribution P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' With this in hand, we are in a position to show Lemma 12, the main statement underlying the proof of Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Proof of Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Let ε := dH(P, LP ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Define ηε = ε2/211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Using Lemma 18, we know that there exists a bracket [u∗, v∗] such that |[u∗, v∗]| ≤ ηε and, almost surely, ˆpt ∈ [u∗, v∗] for all large enough t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' But observe 33 that H[]([u∗, v∗], ε2/211) = 0, since the size of [u∗, v∗] is already ηε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Further, since LP ∈ [u∗, v∗], by the triangle inequality, dH(P, [u∗, v∗]) = inf Q∈[uη,vη] dH(P, Q) ≥ dH(P, LP ) − |[u∗, v∗]| ≥ ε(1 − 2−11) ≥ ε · � 24/25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Let us define �σt(P) := inf q∈[u∗,v∗] � s≤t log p(Xs) − log q(Xs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' By exploiting the above observations, Lemma 17 yields that for every t, P ∞ � �σt(P) ≤ tε2/25 � ≤ 4 exp � −Ctε2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Note further that if ˆpt ∈ [u∗, v∗], then since ˆpt is a maximum likelhood estimate, it must hold that σt(P) = �σt(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Let Es := {∀t ≥ s, ˆpt ∈ [u∗, v∗]} be the event that ˆpt lies in the small bracket after time s, and At := {σt(P)/tε2 ≥ 1/25} be the event that σt(P) is larger than tε2/25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' By Lemma 17, for every fixed time s and t ≥ s, P ∞(Ac t ∩ Es) ≤ 4 exp � −Ctε2� , and since this upper bound is summable, by the Borel-Cantelli Lemma 0 = P ∞ � lim sup t (Ac t ∩ Es) � = P ∞ � (lim sup t Ac t) ∩ Es � , and so for any time s, P ∞(lim sup t Ac t) ≤ P ∞(lim sup t Ac t ∩ Es) + P ∞(Ec s) = P ∞(Ec s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' By Lemma 18, ˆpt must eventually almost surely fall in [u∗, v∗], lims→∞ P ∞(Ec s) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Further notice that lim sup t Ac t = {σt(P)/tε2 < 1/25 infinitely often} = {lim inf σt(P)/tε2 < 1/25}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Putting the observations together, we conclude upon sending s → ∞ that P ∞(lim inf σt(P)/tε2 < 1/25) ≤ lim s→∞ P ∞(Ec s) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='2 Proofs Underlying the Power Analysis We shall begin by stating the key lemmata underlying our argument, which exploit our bracketing entropy control from Lemma 13 along with results in the literature that bound the maximum value attained by a log-concave density in order to make the same effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We then prove the main result, and conclude by proving Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='1 Controlling the Maximum Value Attained by the Log-Concave MLE The rate analysis quantitatively exploits Lemma 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' To do so, we first need bracketing entropy bounds for log-concave laws, which is precisely the subject of Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We recall that this controls the bracketing entropy of the class Ld,B of log-concave laws with densities supported on [−1, 1]d that are bounded from above by B, showing that H[](Ld,B, ζ) = �O((B/ζ)max(d/2,(d−1))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The role of B in the above is quantitatively unimportant as long as this constant does not scale with relevant parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This fact is assured for log-concave laws with near identity covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Intuitively, since the covariance is lower bounded in all directions, the laws cannot concentrate too much, and thus the value of the density at the mode cannot be too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This observation is encapsulated in the following result, which follows trivially from the work of Kim & Samworth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 34 Lemma 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' [KS16, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 6] Let Lγ d denote the set of log-concave laws distributed on [−1, 1]d with covariances lower bounded in the positive semidefinite order by γI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Then there exists a dimension dependent constant Cd such that for any f ∈ Lγ d, max x∈[−1,1]d f(x) ≤ γ−d/2Cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Of course, our bounds in Theorem 14 depend on ∆P , which roughly speaking only controls that the covariance of the underlying law P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The relevance of this quantity arises from the following observation, due to Barber and Samworth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Lemma 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' [BS21, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 8] Let P ∈ D1 be a law supported on [−1, 1]d such that ∆P := min v:∥v∥=1 Ep[|⟨v, X − Ep[X]⟩|] > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Then there exists a dimension dependent constant cd such that Cov(LP ) ⪰ cd∆2 P I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Further, there exists a dimension-independent constant C such that for any t ≥ 2Cd3/∆2 P , it holds with probability at least 1 − 2 exp � −Ct∆2 P /d2� that Cov(ˆpt) ⪰ cd∆2 P 4 I for the log-concave MLE ˆpt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Proof of Lemma 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The first observation is a direct restatement of Lemma 7 of Barber and Samworth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The second statement follows from the fact that over v : ∥v∥ = 1, v �→ ⟨v, X − EP [X]⟩ is bounded by 2 √ d, and is clearly continuous in v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Thus exploiting standard subGaussian concentration results over the unit ball, it follows that with probability at least 1−2 exp � −Ct∆2 P /d2� , it holds that for the empirical law pt = 1 t � s≤t δXs, min v:∥v∥=1 Ept[|⟨v, X − Ept[X]⟩|] ≥ ∆P /2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' But notice that ˆpt = Lpt, from which the claim follows by the first part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Merging Lemmas 19 and 20 immediately yields the following observation, which serves as a concrete bound for the scale of B we need to employ in Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' There exists a constant Cd depending only on d such that for any t ≥ 2Cdd3/∆2 P , it holds with probability at least 1 − 2 exp � −Cdt∆2 P /d2� that max x∈[−1,1]d ˆpt(x) ≤ Cd∆−d P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Employing Lemma 19, we observe that {max ˆpt ≤ (cd∆2 P /4)−d/2} ⊂ {Cov(ˆpt) ⪯ cd∆2 P /4I}, and the latter has probability at least 1 − 2 exp � −t(C∆2 P /d2) � for t ≥ 2Cd3/∆2 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Take Cd = max(C, c−d/2 d ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='2 Proof of Bounds on Rejection Times With the above in hand, we may proceed with the main argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Proof of Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Recall the definition σt := � s≤t log p(Xs) − log ˆpt(Xs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We shall first lower bound σt with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Let B be a quantity that we will choose later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Let εt denote the solution to the fixed point equation from Lemma 17, instantiated with the bracketing entropy of Ld,B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Further, let define the event Et := {ˆpt ∈ Ld,B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For any t, provided that such that εt ≤ dH(p, L) and ˆpt ∈ Ld,B, Lemma 17 yields that σt ≥ inf q∈Ld,B:dH(p,q)≥dH(p,L) log � s≤t p(Xs) q(Xs) ≥ td2 H(p, L) 24 (5) with probability at least 1 − exp � −Ctd2 H(p, L) � − P ∞(Ec t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 35 In the rest of the proof, we will determine the range of t that leads to a small enough value for εt to ensure that the condition εt ≤ dH(p, L) is met and, at the same time, control P ∞(Ec t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' To this end, we deploy Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' First, observe that for d ≥ 3 and for any positive constants c and C � Cε cε2 � ˜O(Bd−1ζ−(d−1))dζ = �O(B(d−1)/2ε−(d−3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Note that polylogarithmic terms do not affect the main growth of the integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='5 Therefore, solving the fixed point equation �O(B(d−1)/2ε−(d−3)) = ε2t1/2, we obtain that for d ≥ 3 εt(B) = �O(B1/2t−1/2(d−1)), where we highlight the dependence on the as yet undetermined quantity B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' A similar argument using the entropy bound ζ−d/2 yields εt(B) = �O(Bd/(d+4)t−2/(d+4)) for d ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Now define T1(B) = inf{t : εt(B) ≤ dH(p, L)} and observe that T1(B) = � �O(B(d−1)(dH(p, L))−2(d−1)) d ≥ 3 �O(Bd/2(dH(p, L)−(4+d)/2)) d ∈ {1, 2} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Finally, by Lemma 21, for B ≥ Cd∆−d P and t ≥ T2 := C∆2 P /d2 the probability of the event Et is at least 1 − 2 exp � −tC∆2 P /d2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Let us set B∗ = Cd∆−d P and let T0 := max(T1(B∗), T2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We obtain that the lower bound σt ≥ td2 H(p, L) 24 holds with probability at least 1−C exp � −tcd2 H(p, L) � −C exp � −tc∆2 P /d2� for t ≥ T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Now, observe that at any time t ≥ max(T0, 600 log(1/α) d2 H(p,L) ), it holds with probability at least 1−πt−C exp � −td2 H(p, L) � −C exp � −Ct∆2 P /d2� that log Rt = σt − ρt ≥ td2 H(p, L) 600 ≥ log(1/α), and thus the probability that the rejection time τα := inf{t : Rt ≥ 1/α} exceeds the above bound is bounded by πt + C exp � −td2 H(p, L) � + C exp � −Ct∆2 P /d2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='3 Proof of Bracketing Entropy Bound on Log-Concave Laws We proceed to show Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We note that the upper bound for d ≤ 3 was shown by Kim and Samworth [KS16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Below we focus on d ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We shall exploit two existing results in the literature regarding convex sets and functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The first is essentially due to Bronshtein (also see [KDR19, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Lemma 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' [Bro76] Let Kd denote the collection of convex sets in [−1, 1]d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For any ζ > 0, there exists a collection of pairs of convex sets Kd,ζ ⊂ Kd × Kd with log |Kd,ζ| = O(ζ−(d−1)/2) such that Every (K, K) ∈ Kd,ζ satisfies Lebd(K \\ K) ≤ ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For every K ∈ Kd, exists (K, K) ∈ Kd,ζ satisfying K ⊂ K ⊂ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In other words, the bracketing entropy of convex sets under the set difference metric is controlled at rate (d − 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Importantly, the bracketing demonstrated above is proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This result may be extended to the following bracketing entropy bound on convex functions as by Gao and Wellner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Lemma 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' [GW17, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='5] Let K be a convex set in [−1, 1]d, and let CK,B be the set of convex functions upper bounded by B over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Then the L2(K) bracketing entropy of CK,B at scale ζ is bounded as O((B/ζ)(d−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 5This can be seen by iterating the relation � xn logm x = xn+1 logm x n+1 − m n+1 � xn logm−1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 36 Above, the L2(K) metric is the usual L2 distance ∥f − g∥L2(K) = ( � K(f − g)2dx)1/2, and the L2(K) bracketing entropy is the bracketing entropy when the size of a bracket [u, v] is |[u, v]| = ∥u − v∥L2(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' With the above in hand, we may proceed with the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Proof of Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For any log-concave law f, let S := {x ∈ Rd : f(x) ≥ ζ3} = {x ∈ Rd : log f(x) ≥ 3 log ζ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Since f is log-concave, the set S is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' As a result, by Lemma 22, there exists some convex set ˜S ∈ Kd,ζ2/B such that Leb(S \\ ˜S) ≤ ζ2/B and ˜S ⊂ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Let ˜C ˜S,ζ,B denote a ζ-bracketing of convex functions bounded by B on ˜S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Since, on ˜S, the function − log f is convex and is upper bounded by − log B, by Lemma 23 there exists a bracket [−u, −l] ∈ ˜C ˜S,ζ/B,log B/ζ3 such that, on ˜S, l ≤ log f ≤ u, and � ˜S(u(x) − l(x))2dx ≤ ζ2/B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Note that, on ˜S, f is lower bounded by −3 log ζ and that, without loss of generality, we may assume that supx∈ ˜S u(x) ≤ log B, since this is already a pointwise upper bound on f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Next, we construct the functions x ∈ [−1, 1]d �→ U(x) := \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 eu(x) x ∈ ˜S B x ∈ S \\ ˜S ζ3 x ∈ [−1, 1]d \\ S , x ∈ [−1, 1]d �→ L(x) := \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 el(x) x ∈ ˜S ζ3 x ∈ S \\ ˜S 0 x ∈ [−1, 1]d \\ S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Observe that U ≥ f ≥ L on [−1, 1]d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Furthermore, for ζ < 2−d, � ( √ U − √ L)2dx = � ˜S (eu(x)/2 − el(x)/2)2dx + � S\\ ˜S Bdx + � [−1,1]d\\S ζ3dx ≤ � ˜S eu(x)(1 − eu(x)−l(x)/2)2dx + B · ζ2/B + 2dζ3 ≤ � ˜S B2(u(x) − l(x))2/4 dx + 2ζ2 ≤ B2 · ζ2/B2 + 2ζ2 = 3ζ2, where we have exploited the fact that z �→ ez/2 is Lipschitz on [−∞, log C], with derivative bounded by e(log B)/2/2 = √ B/2 to argue that e(u(x)−l(x))/2 − e0 ≤ √ B|u(x) − l(x) − 0|/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Since this construction can be carried out for any f, we conclude that we can construct a bracketing cover of Ld,B at scale O(ζ) as the union of the bracketing covers of convex functions on each of the smaller sets in Kd,ζ2/B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' By Lemmas 22 and 23, the size of this cover is exp � O((B/ζ)d−1) � exp � O((log B/ζ3)(B)/ζ)d−1� = exp � �O((B/ζ)d−1) � , and the claim now follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Let us again observe that the resulting cover is improper, in that the maps U(x) and L(x) are not log-concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content='3 Regret Control for Bounded Lipschitz Laws on the Unit Box As this subsection demonstrates, both Corollaries 15 and 11 rely on arguing that laws in DBox,Lip,B can be estimated in a low-regret manner online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We argue this by exploiting the following result, which follows as a simplification of the results of Wong and Shen on sieve estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Lemma 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' (Adaptation of [WS95, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 1 & Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 6])For every P ∈ DBox,Lip,B and t ≥ 1, there exists a sieve MLE ˆq(·) = ˆq(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Xt 1) and a constant A > 1 depending only on B such that for every ζ ≥ ζt, P ∞ � KL(p∥ˆq) > 1 Aζ2 log(1/ζ) � ≤ A exp � −t ζ2 A log(1/ζ) � , where ζt = �O(t−1/2(d+2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 37 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The cited results of Wong and Shen apply because densities of laws in DBox,Lip,B are uniformly upper bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This directly yields the entirety of the statement, barring the scale bound on ζt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This scale is determined by the same entropy integral fixed point equation that appears in Lemma 17, and for this instance, the bound can be derived by using the standard fact that the Hellinger bracketing entropy of Lipschitz functions on a box at scale η are controlled as O(η−(d+1)) [Vaa94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The sieve estimators in this result can be taken with a fair bit of lassitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In particular, one explicit choice is to construct for each ζ > 0 a bracket of the class DBox,Lip,B at scale ζ, and choose a representative density within each bracket of the class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The sieve MLE then involves choosing a ζ at each time, and estimating the law as the maximiser of likelihood amongst the aforementioned representative densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Importantly for us, the lower brackets in these bracketings can be taken to be uniformly larger than 1/B, and the upper brackets smaller than B, since p ∈ [1/B, B], and as a result the sieve estimates are uniformly bounded between 1/B and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Below we first show Corollary 15 using the above results, and then show Corollary 11 follows as a simple consequence of this argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Proof of Corollary 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' As argued in the main text, the expected rejection time is bounded as E[τ] ≤ � πt + O(T0), where T0 = o(dH(p, L)−2(d+3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We thus only need to show that a sequence πt exists such that � πt is appropriately small, and that for any t, P ∞ � ρt(E ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' p) td2 H(p, L) ≥ 1 25 � ≤ πt, where p ∈ DBox,Lip,B and E are sieve estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We proceed to do so below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' For succinctness, we shall define ε = dH(p, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Let A be the constant from Lemma 24, and set T1 := min{t : ζ2 t log(1/ζt)/A < ε2/200, ζt < 1/√e}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Further let ζ(ε) := max{ζ ∈ [0, 1/√e] : ζ2 log(1/ζ) ≤ Aε2/200}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' In the subsequent proof, we shall use Lemma 24 with ζ = ζ(ε) ≥ ζT1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' To this end, we note that if ζ(ε) < 1/√e ⇐⇒ Aε2/200 < 1/2e, and in this case the equality ζ(ε)2 log(1/ζ(ε)) = Aε2/200 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' From this, we may derive6 that ζ(ε)2 > aε2/ log(1/ε) for some small enough constant a, and so that the exponent of the upper bound of Lemma 24 is ζ(ε)2 A log(1/ζ(ε)) = 400ζ4 A2ε2 ≥ ε2 A′ log(1/ε) for some large enough constant A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We shall also assume that A′ ≥ max(1, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Let E be a choice of sieve estimators such that for every t, x ∈ [−1, 1]d, ˆqt−1(x) ∈ [1/B, B], which can be ensured due to the discussion above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Notice, by the independence of the data {Xt}, that for any t, E[log p(Xt)/ˆqt−1(Xt)|Ft−1] = KL(p∥ˆqt−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Let θ ∈ (0, 1) and M ≥ 0 be two parameters of argument that we shall set later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Let us consider the case of t = T1 + τ for some τ ≥ MT1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Since for each τ > 0, ζT1+θτ ≤ ζT1 ≤ ζ(ε), the bound of Lemma 24 is effective at each time s ∈ [T1 + θτ : T1 + τ] with ζ = ζ(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' As a result, applying Lemma 24 to each s in this range, and exploiting the behaviour of ζ(ε)2 established above, P ∞(KL(p∥ˆqs−1) > ε2/200) ≤ A′ exp � −s ε2 A′ log(1/ε) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 6This equation is equivalent to x log x = y for x = ζ(ε)2, y = Aε2/100 in the range 0 < x < 1/e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The claim follows by noting that the map x �→ x log(1/x) is monotonically increasing on [0,1/e], and verifying that for y ∈ [0, 1/2e], y 2 log(1/y) · log(2 log(1/y)/y) < y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Indeed, this inequality is equivalent to arguing that log(2 log(1/y)) < log(1/y) ⇐⇒ y log(1/y) < 1/2, which holds since the maximum value of y �→ y log(1/y) is 1/e < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 38 Next, by applying the union bound over s ∈ [T1 + θτ : T1 + τ] in the above result, we conclude that P ∞ � ∃s ∈ [T1 + θτ : T1 + τ] : KL(p∥ˆqs−1) > ε2 200 � ≤ T1+τ � s=T1+θτ A′ exp � −s ε2 A′ log(1/ε) � = A′ exp � −(T1 + θτ) ε2 A′ log(1/ε) � 1 1 − exp (−ε2/(A′ log(1/ε)), ≤ A′2 log(1/ε) ε2 exp � −θτ ε2 A′ log(1/ε) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' where the equality sums over the geometric series, and the final inequality uses that T1 ≥ 0 and that for u < 1, 1/(1 − e−u) ≤ 2 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Next, observe that since for any x ∈ [−1, 1]d, 1 B2 ≤ p(x) ˆqt−1(x) ≤ B2, we have the bound | log(p(Xt)/ˆqt−1(Xt))| ≤ 2 log B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Therefore, the Azuma-Hoeffding inequality is applicable, and yields that for every τ ≥ 1, δ > 0 P ∞ � T1+τ � s=T1+θτ log p(Xs) ˆqs−1(Xs) > T1+τ � s=T1+θτ KL(p∥ˆqs−1) + (τ − θτ)δ � ≤ exp � −(τ − θτ)δ2/8 log2 B � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We proceed by setting δ = ε2/200 in the above, and applying the union bound, to conclude that there exists a constant C such that P ∞ � T1+τ � s=T1+θτ log p(Xs) ˆqs−1(Xs) > (1 − θ)τε2 100 � ≤ exp � −(1 − θ)τε4 C log2 B � + C log(1/ε) ε2 exp � −θτ ε2 C log(1/ε) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' (6) Let us call the right hand side of (6) π(τ, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' By the definition of ρt, and the boundedness of log p(x) ˆqs−1(x) for every s, it follows that with probability at least 1 − π(τ, θ), ρT1+τ(E ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' p) = T1+τ � s=1 log p(Xs) ˆqs−1(Xs) ≤ 2(T1 + θτ) log B + (1 − θ)τε2 100 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' So long as we can choose θ, M such that the upper bound above is smaller than (τ +T1)ε2/25, the inequality (6) will limit the probability that ρT1+τ > ε2(T1 + τ)/25, which is precisely our goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' But observe that this indeed occurs if (θ + 1/M) ≤ 3ε2/(200 log B), since in such a case 2(T1 + θτ) log B + (1 − θ)τε2 100 ≤ τ � 2(1/M + θ) log B + ε2 100 � ≤ τ � 3ε2 200 log B · 2 log B + ε2 100 � = τε2 25 ≤ (T1 + τ)ε2 25 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' So, we may set θ = min(1/2, ε2/(100 log B)) and M = max(1, (200 log B)/ε2), and conclude that for any τ ≥ MT1, it holds that P ∞(ρT1+τ(E ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' p)/tε2 > 1/25) ≤ π(τ), where, for a constant C′, π(τ) = exp � − τε4 C′ log2 B � + C′ log(1/ε) ε2 exp � − τε2 C′ log(1/ε) · log B � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 39 Note that in the terminology of Theorem 14, πt = π(t − T1) for t ≥ (M + 1)T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Of course we can always provide the trivial bound πt ≤ 1 for t < (M + 1)T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' It remains to compute the resulting bound on expected rejection time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' To this end, observe by summing the appropriate geometric series that � t≥1 πt ≤ (M + 1)T1 + ∞ � τ=MT1 π(τ) ≤ (M + 1)T1 + 1 1 − exp � −ε4/C′ log2 B � + C′ log(1/ε) ε2(1 − exp (−ε2/C′ log(1/ε) · log B) ≤ O � 1 ε2 � T1 + �O � 1 ε4 � , where the O bounds are as ε → 0, and we have hidden the dependence on B and log(1/ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' But, since in Lemma 24, ζt = �O(t−1/2(d+2)), and since T1 is the first time that ζ2 t log2(1/ζt) ≤ Aε2/200, we may conclude that T1 = �O(ε−2(d+2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' The claim follows upon noticing that O(ε−2) · T1 = �O(ε−2(d+3)), and recalling that ε = dH(p, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' We conclude with a brief proof of Corollary 11 that exploits the bounds developed in the argument above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' Proof of Corollary 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' It suffices to argue that using the estimators in the proof of Corollary 15, for any P ∈ DBox,Lip,B, P ∞ � lim sup ρt(E ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' p) td2 H(p, L) ≤ 1 25 � = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' This follows since for each t, P ∞ � ρt(E ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' p) td2 H(p, L) > 1 25 � ≤ πt, and � πt < ∞, which yields precisely the above relation by the Borel-Cantelli Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} +page_content=' 40' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E1T4oBgHgl3EQf8QUL/content/2301.03542v1.pdf'} diff --git a/A9E1T4oBgHgl3EQfpAX1/content/2301.03328v1.pdf b/A9E1T4oBgHgl3EQfpAX1/content/2301.03328v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..8d8e08838abc9e91fd1ece579f0b4ac8920f9596 --- /dev/null +++ b/A9E1T4oBgHgl3EQfpAX1/content/2301.03328v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:565b0f71d8fe8c09073a97277847b6757666618cdf4c18fb427ef33529c4b377 +size 648515 diff --git a/A9E1T4oBgHgl3EQfpAX1/vector_store/index.faiss 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LATEX template +A massive quiescent galaxy at redshift 4.658 +Adam C. Carnall1*, Ross J. McLure1, James S. Dunlop1, Derek J. +McLeod1, Vivienne Wild2, Fergus Cullen1, Dan Magee3, Ryan +Begley1, Andrea Cimatti4,5, Callum T. Donnan1, Massissilia L. +Hamadouche1, Sophie M. Jewell1 and Sam Walker1 +1Institute for Astronomy, School of Physics & Astronomy, University of +Edinburgh, Royal Observatory, Edinburgh, EH9 3HJ, UK. +2School of Physics & Astronomy, University of St Andrews, North +Haugh, St Andrews, KY16 9SS, UK. +3Department of Astronomy and Astrophysics, UCO/Lick Observatory, +University of California, Santa Cruz, CA 95064, USA. +4Department of Physics and Astronomy (DIFA), University of Bologna, +Via Gobetti 93/2, I-40129, Bologna, Italy. +5INAF, Osservatorio di Astrofisica e Scienza dello Spazio, Via Piero +Gobetti 93/3, I-40129, Bologna, Italy. +*Corresponding author email: adam.carnall@ed.ac.uk +Abstract +We report the spectroscopic confirmation of a massive quiescent galaxy, +GS-9209 at a new redshift record of z = 4.658, just 1.25 Gyr after +the Big Bang, using new deep continuum observations from JWST NIR- +Spec. From our full-spectral-fitting analysis, we find that this galaxy +formed its stellar population over a ≃ 200 Myr period, approximately +600 − 800 Myr after the Big Bang (zform = 7.3 ± 0.2), before quench- +ing at zquench = 6.7 ± 0.3. GS-9209 demonstrates unambiguously that +massive galaxy formation was already well underway within the first bil- +lion years of cosmic history, with this object having reached a stellar +mass of log10(M∗/M⊙) +> +10.3 by z = 7. This galaxy also clearly +demonstrates that the earliest onset of galaxy quenching was no later +than ≃ 800 Myr after the Big Bang. We estimate the iron abundance +and α-enhancement of GS-9209, finding [Fe/H] = −0.97+0.06 +−0.07 and +[α/Fe] = 0.67+0.25 +−0.15, suggesting the stellar mass vs iron abundance rela- +tion at z ≃ 7, when this object formed most of its stars, was ≃ 0.4 dex +lower than at z ≃ 3.5. Whilst its spectrum is dominated by stellar emis- +sion, GS-9209 also exhibits broad Hα emission, indicating that it hosts +an active galactic nucleus (AGN), for which we measure a black-hole +1 +arXiv:2301.11413v1 [astro-ph.GA] 26 Jan 2023 + +Springer Nature 2021 LATEX template +2 +A massive quiescent galaxy at redshift 4.658 +mass of log10(M•/M⊙) = 8.7 ± 0.1. Although large-scale star forma- +tion in GS-9209 has been quenched for almost half a billion years, the +significant integrated quantity of accretion implied by this large black- +hole mass suggests AGN feedback plausibly played a significant role +in quenching star formation in this galaxy. GS-9209 is also extremely +compact, with an effective radius of just 215 ± 20 parsecs. This intrigu- +ing object offers perhaps our deepest insight yet into massive galaxy +formation and quenching during the first billion years of cosmic history. +1 Summary +The discovery of massive galaxies with old stellar populations at early cosmic +epochs has historically acted as a key constraint on models for both galaxy for- +mation physics and cosmology [1–4]. Today, the extremely rapid assembly of +the earliest galaxies during the first billion years of cosmic history continues to +challenge our understanding of galaxy formation physics [5, 6]. The advent of +the James Webb Space Telescope (JWST) has exacerbated this issue by con- +firming the existence of galaxies in significant numbers as early as the first few +hundred million years [7–9]. Perhaps even more surprisingly, in some galaxies, +this initial highly efficient star formation rapidly shuts down, or quenches, giv- +ing rise to massive quiescent galaxies as little as ∼ 1.5 billion years after the +Big Bang, at redshifts up to z ≃ 4 [4, 10]. Due to their faintness and red colour, +it has proven extremely challenging to learn about these extreme quiescent +galaxies, or to confirm whether any exist at earlier times. Here, we report the +spectroscopic confirmation of a quiescent galaxy, GS-9209, at a new redshift +record of 4.658, just 1.25 billion years after the Big Bang, using the NIRSpec +instrument on JWST. The transformative power of JWST allows us to char- +acterise the physical properties of this early massive galaxy in unprecedented +detail. GS-9209 has a stellar mass of M∗ = 4.1 ± 0.2 × 1010 M⊙, and quenched +star formation at z = 6.7 ± 0.3, when the Universe was ≃ 800 million years +old. This intriguing object offers perhaps our deepest insight yet into massive +galaxy formation and quenching during the first billion years of cosmic history. +2 Results +GS-9209 was first highlighted in the early 2000s as an object with red optical +to near-infrared colours and a photometric redshift of z ≃ 4.5 [11]. An optical +spectrum was taken in the mid-2010s as part of the VIMOS Ultra Deep Sur- +vey (VUDS) [12], showing tentative evidence for a Lyman break at λ ≃ 7000˚A, +but no Lyman α emission. During the past 5 years, several studies have iden- +tified GS-9209 as a candidate high-redshift massive quiescent galaxy [13, 14], +based on its blue colours at wavelengths, λ = 2 − 8µm and non-detection at +millimetre wavelengths [15]. GS-9209 is also not detected in X-rays [16], at +radio wavelengths [17], or at λ = 24µm [18]. The faint, red nature of the source +(with magnitudes HAB = 24.7 and KAB = 23.6) means that near-infrared +spectroscopy with ground-based instrumentation is prohibitively expensive. + +Springer Nature 2021 LATEX template +A massive quiescent galaxy at redshift 4.658 +3 +2.0 +2.5 +3.0 +3.5 +4.0 +4.5 +5.0 +Observed Wavelength / µm +0.0 +0.3 +0.6 +0.9 +1.2 +1.5 +fλ / 10−19 erg s−1 cm−2 ˚A−1 +F170LP + G235M +F290LP + G395M +2.0 +2.2 +2.4 +2.6 +λ / µm +0.3 +0.6 +0.9 +1.2 +fλ / 10−19 erg s−1 cm−2 ˚A−1 +Hγ +Hδ +Hζ +Hη +Ca k +Ca h +Hϵ ++ +[O ii] +Fitted model +0.4 +0.5 +0.6 +0.7 +0.8 +Rest-frame Wavelength / µm +Fig. 1 JWST NIRSpec observations of GS-9209. Data were taken using the G235M and +G395M gratings (R = 1000), providing wavelength coverage from λ = 1.7 − 5.1µm. The +galaxy is at z = 4.658, and exhibits extremely deep Balmer absorption lines, similar to lower +redshift post-starburst galaxies, clearly indicating this galaxy experienced a significant, rapid +drop in star-formation rate (SFR) within the past few hundred million years. The spectral +region from λ = 2.6 − 4.0µm, containing Hβ and Hα, is shown at a larger scale in Fig. 2. +2.1 Spectroscopic data +On 16th November 2022, we obtained medium-resolution spectroscopy (R = +λ/∆λ = 1000) through the JWST NIRSpec fixed slit, integrating for 3 hours +with the G235M grism and 2 hours with the G395M grism, providing con- +tinuous wavelength coverage from λ = 1.7 − 5.1µm. These data, shown in +Fig. 1, reveal a full suite of extremely deep Balmer absorption features, from +which we measure a spectroscopic redshift of 4.6582 ± 0.0002, consistent with +previous photometric data and the VUDS spectrum. The spectrum strongly +resembles that of an A-type star, and is reminiscent of lower-redshift post- +starburst galaxies [19–21], with a Hδ equivalent width (EW), as measured by +the HδA Lick index, of 7.9 ± 0.3˚A, comparable to the most extreme values +observed in the local Universe [22]. These spectral features strongly indicate +this galaxy has undergone a sharp decline in star-formation rate (SFR) during +the preceding few hundred Myr. +The observed continuum is relatively smooth, as is the case for A-type +stars, with only two clearly detected metal absorption features: the Ca k line +at 3934˚A and the Na d feature at 5895˚A. The Ca h line at 3969˚A is blended +with the much stronger Hϵ Balmer line. The spectrum exhibits only the merest +suspicion of [O ii] 3727˚A and [O iii] 4959˚A, 5007˚A emission, and no apparent +infilling of Hβ or any of the higher-order Balmer absorption lines. However, +as can be seen in Fig. 2, both Hα and [Nii] 6584˚A are clearly albeit weakly +detected in emission, with Hα also exhibiting an obvious broad component. +This broad component, along with the relative strength of [N ii] compared +with the narrow Hα line indicate the presence of an accreting supermassive + +Springer Nature 2021 LATEX template +4 +A massive quiescent galaxy at redshift 4.658 +2.6 +2.8 +3.0 +3.2 +3.4 +3.6 +3.8 +4.0 +Observed Wavelength / µm +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +fλ / 10−19 erg s−1 cm−2 ˚A−1 +Hβ +Hα +Mg i +Na d +[N ii] +Fe i +[O iii] +[O iii] +Bagpipes full fitted model +Bagpipes AGN component +Narrow line model +0.50 +0.55 +0.60 +0.65 +0.70 +Rest-frame Wavelength / µm +Observed fluxes +Observed flux errors +Fig. 2 JWST NIRSpec observations of GS-9209: zoom in on Hβ and Hα. Data are shown +in blue, with their associated uncertainties visible at the bottom in purple. The full Bagpipes +fitted model is shown in black, with the AGN component shown in red. The narrow Hα and +[N ii] lines were masked during the Bagpipes fitting process, and subsequently fitted with +Gaussian functions, shown in green. Key emission and absorption features are also marked. +black hole: an active galactic nucleus (AGN). However, the extreme EWs of +the observed Balmer absorption features indicate that the continuum emission +must be strongly dominated by the stellar component. Nevertheless, the AGN +contribution to GS-9209 must be carefully modelled when fitting the spectrum +of this source to extract reliable stellar population properties (see Section 4.3). +2.2 Full spectral fitting +To measure the stellar population properties of GS-9209, we perform full spec- +trophotometric fitting using the Bagpipes code. Full details of the methodology +we employ are given in Section 4.3. Briefly, we combine our spectroscopic +data with previously available CANDELS photometry, as well as new JWST +NIRCam medium-band imaging in 5 filters from the Ultra Deep Field +Medium-Band Survey (Programme ID: 1963; PI: Williams). We first mask the +wavelengths corresponding to [O ii], [O iii], narrow Hα and [N ii], due to likely +AGN contributions. We discuss the properties of these lines and their likely +origin in Section 2.5. We then fit a 22-parameter model for the stellar, dust, +nebular and AGN components, as well as spectrophotometric calibration. +The resulting posterior median model is shown in black in Figs 1 and 2. We +obtain a stellar mass of log10(M∗/M⊙) = 10.61±0.02, under the assumption of +a Kroupa initial mass function (IMF) [23]. We additionally recover a very low +level of dust attenuation, with AV = 0.04+0.05 +−0.03. The SFR we measure averaged +over the past 100 Myr is consistent with zero, with a very stringent upper +bound, though this is largely a result of our chosen star-formation history +(SFH) parameterisation [24]. We report a more-realistic upper bound on the +SFR in Section 2.5 based on the narrow Hα line. + +Springer Nature 2021 LATEX template +A massive quiescent galaxy at redshift 4.658 +5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +Age of Universe / Gyr +0.0 +1.0 +2.0 +3.0 +log10(SFR/yr−1) +SFRpeak = 530+840 +−310 M⊙ yr−1 +tform = 0.71+0.3 +−0.2 Gyr +2σ +1σ +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +Age of Universe / Gyr +9.0 +9.5 +10.0 +10.5 +11.0 +log10(M∗/M⊙) +Labbe et al. (2022) +5 +6 +8 +12 +30 +Redshift +5 +6 +8 +12 +30 +Redshift +Fig. 3 The star-formation history of GS-9209. The SFR as a function of time is shown in +the left panel, with the stellar mass as a function of time shown in the right panel. The blue +lines show the posterior medians, with the darker and lighter shaded regions showing the 1σ +and 2σ confidence intervals respectively. We find a formation redshift, zform = 7.3 ± 0.2 and +a quenching redshift, zquench = 6.7 ± 0.3. The sample of massive z ≃ 8 galaxy candidates +from JWST CEERS reported by [7] is also shown in the right panel, demonstrating that +these candidates are plausible progenitors for GS-9209. +2.3 Star-formation history +The star-formation history (SFH) we recover is shown in Fig. 3. We find that +GS-9209 formed its stellar population largely during a ≃ 200 Myr period, from +around 600 − 800 Myr after the Big Bang (z ≃ 7 − 8). We recover a mass- +weighted mean formation time, tform = 0.71+0.03 +−0.02 Gyr after the Big Bang, +corresponding to a formation redshift, zform = 7.3 ± 0.2. This is the redshift +at which GS-9209 would have had half its current stellar mass, approximately +log10(M∗/M⊙) = 10.3. We find that GS-9209 quenched (which we define as +the time at which its sSFR fell below 0.2 divided by the Hubble time, e.g., +[25]) at time tquench = 0.79+0.06 +−0.04 Gyr after the Big Bang, corresponding to a +quenching redshift, zquench = 6.7 ± 0.3. +Our model predicts that the peak historical SFR for GS-9209 (at approx- +imately zform) was within the range SFRpeak = 530+840 +−310 M⊙ yr−1. This is +similar to the SFRs of bright submillimetre galaxies (SMGs). The number den- +sity of SMGs with SFR > 300 M⊙ yr−1 at 5 < z < 6 has been estimated to +be ≃ 3×10−6 Mpc−3 [26]. Extrapolation then suggests that the SMG number +density at z ≃ 7 is ≃ 1 × 10−6 Mpc−3, which equates to ≃ 1 SMG at z ≃ 7 +over the ≃ 400 square arcmin area from which GS-9209 and one other z > 4 +quiescent galaxy were selected [14]. This broadly consistent number density +suggests it is entirely plausible that GS-9209 went through a SMG phase at +z ≃ 7, shortly before quenching. +In the right panel of Fig. 3, we show the positions of the massive, high- +redshift galaxies recently reported by [7] in the first imaging release from the +JWST CEERS survey. It can be seen that the positions of these galaxies are + +Springer Nature 2021 LATEX template +6 +A massive quiescent galaxy at redshift 4.658 +broadly consistent with the SFH of GS-9209 at z ≃ 8. It should however be +noted that, as previously discussed, GS-9209 was selected as one of only two +robustly identified z > 4 massive quiescent galaxies in an area roughly 10 times +the size of the initial CEERS imaging area [14]. It therefore seems unlikely +that a large fraction of the objects reported by [7] will evolve in a similar way +to GS-9209 over the redshift interval from z ≃ 5 − 8. +2.4 Stellar metallicity +We obtain a relatively low stellar metallicity for GS-9209 of log10(Z∗/Z⊙) = +−0.97+0.06 +−0.07 (where we adopt a value of Z⊙=0.0142 [27]). By re-running our +fitting procedure at a range of fixed metallicity values, we find that metallicity +is constrained mainly by the shape of the stellar continuum emission above +the Balmer break (the λ = 2.0 − 2.6µm region shown in the inset panel of +Fig. 1), which is strongly incompatible with models at higher metallicities. +This UV continuum shape is mostly sensitive to the Fe abundance [28, 29], +and we therefore associate our measured Z∗ value with the Fe abundance, +[Fe/H] = −0.97+0.06 +−0.07. This is ≃ 0.4 dex below the mean z ≃ 3.5 stellar mass vs +iron abundance relationship for star-forming galaxies [30]. Given that GS-9209 +formed its stellar population at z ≃ 7, our result suggests that the stellar mass +vs iron abundance relation continues to trend downwards over the redshift +interval from z ≃ 3.5−7, as is observed between the local Universe and z ≃ 3.5. +As can be seen from Figs 1 and 2, we do not obtain a good fit to either the +Ca k or Na d absorption features, with our model significantly under-predicting +the depths of both. Stellar populations that form and quench rapidly are known +to be α-enhanced [31], whereas the stellar population models we fit assume a +fixed scaled-Solar abundance pattern (see Section 4.3). We therefore provision- +ally attribute the failure of our model to reproduce these α-element absorption +features to significant α-enhancement in GS-9209. It should be noted however +that both of these features (in particular Na d) can also arise from interstellar +medium (ISM) absorption, though the low dust attenuation we infer from our +spectral fit might be taken to suggest this effect should be small. +Unfortunately, reliable empirical α-enhanced models are not currently +available for stellar populations with ages less than 1 Gyr. Therefore, to test +this α-enhancement hypothesis, we first measure the EWs of these two fea- +tures from our data (see Section 4), obtaining a Ca k EW of 2.15 ± 0.25˚A, +and a Na d EW of 2.09 ± 0.46˚A. For comparison, our posterior median model +predicts values of 1.12˚A and 0.41˚A respectively. We then scale up the metallic- +ity of our model, keeping all other parameters fixed, until the predicted EWs +match our data. By this process, we obtain [Ca/Fe] = 0.67+0.25 +−0.15. We are how- +ever unable to reproduce the observed depth of Na d via this process, which +we attribute to the known strong ISM component of this absorption feature +[29, 32]. The Ca abundance we calculate is however fully consistent with both +theoretical predictions [33] and observational evidence [34] for α-enhancement +in extreme stellar populations. In particular, [3] report a consistent value of +[Ca/Fe] = 0.59 ± 0.07 for an extreme massive quiescent galaxy at z = 2.1. + +Springer Nature 2021 LATEX template +A massive quiescent galaxy at redshift 4.658 +7 +We therefore adopt our measured Ca abundance as our best estimate of the +α-enhancement of GS-9209, [α/Fe] = 0.67+0.25 +−0.15. This extreme α-enhancement +supports our finding of an extremely short, ≲ 200 Myr formation timescale +[31], as shown in Fig. 3. We caution however that this value could be artificially +boosted by an ISM contribution to the Ca k absorption line. +2.5 Evidence for AGN activity +From our Bagpipes full spectral fit, we measure an observed broad Hα flux of +fHα, broad = 1.26±0.08×10−17 = erg s−1 cm−2 and full width at half maximum +(FWHM) of 10800±600 km s−1 in the rest frame. This line width, whilst very +broad, is consistent with rest-frame UV broad line widths measured for some +z = 6 quasars (e.g., [35, 36]). +We also recover an observed AGN continuum flux at rest-frame wave- +length, λrest = 5100˚A of f5100 = 0.040 ± 0.004 × 10−19 erg s−1 cm−2 ˚A−1. +This is approximately 5 per cent of the total observed flux from GS-9209 at +λ = 2.9µm. We measure a power-law index for the AGN continuum emission +of αλ = −1.36±0.08 at λrest < 5000˚A, and αλ = 0.69±0.14 at λrest > 5000˚A. +These indices are broadly consistent with the average values observed for local +quasars [37]. In combination with the non-detection of GS-9209 at longer wave- +lengths (see Section 2), this suggests the AGN component in GS-9209 is not +significantly reddened. The AGN contribution to the continuum flux from GS- +9209 rises to ≃ 15 per cent at the blue end of our spectrum (λ = 1.7µm), +and ≃ 20 per cent at the red end (λ = 5µm). Just above the Lyman break at +λ ≃ 7000˚A, the AGN contribution is ≃ 35 per cent of the observed flux. +Given our measured fHα, broad, which is more direct than our AGN con- +tinuum measurement, the average relation for local AGN presented by [38] +predicts f5100 to be ≃ 0.4 dex brighter than we measure. However, given the +intrinsic scatter of 0.2 dex they report, our measured f5100 is only 2σ below +the mean relation. The extreme equivalent widths of the observed Balmer +absorption features firmly disfavour stronger AGN continuum emission. +We fit the narrow Hα and [N ii] lines in our spectrum as follows. We first +subtract from our observed spectrum the posterior median Bagpipes model +from our full spectral fitting, described in Section 2.2. We then simultaneously +fit Gaussian components to both lines, assuming the same velocity width for +both, which is allowed to vary. This process is visualised in Fig. 2. We also +show the broad Hβ line in our AGN model, for which we assume the same +width as broad Hα, as well as Case B recombination. It can be seen that the +broad Hβ line peaks at around the noise level in our spectrum, and is hence +too weak to be clearly observed in our data. +We obtain a Hα narrow-line flux of 1.58 ± 0.10 × 10−18 erg s−1 cm−2 +and a [N ii] flux of 1.56 ± 0.10 × 10−18 erg s−1 cm−2, giving a line ratio of +log10([N ii]/Hα) = −0.01 ± 0.04. This line ratio is significantly higher than +would be expected as a result of ongoing star formation, and is consistent +with excitation due to an AGN or shocks resulting from galactic outflows [39]. +Such outflows are commonly observed in post-starburst galaxies at z ≳ 1 [40] + +Springer Nature 2021 LATEX template +8 +A massive quiescent galaxy at redshift 4.658 +Fig. 4 JWST NIRCam imaging of GS-9209. Each cutout image shows an area of 1.5′′×1.5′′. +The RGB image in the first (leftmost) panel is constructed with F430M as red, F210M as +green and F182M as blue. The second panel shows the F210M image, with our posterior +median PetroFit model shown in the third panel. The residuals between model and data are +shown in the right panel, on the same colour scale as the middle two panels. +without corresponding AGN signatures, suggesting either that these outflows +are driven by stellar feedback, or that the AGN activity responsible for the +outflow has since shut down. +Even if we assume all the narrow Hα emission is driven by ongoing +star formation, we obtain SFR = 1.9 ± 0.1 M⊙ yr−1 [41], corresponding to +log10(sSFR/yr−1) = −10.3±0.1. This is under the assumption that dust atten- +uation is negligible, based on our finding of a very low AV from full spectral +fitting in Section 2.2. This is well below the commonly applied sSFR threshold +for defining quiescent galaxies at this redshift [25], log10(sSFRthreshold/yr−1) = +0.2/tH = −9.8, where tH is the age of the Universe. Given the multiple lines +of evidence we uncover for a significant non-stellar component to this line, it +is likely that the SFR of GS-9209 is considerably lower than this estimate. +We estimate the black-hole mass for GS-9209, M•, from our combined +Hα flux and broad-line width, using the relation presented in Equation 6 +of [38], obtaining log10(M•/M⊙) = 8.7 ± 0.1. From our Bagpipes full spec- +tral fit, we infer a stellar velocity dispersion, σ = 247 ± 16 km s−1 for +GS-9209, after correcting for the intrinsic dispersion of our template set, +as well as instrumental dispersion. Given this measurement, the relationship +between velocity dispersion and black-hole mass presented by [42] predicts +log10(M•/M⊙) = 8.9 ± 0.1. +Given the broad agreement between these estimators, it seems reasonable +to conclude that GS-9209 contains a supermassive black hole with a mass of +approximately half a billion to a billion Solar masses. It is interesting to note +that this is ≃ 4 − 5 times the black-hole mass that would be expected given +the stellar mass of the galaxy, assuming this is equivalent to the bulge mass. +This is consistent with the observed increase in the average black-hole to bulge +mass ratio for massive galaxies from 0 < z < 2 [43]. This large amount of +historical AGN accretion relative to star formation strongly implies that AGN +feedback may be responsible for quenching this galaxy. +2.6 Size measurement and dynamical mass +GS-9209 is an extremely compact source, which is only marginally resolved in +the highest-resolution available imaging data. The CANDELS/3DHST team + +RGB +F210M Data +Model +Residual +1.5"×Springer Nature 2021 LATEX template +A massive quiescent galaxy at redshift 4.658 +9 +[44] measured an effective radius, re = 0.029 ± 0.002′′ for GS-9209 in the HST +F125W filter via S´ersic fitting, along with a S´ersic index, n = 6.0 ± 0.8. At +z = 4.658, this corresponds to re = 189 ± 13 parsecs. +We update this size measurement using the newly available JWST NIR- +Cam F210M-band imaging, which has a FWHM of ≃ 0.07′′ (see Section 4.4). +Accounting for the AGN point-source contribution, we measure an effective +radius, re = 0.033 ± 0.003′′ for the stellar component of GS-9209, along with +a S´ersic index, n = 2.3 ± 0.3. At z = 4.658, this corresponds to re = 215 ± 20 +parsecs. This is consistent with the CANDELS/3DHST measurement, and is +≃ 0.7 dex below the mean relationship between re and stellar mass for qui- +escent galaxies at z ≃ 1 [44, 45]. This is interesting given that post-starburst +galaxies z ≃ 1 are known to be more compact than is typical for the wider +quiescent population [46]. We calculate a stellar-mass surface density within +re of log10(Σeff/M⊙ kpc−2) = 11.15 ± 0.08, consistent with the densest stel- +lar systems in the Universe [47]. We show the F210M data for GS-9209, along +with our posterior-median model in Fig. 4. +We estimate the dynamical mass using our size and velocity dispersion +measurements (e.g., [40]), obtaining a value of log10(Mdyn/M⊙) = 10.3 ± 0.1. +This is ≃ 0.3 dex lower than the stellar mass we measure. As GS-9209 is only +marginally resolved, even in JWST imaging data, and due to the presence +of the AGN component, it is plausible that our measured re may be subject +to systematic uncertainties. Deeper imaging data in the F200W or F277W +bands (e.g., from the JWST Advanced Deep Extragalactic Survey; JADES) +will provide a useful check on this, particularly given the lower AGN fraction +in the F277W band. Furthermore, since the pixel scale of NIRSpec is 0.1′′, +our velocity dispersion measurement may not accurately represent the central +velocity dispersion of GS-9209, leading to an underestimated dynamical mass. +It should also be noted that the stellar mass we measure is strongly dependent +on our assumed IMF. +A final, intriguing possibility would be a high level of rotational support in +GS-9209, as has been observed for quiescent galaxies at 2 < z < 3 [48]. Unfor- +tunately, the extremely compact nature of the source makes any attempt at +resolved studies extremely challenging, even with the JWST NIRSpec integral +field unit. Resolved kinematics for this galaxy would be a clear use case for the +High Angular Resolution Monolithic Optical and Near-infrared Integral field +spectrograph (HARMONI) planned for the Extremely Large Telescope (ELT). +3 Conclusion +We report the spectroscopic confirmation of a massive quiescent galaxy, GS- +9209 at a new redshift record of z = 4.6582 ± 0.002, with a stellar mass +of log10(M∗/M⊙) = 10.61 ± 0.02. This galaxy formed its stellar popula- +tion over a ≃ 200 Myr period, approximately 600 − 800 Myr after the Big +Bang (zform = 7.3 ± 0.2), before quenching at zquench = 6.7 ± 0.3. GS-9209 +demonstrates unambiguously that massive galaxy formation was already well + +Springer Nature 2021 LATEX template +10 +A massive quiescent galaxy at redshift 4.658 +underway within the first billion years of cosmic history, with this object having +reached log10(M∗/M⊙) > 10.3 by z = 7. This galaxy also clearly demonstrates +that the earliest onset of galaxy quenching was no later than ≃ 800 Myr after +the Big Bang. +We estimate the iron abundance and α-enhancement of GS-9209, finding +[Fe/H] = −0.97+0.06 +−0.07 and [α/Fe] = 0.67+0.25 +−0.15, suggesting the stellar mass vs +iron abundance relation at z ≃ 7, when this object formed most of its stars, +was ≃ 0.4 dex lower than at z ≃ 3.5 [30]. Whilst its spectrum is dominated by +stellar emission, GS-9209 also hosts an AGN, for which we measure a black-hole +mass of log10(M•/M⊙) = 8.7 ± 0.1 from the observed broad and narrow Hα +emission [38]. We also predict a consistent value of log10(M•/M⊙) = 8.9 ± 0.1 +based on the stellar velocity dispersion of GS-9209 [42]. Whilst large-scale star +formation in GS-9209 has been quenched for almost half a billion years, the +significant integrated quantity of AGN accretion implied by this large black- +hole mass (≃ 4 − 5 times what would be expected given the stellar mass of +this galaxy) suggests that AGN activity plausibly played a significant role in +quenching star formation in this galaxy. +Based on the properties we measure, GS-9209 seems likely to be associated +with the most extreme galaxy populations currently known at z > 5, such as +the highest-redshift submillimetre galaxies and quasars (e.g., [36, 49, 50]). GS- +9209 is also plausibly descended from an object similar to the z ≃ 8 massive +galaxy candidates recently reported in the first data from the JWST CEERS +programme [7], though the number density of these candidates is significantly +higher than that of z > 4 quiescent galaxies. GS-9209 and similar objects (e.g., +[9]) are also likely progenitors for the dense, ancient cores of the most massive +galaxies in the local Universe. +This study, which makes use of just 5 hours of on-source integration time, +demonstrates the huge potential of JWST for revolutionising our understand- +ing of the high-redshift Universe. It seems clear that this work will be followed +rapidly by the confirmation and detailed spectroscopic exploration of large +samples of z > 4 quiescent galaxies, to build up a detailed understanding of +massive galaxy formation and quenching during the first billion years. +4 Methods +4.1 Spectroscopic data reduction +We reduce our NIRSpec data using the JWST Science Calibration Pipeline +v1.8.4, using version 1017 of the JWST calibration reference data. To improve +the spectrophotometric calibration of our data, we also reduce observations +of the A-type standard star 2MASS J18083474+6927286 [51], taken as part +of JWST commissioning programme 1128 (PI: L¨utzgendorf) [52] using the +same instrument modes. We compare the resulting stellar spectrum against +a spectral model for this star from the CALSPEC library [53] to construct a +calibration function, which we then apply to our observations of GS-9209. + +Springer Nature 2021 LATEX template +A massive quiescent galaxy at redshift 4.658 +11 +4.2 Photometric data reduction +The majority of our photometric data are taken directly from the CANDELS +GOODS South catalogue [54]. We supplement this with new JWST NIRCam +photometric data taken as part of the Ultra Deep Field Medium-Band Survey +[55] (Programme ID: 1963; PI: Williams). Data are available in the F182M, +F210M, F430M, F460M and F480M bands. We reduce these data using the +PRIMER Enhanced NIRCam Image-processing Library (PENCIL, e.g., [8]), a +custom version of the JWST Science Calibration Pipeline (v1.8.0), and using +version 1011 of the JWST calibration reference data. We measure photometric +fluxes for GS-9209 in large, 1′′-diameter apertures to ensure we measure the +total flux in each band (the object is isolated, with no other sources within +this radius, see Fig. 4). We measure uncertainties as the standard deviation of +flux values in the nearest 100 blank-sky apertures, masking out nearby objects +(e.g., [56]). +4.3 Bagpipes full spectral fitting +We fit the available photometry in parallel with our new spectroscopic data +using the Bagpipes code [57]. Our model has a total of 22 free parameters, +describing the stellar, dust, nebular and AGN components of the spectrum. +A full list of these parameters, along with their associated priors, is given in +Table 1. We fit our model to the data using the MultiNest nested sampling +algorithm [58–60]. +We use the 2016 updated version of the BC03 [61, 62] stellar population +models, using the MILES stellar spectral library [63] and updated stellar evolu- +tionary tracks [64, 65]. We assume a double-power-law star-formation-history +model (e.g., [24, 57]). We allow the logarithm of the stellar metallicity, Z∗ to +vary freely from log10(Z∗/Z⊙) = −2.45 to 0.55. These are the limits of the +range spanned by the BC03 model grid relative to our adopted Solar metallicity +value (Z⊙ = 0.0142 [27]). +We mask out the narrow emission lines in our spectrum during our Bag- +pipes fitting due to likely AGN contributions, whereas Bagpipes is only capable +of modelling emission lines from star-forming regions. We do however still +include a nebular model in our Bagpipes fit to allow for the possibility of +nebular continuum emission from star-forming regions. We assume a stellar- +birth-cloud lifetime of 10 Myr, and vary the logarithm of the ionization +parameter, U, from log10(U) = −4 to −2. We also allow the logarithm of the +gas-phase metallicity, Zg, to vary freely from log10(Zg/Z⊙) = −2.45 to 0.55. +Because our eventual fitted model only includes an extremely small amount +of star formation within the last 10 Myr for GS-9209, this nebular component +makes a negligible contribution to the fitted model spectrum. +We model attenuation of the above components by dust using the model +of [66, 67], which is parameterised as a power-law deviation from the Calzetti +dust attenuation law [68], and also includes a Drude profile to model the 2175˚A +bump. We allow the V −band attenuation, AV to vary from 0 − 4 magnitudes. + +Springer Nature 2021 LATEX template +12 +A massive quiescent galaxy at redshift 4.658 +Table 1 The 22 free parameters of the Bagpipes model we fit to our spectroscopic and photometric data (see Sections 2.2 and 4.3), along with their +associated prior distributions. The upper limit on τ, tobs, is the age of the Universe as a function of redshift. Logarithmic priors are all applied in +base ten. For parameters with Gaussian priors, the mean is µ and the standard deviation is σ. +Component Parameter +Symbol / Unit +Range +Prior +Hyper-parameters +General +Redshift +z +(4.6, 4.7) +Gaussian +µ = 4.66 +σ = 0.01 +Stellar velocity dispersion +σ / km s−1 +(50, 500) +Logarithmic +SFH +Total stellar mass formed +M∗ / M⊙ +(1, 1013) +Logarithmic +Stellar metallicity +Z∗ / Z⊙ +(0.00355, 3.55) +Logarithmic +Double-power-law falling slope +α +(0.01, 1000) +Logarithmic +Double-power-law rising slope +β +(0.01, 1000) +Logarithmic +Double-power-law turnover time +τ / Gyr +(0.1, tobs) +Uniform +Dust +V −band attenuation +AV / mag +(0, 4) +Uniform +Deviation from Calzetti slope +δ +(−0.3, 0.3) +Gaussian +µ = 0 +σ = 0.1 +Strength of 2175˚A bump +B +(0, 5) +Uniform +Attenuation ratio for birth clouds ϵ +(1, 5) +Uniform +AGN +Power law slope (λ < 5000˚A) +αλ<5000˚ +A +(−2.5, −0.5) +Gaussian +µ = −1.5 σ = 0.1 +Power law slope (λ > 5000˚A) +αλ>5000˚ +A +(−0.5, 1.5) +Gaussian +µ = 0.5 +σ = 0.2 +Hα broad-line flux +fHα, broad / erg s−1 cm−2 +(0, 2.5 × 10−17) Uniform +Hα broad-line velocity dispersion +σHα, broad / km s−1 +(1000, 5000) +Logarithmic +Continuum flux at λ = 5100˚A +f5100 / erg s−1 cm−2 ˚A−1 (0, 10−19) +Uniform +Nebular +Ionization parameter +U +(10−4, 10−2) +Logarithmic +Gas-phase metallicity +Zg / Z⊙ +(0.00355, 3.55) +Logarithmic +Calibration Zero order +P0 +(0.75, 1.25) +Gaussian +µ = 1 +σ = 0.1 +First order +P1 +(−0.25, 0.25) +Gaussian +µ = 0 +σ = 0.1 +Second order +P2 +(−0.25, 0.25) +Gaussian +µ = 0 +σ = 0.1 +Noise +White noise scaling +a +(0.1, 10) +logarithmic + +Springer Nature 2021 LATEX template +A massive quiescent galaxy at redshift 4.658 +13 +We further assume that attenuation is multiplied by an additional factor for +all stars with ages below 10 Myr, and resulting nebular emission. This factor +is commonly assumed to be 2, however we allow this to vary from 1 to 5. +We allow redshift to vary, using a narrow Gaussian prior with a mean of 4.66 +and standard deviation of 0.01. We additionally convolve the spectral model +with a Gaussian kernel in velocity space, to account for velocity dispersion in +our target galaxy. The width of this kernel is allowed to vary with a logarithmic +prior across a range from 50 − 500 km s−1. +Separately from the above components, we also include a model for AGN +continuum, broad Hα and Hβ emission. Following [37], we model AGN contin- +uum emission with a broken power law, with two spectral indices and a break +at λrest = 5000˚A in the rest frame. We vary the spectral index at λrest < 5000˚A +using a Gaussian prior with a mean value of αλ = −1.5 (αν = −0.5) and stan- +dard deviation of 0.1. We also vary the spectral index at λrest > 5000˚A using +a Gaussian prior with a mean value of αλ = 0.5 (αν = −2.5) and standard +deviation of 0.2. We parameterise the normalisation of the AGN continuum +component using f5100, the flux at rest-frame 5100˚A, which we allow to vary +with a linear prior from 0 to 10−19 erg s−1 cm−2 ˚A−1. +We model broad Hα with a Gaussian component, varying the normalisation +from 0 to 2.5 × 10−17 erg s−1 cm−2 using a linear prior, and the velocity +dispersion from 1000 − 5000 km s−1 in the rest frame using a logarithmic +prior. We also include a broad Hβ component in the model, which has the +same parameters as the broad Hα line, but with normalisation divided by the +standard 2.86 ratio from Case B recombination theory. However, as shown in +Fig. 2, this Hβ model peaks at around the noise level in our spectrum, and +the line is therefore plausible in not being obviously detected in the observed +spectrum. +We include intergalactic medium (IGM) absorption using the model of +[69]. To allow for imperfect spectrophotometric calibration of our spectroscopic +data, we also include a second-order Chebyshev polynomial (e.g., [70, 71]), +which the above components of our combined model are all divided by before +being compared with our spectroscopic data. We finally fit an additional white +noise term, which multiplies the spectroscopic uncertainties from the JWST +pipeline by a factor, a, which we vary with a logarithmic prior from 1 − 10. +4.4 Size measurement from F210M-band imaging +We model the light distribution of GS-9209 in the JWST NIRCam F210M +imaging data using PetroFit [72]. We fit these PetroFit models to our data +using the MultiNest nested sampling algorithm [58–60]. We use F210M in +preference to the F182M band due to the smaller AGN contribution in +F210M and the fact that it sits above the Balmer break, therefore being +more representative of the stellar mass present rather than any ongoing star +formation. + +Springer Nature 2021 LATEX template +14 +A massive quiescent galaxy at redshift 4.658 +As our spectroscopic data contains strong evidence for an AGN, we fit both +S´ersic and delta-function components simultaneously, convolved by an empir- +ically estimated PSF, derived by stacking bright stars. In preliminary fitting, +we find that the relative fluxes of these two components are entirely degen- +erate with the S´ersic parameters. We therefore predict the AGN contribution +to the flux in this band based on our full-spectral-fitting result, obtaining a +value of 8 ± 1 per cent. We then impose this as a Gaussian prior on the rela- +tive contributions from the S´ersic and delta function components. The 11 free +parameters of our model are the overall flux normalisation, which we fit with a +logarithmic prior, the effective radius, re, S´ersic index, n, ellipticity and posi- +tion angle of the S´ersic component, the x and y centroids of both components, +the position angle of the point spread function, and the fraction of light in the +delta-function component, which we fit with a Gaussian prior with a mean of +8 per cent and standard deviation of 1 per cent, based on our full spectral +fitting result. +Acknowledgements +The authors would like to thank James Aird for helpful discussions. A. C. +Carnall thanks the Leverhulme Trust for their support via a Leverhulme +Early Career Fellowship. R. J. McLure, J. S. Dunlop, D. J. McLeod, V. Wild, +R. Begley, C. T. Donnan and M. L. Hamadouche acknowledge the support +of the Science and Technology Facilities Council. F. Cullen acknowledges +support from a UKRI Frontier Research Guarantee Grant (grant reference +EP/X021025/1). A. Cimatti acknowledges support from the grant PRIN +MIUR 2017 - 20173ML3WW 001. +Statement of Author Contributions +ACC led the preparation of the observing proposal, reduction and analysis of +the data, and preparation of the manuscript. RJM, JSD, VW, FC and AC +provided advice and assistance with data reduction, analysis and interpreta- +tion, as well as consulting on the preparation of the observing proposal. DJM, +DM, RB and CTD reduced the JWST imaging data and prepared the empir- +ical PSF. DJM, MLH and SMJ assisted with measurement of the size and +morphology of GS-9209. SW assisted with selection of GS-9209 from the CAN- +DELS catalogues prior to the observing proposal being submitted. All authors +assisted with preparation of the final published manuscript. +References +[1] Dunlop, J., Peacock, J., Spinrad, H., Dey, A., Jimenez, R., Stern, D., +Windhorst, R.: A 3.5-Gyr-old galaxy at redshift 1.55. 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The Astronomical Journal 163(5), +202 (2022). https://doi.org/10.3847/1538-3881/ac5908 + diff --git a/AtFIT4oBgHgl3EQf_CxR/content/tmp_files/load_file.txt b/AtFIT4oBgHgl3EQf_CxR/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d4dae9a86143b82020468f5cf15761ae336ecf31 --- /dev/null +++ b/AtFIT4oBgHgl3EQf_CxR/content/tmp_files/load_file.txt @@ -0,0 +1,2318 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf,len=2317 +page_content='Springer Nature 2021 LATEX template A massive quiescent galaxy at redshift 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='658 Adam C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Carnall1*, Ross J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' McLure1, James S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Dunlop1, Derek J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' McLeod1, Vivienne Wild2, Fergus Cullen1, Dan Magee3, Ryan Begley1, Andrea Cimatti4,5, Callum T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Donnan1, Massissilia L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Hamadouche1, Sophie M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Jewell1 and Sam Walker1 1Institute for Astronomy, School of Physics & Astronomy, University of Edinburgh, Royal Observatory, Edinburgh, EH9 3HJ, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' 2School of Physics & Astronomy, University of St Andrews, North Haugh, St Andrews, KY16 9SS, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' 3Department of Astronomy and Astrophysics, UCO/Lick Observatory, University of California, Santa Cruz, CA 95064, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' 4Department of Physics and Astronomy (DIFA), University of Bologna, Via Gobetti 93/2, I-40129, Bologna, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' 5INAF, Osservatorio di Astrofisica e Scienza dello Spazio, Via Piero Gobetti 93/3, I-40129, Bologna, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Corresponding author email: adam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='carnall@ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='uk Abstract We report the spectroscopic confirmation of a massive quiescent galaxy, GS-9209 at a new redshift record of z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='658, just 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='25 Gyr after the Big Bang, using new deep continuum observations from JWST NIR- Spec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' From our full-spectral-fitting analysis, we find that this galaxy formed its stellar population over a ≃ 200 Myr period, approximately 600 − 800 Myr after the Big Bang (zform = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='2), before quench- ing at zquench = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' GS-9209 demonstrates unambiguously that massive galaxy formation was already well underway within the first bil- lion years of cosmic history, with this object having reached a stellar mass of log10(M∗/M⊙) > 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='3 by z = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' This galaxy also clearly demonstrates that the earliest onset of galaxy quenching was no later than ≃ 800 Myr after the Big Bang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We estimate the iron abundance and α-enhancement of GS-9209, finding [Fe/H] = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='97+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='07 and [α/Fe] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='67+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='15, suggesting the stellar mass vs iron abundance rela- tion at z ≃ 7, when this object formed most of its stars, was ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='4 dex lower than at z ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Whilst its spectrum is dominated by stellar emis- sion, GS-9209 also exhibits broad Hα emission, indicating that it hosts an active galactic nucleus (AGN), for which we measure a black-hole 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='11413v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='GA] 26 Jan 2023 Springer Nature 2021 LATEX template 2 A massive quiescent galaxy at redshift 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='658 mass of log10(M•/M⊙) = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Although large-scale star forma- tion in GS-9209 has been quenched for almost half a billion years, the significant integrated quantity of accretion implied by this large black- hole mass suggests AGN feedback plausibly played a significant role in quenching star formation in this galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' GS-9209 is also extremely compact, with an effective radius of just 215 ± 20 parsecs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' This intrigu- ing object offers perhaps our deepest insight yet into massive galaxy formation and quenching during the first billion years of cosmic history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' 1 Summary The discovery of massive galaxies with old stellar populations at early cosmic epochs has historically acted as a key constraint on models for both galaxy for- mation physics and cosmology [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Today, the extremely rapid assembly of the earliest galaxies during the first billion years of cosmic history continues to challenge our understanding of galaxy formation physics [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' The advent of the James Webb Space Telescope (JWST) has exacerbated this issue by con- firming the existence of galaxies in significant numbers as early as the first few hundred million years [7–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Perhaps even more surprisingly, in some galaxies, this initial highly efficient star formation rapidly shuts down, or quenches, giv- ing rise to massive quiescent galaxies as little as ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='5 billion years after the Big Bang, at redshifts up to z ≃ 4 [4, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Due to their faintness and red colour, it has proven extremely challenging to learn about these extreme quiescent galaxies, or to confirm whether any exist at earlier times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Here, we report the spectroscopic confirmation of a quiescent galaxy, GS-9209, at a new redshift record of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='658, just 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='25 billion years after the Big Bang, using the NIRSpec instrument on JWST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' The transformative power of JWST allows us to char- acterise the physical properties of this early massive galaxy in unprecedented detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' GS-9209 has a stellar mass of M∗ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='2 × 1010 M⊙, and quenched star formation at z = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='3, when the Universe was ≃ 800 million years old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' This intriguing object offers perhaps our deepest insight yet into massive galaxy formation and quenching during the first billion years of cosmic history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' 2 Results GS-9209 was first highlighted in the early 2000s as an object with red optical to near-infrared colours and a photometric redshift of z ≃ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='5 [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' An optical spectrum was taken in the mid-2010s as part of the VIMOS Ultra Deep Sur- vey (VUDS) [12], showing tentative evidence for a Lyman break at λ ≃ 7000˚A, but no Lyman α emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' During the past 5 years, several studies have iden- tified GS-9209 as a candidate high-redshift massive quiescent galaxy [13, 14], based on its blue colours at wavelengths, λ = 2 − 8µm and non-detection at millimetre wavelengths [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' GS-9209 is also not detected in X-rays [16], at radio wavelengths [17], or at λ = 24µm [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' The faint, red nature of the source (with magnitudes HAB = 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='7 and KAB = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='6) means that near-infrared spectroscopy with ground-based instrumentation is prohibitively expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Springer Nature 2021 LATEX template A massive quiescent galaxy at redshift 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='658 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='0 Observed Wavelength / µm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='5 fλ / 10−19 erg s−1 cm−2 ˚A−1 F170LP + G235M F290LP + G395M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='6 λ / µm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='2 fλ / 10−19 erg s−1 cm−2 ˚A−1 Hγ Hδ Hζ Hη Ca k Ca h Hϵ + [O ii] Fitted model 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='8 Rest-frame Wavelength / µm Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' 1 JWST NIRSpec observations of GS-9209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Data were taken using the G235M and G395M gratings (R = 1000), providing wavelength coverage from λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='7 − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='1µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' The galaxy is at z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='658, and exhibits extremely deep Balmer absorption lines, similar to lower redshift post-starburst galaxies, clearly indicating this galaxy experienced a significant, rapid drop in star-formation rate (SFR) within the past few hundred million years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' The spectral region from λ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='6 − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='0µm, containing Hβ and Hα, is shown at a larger scale in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='1 Spectroscopic data On 16th November 2022, we obtained medium-resolution spectroscopy (R = λ/∆λ = 1000) through the JWST NIRSpec fixed slit, integrating for 3 hours with the G235M grism and 2 hours with the G395M grism, providing con- tinuous wavelength coverage from λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='7 − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='1µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' These data, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' 1, reveal a full suite of extremely deep Balmer absorption features, from which we measure a spectroscopic redshift of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='6582 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='0002, consistent with previous photometric data and the VUDS spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' The spectrum strongly resembles that of an A-type star, and is reminiscent of lower-redshift post- starburst galaxies [19–21], with a Hδ equivalent width (EW), as measured by the HδA Lick index, of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='3˚A, comparable to the most extreme values observed in the local Universe [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' These spectral features strongly indicate this galaxy has undergone a sharp decline in star-formation rate (SFR) during the preceding few hundred Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' The observed continuum is relatively smooth, as is the case for A-type stars, with only two clearly detected metal absorption features: the Ca k line at 3934˚A and the Na d feature at 5895˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' The Ca h line at 3969˚A is blended with the much stronger Hϵ Balmer line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' The spectrum exhibits only the merest suspicion of [O ii] 3727˚A and [O iii] 4959˚A, 5007˚A emission, and no apparent infilling of Hβ or any of the higher-order Balmer absorption lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' However, as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' 2, both Hα and [Nii] 6584˚A are clearly albeit weakly detected in emission, with Hα also exhibiting an obvious broad component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' This broad component, along with the relative strength of [N ii] compared with the narrow Hα line indicate the presence of an accreting supermassive Springer Nature 2021 LATEX template 4 A massive quiescent galaxy at redshift 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='658 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='0 Observed Wavelength / µm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='0 fλ / 10−19 erg s−1 cm−2 ˚A−1 Hβ Hα Mg i Na d [N ii] Fe i [O iii] [O iii] Bagpipes full fitted model Bagpipes AGN component Narrow line model 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='70 Rest-frame Wavelength / µm Observed fluxes Observed flux errors Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' 2 JWST NIRSpec observations of GS-9209: zoom in on Hβ and Hα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Data are shown in blue, with their associated uncertainties visible at the bottom in purple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' The full Bagpipes fitted model is shown in black, with the AGN component shown in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' The narrow Hα and [N ii] lines were masked during the Bagpipes fitting process, and subsequently fitted with Gaussian functions, shown in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Key emission and absorption features are also marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' black hole: an active galactic nucleus (AGN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' However, the extreme EWs of the observed Balmer absorption features indicate that the continuum emission must be strongly dominated by the stellar component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Nevertheless, the AGN contribution to GS-9209 must be carefully modelled when fitting the spectrum of this source to extract reliable stellar population properties (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='2 Full spectral fitting To measure the stellar population properties of GS-9209, we perform full spec- trophotometric fitting using the Bagpipes code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Full details of the methodology we employ are given in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Briefly, we combine our spectroscopic data with previously available CANDELS photometry, as well as new JWST NIRCam medium-band imaging in 5 filters from the Ultra Deep Field Medium-Band Survey (Programme ID: 1963;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' PI: Williams).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We first mask the wavelengths corresponding to [O ii], [O iii], narrow Hα and [N ii], due to likely AGN contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We discuss the properties of these lines and their likely origin in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We then fit a 22-parameter model for the stellar, dust, nebular and AGN components, as well as spectrophotometric calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' The resulting posterior median model is shown in black in Figs 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We obtain a stellar mass of log10(M∗/M⊙) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='61±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='02, under the assumption of a Kroupa initial mass function (IMF) [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We additionally recover a very low level of dust attenuation, with AV = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='04+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' The SFR we measure averaged over the past 100 Myr is consistent with zero, with a very stringent upper bound, though this is largely a result of our chosen star-formation history (SFH) parameterisation [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We report a more-realistic upper bound on the SFR in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='5 based on the narrow Hα line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Springer Nature 2021 LATEX template A massive quiescent galaxy at redshift 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='658 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='2 Age of Universe / Gyr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='0 log10(SFR/yr−1) SFRpeak = 530+840 −310 M⊙ yr−1 tform = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='71+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='2 Gyr 2σ 1σ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='2 Age of Universe / Gyr 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='0 log10(M∗/M⊙) Labbe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' (2022) 5 6 8 12 30 Redshift 5 6 8 12 30 Redshift Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' 3 The star-formation history of GS-9209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' The SFR as a function of time is shown in the left panel, with the stellar mass as a function of time shown in the right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' The blue lines show the posterior medians, with the darker and lighter shaded regions showing the 1σ and 2σ confidence intervals respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We find a formation redshift, zform = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='2 and a quenching redshift, zquench = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' The sample of massive z ≃ 8 galaxy candidates from JWST CEERS reported by [7] is also shown in the right panel, demonstrating that these candidates are plausible progenitors for GS-9209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='3 Star-formation history The star-formation history (SFH) we recover is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We find that GS-9209 formed its stellar population largely during a ≃ 200 Myr period, from around 600 − 800 Myr after the Big Bang (z ≃ 7 − 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We recover a mass- weighted mean formation time, tform = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='71+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='02 Gyr after the Big Bang, corresponding to a formation redshift, zform = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' This is the redshift at which GS-9209 would have had half its current stellar mass, approximately log10(M∗/M⊙) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We find that GS-9209 quenched (which we define as the time at which its sSFR fell below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='2 divided by the Hubble time, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=', [25]) at time tquench = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='79+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='04 Gyr after the Big Bang, corresponding to a quenching redshift, zquench = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Our model predicts that the peak historical SFR for GS-9209 (at approx- imately zform) was within the range SFRpeak = 530+840 −310 M⊙ yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' This is similar to the SFRs of bright submillimetre galaxies (SMGs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' The number den- sity of SMGs with SFR > 300 M⊙ yr−1 at 5 < z < 6 has been estimated to be ≃ 3×10−6 Mpc−3 [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Extrapolation then suggests that the SMG number density at z ≃ 7 is ≃ 1 × 10−6 Mpc−3, which equates to ≃ 1 SMG at z ≃ 7 over the ≃ 400 square arcmin area from which GS-9209 and one other z > 4 quiescent galaxy were selected [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' This broadly consistent number density suggests it is entirely plausible that GS-9209 went through a SMG phase at z ≃ 7, shortly before quenching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' In the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' 3, we show the positions of the massive, high- redshift galaxies recently reported by [7] in the first imaging release from the JWST CEERS survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' It can be seen that the positions of these galaxies are Springer Nature 2021 LATEX template 6 A massive quiescent galaxy at redshift 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='658 broadly consistent with the SFH of GS-9209 at z ≃ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' It should however be noted that, as previously discussed, GS-9209 was selected as one of only two robustly identified z > 4 massive quiescent galaxies in an area roughly 10 times the size of the initial CEERS imaging area [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' It therefore seems unlikely that a large fraction of the objects reported by [7] will evolve in a similar way to GS-9209 over the redshift interval from z ≃ 5 − 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='4 Stellar metallicity We obtain a relatively low stellar metallicity for GS-9209 of log10(Z∗/Z⊙) = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='97+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='07 (where we adopt a value of Z⊙=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='0142 [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' By re-running our fitting procedure at a range of fixed metallicity values, we find that metallicity is constrained mainly by the shape of the stellar continuum emission above the Balmer break (the λ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='0 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='6µm region shown in the inset panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' 1), which is strongly incompatible with models at higher metallicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' This UV continuum shape is mostly sensitive to the Fe abundance [28, 29], and we therefore associate our measured Z∗ value with the Fe abundance, [Fe/H] = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='97+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' This is ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='4 dex below the mean z ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='5 stellar mass vs iron abundance relationship for star-forming galaxies [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Given that GS-9209 formed its stellar population at z ≃ 7, our result suggests that the stellar mass vs iron abundance relation continues to trend downwards over the redshift interval from z ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='5−7, as is observed between the local Universe and z ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' As can be seen from Figs 1 and 2, we do not obtain a good fit to either the Ca k or Na d absorption features, with our model significantly under-predicting the depths of both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Stellar populations that form and quench rapidly are known to be α-enhanced [31], whereas the stellar population models we fit assume a fixed scaled-Solar abundance pattern (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We therefore provision- ally attribute the failure of our model to reproduce these α-element absorption features to significant α-enhancement in GS-9209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' It should be noted however that both of these features (in particular Na d) can also arise from interstellar medium (ISM) absorption, though the low dust attenuation we infer from our spectral fit might be taken to suggest this effect should be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Unfortunately, reliable empirical α-enhanced models are not currently available for stellar populations with ages less than 1 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Therefore, to test this α-enhancement hypothesis, we first measure the EWs of these two fea- tures from our data (see Section 4), obtaining a Ca k EW of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='25˚A, and a Na d EW of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='46˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' For comparison, our posterior median model predicts values of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='12˚A and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='41˚A respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We then scale up the metallic- ity of our model, keeping all other parameters fixed, until the predicted EWs match our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' By this process, we obtain [Ca/Fe] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='67+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We are how- ever unable to reproduce the observed depth of Na d via this process, which we attribute to the known strong ISM component of this absorption feature [29, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' The Ca abundance we calculate is however fully consistent with both theoretical predictions [33] and observational evidence [34] for α-enhancement in extreme stellar populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' In particular, [3] report a consistent value of [Ca/Fe] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='07 for an extreme massive quiescent galaxy at z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Springer Nature 2021 LATEX template A massive quiescent galaxy at redshift 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='658 7 We therefore adopt our measured Ca abundance as our best estimate of the α-enhancement of GS-9209, [α/Fe] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='67+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' This extreme α-enhancement supports our finding of an extremely short, ≲ 200 Myr formation timescale [31], as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We caution however that this value could be artificially boosted by an ISM contribution to the Ca k absorption line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='5 Evidence for AGN activity From our Bagpipes full spectral fit, we measure an observed broad Hα flux of fHα, broad = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='26±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='08×10−17 = erg s−1 cm−2 and full width at half maximum (FWHM) of 10800±600 km s−1 in the rest frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' This line width, whilst very broad, is consistent with rest-frame UV broad line widths measured for some z = 6 quasars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=', [35, 36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We also recover an observed AGN continuum flux at rest-frame wave- length, λrest = 5100˚A of f5100 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='040 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='004 × 10−19 erg s−1 cm−2 ˚A−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' This is approximately 5 per cent of the total observed flux from GS-9209 at λ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='9µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We measure a power-law index for the AGN continuum emission of αλ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='36±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='08 at λrest < 5000˚A, and αλ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='69±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='14 at λrest > 5000˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' These indices are broadly consistent with the average values observed for local quasars [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' In combination with the non-detection of GS-9209 at longer wave- lengths (see Section 2), this suggests the AGN component in GS-9209 is not significantly reddened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' The AGN contribution to the continuum flux from GS- 9209 rises to ≃ 15 per cent at the blue end of our spectrum (λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='7µm), and ≃ 20 per cent at the red end (λ = 5µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Just above the Lyman break at λ ≃ 7000˚A, the AGN contribution is ≃ 35 per cent of the observed flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Given our measured fHα, broad, which is more direct than our AGN con- tinuum measurement, the average relation for local AGN presented by [38] predicts f5100 to be ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='4 dex brighter than we measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' However, given the intrinsic scatter of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='2 dex they report, our measured f5100 is only 2σ below the mean relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' The extreme equivalent widths of the observed Balmer absorption features firmly disfavour stronger AGN continuum emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We fit the narrow Hα and [N ii] lines in our spectrum as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We first subtract from our observed spectrum the posterior median Bagpipes model from our full spectral fitting, described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We then simultaneously fit Gaussian components to both lines, assuming the same velocity width for both, which is allowed to vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' This process is visualised in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We also show the broad Hβ line in our AGN model, for which we assume the same width as broad Hα, as well as Case B recombination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' It can be seen that the broad Hβ line peaks at around the noise level in our spectrum, and is hence too weak to be clearly observed in our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We obtain a Hα narrow-line flux of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='10 × 10−18 erg s−1 cm−2 and a [N ii] flux of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='10 × 10−18 erg s−1 cm−2, giving a line ratio of log10([N ii]/Hα) = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' This line ratio is significantly higher than would be expected as a result of ongoing star formation, and is consistent with excitation due to an AGN or shocks resulting from galactic outflows [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Such outflows are commonly observed in post-starburst galaxies at z ≳ 1 [40] Springer Nature 2021 LATEX template 8 A massive quiescent galaxy at redshift 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='658 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' 4 JWST NIRCam imaging of GS-9209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Each cutout image shows an area of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='5′′×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='5′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' The RGB image in the first (leftmost) panel is constructed with F430M as red, F210M as green and F182M as blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' The second panel shows the F210M image, with our posterior median PetroFit model shown in the third panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' The residuals between model and data are shown in the right panel, on the same colour scale as the middle two panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' without corresponding AGN signatures, suggesting either that these outflows are driven by stellar feedback, or that the AGN activity responsible for the outflow has since shut down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Even if we assume all the narrow Hα emission is driven by ongoing star formation, we obtain SFR = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='1 M⊙ yr−1 [41], corresponding to log10(sSFR/yr−1) = −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' This is under the assumption that dust atten- uation is negligible, based on our finding of a very low AV from full spectral fitting in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' This is well below the commonly applied sSFR threshold for defining quiescent galaxies at this redshift [25], log10(sSFRthreshold/yr−1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='2/tH = −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='8, where tH is the age of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Given the multiple lines of evidence we uncover for a significant non-stellar component to this line, it is likely that the SFR of GS-9209 is considerably lower than this estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We estimate the black-hole mass for GS-9209, M•, from our combined Hα flux and broad-line width, using the relation presented in Equation 6 of [38], obtaining log10(M•/M⊙) = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' From our Bagpipes full spec- tral fit, we infer a stellar velocity dispersion, σ = 247 ± 16 km s−1 for GS-9209, after correcting for the intrinsic dispersion of our template set, as well as instrumental dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Given this measurement, the relationship between velocity dispersion and black-hole mass presented by [42] predicts log10(M•/M⊙) = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Given the broad agreement between these estimators, it seems reasonable to conclude that GS-9209 contains a supermassive black hole with a mass of approximately half a billion to a billion Solar masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' It is interesting to note that this is ≃ 4 − 5 times the black-hole mass that would be expected given the stellar mass of the galaxy, assuming this is equivalent to the bulge mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' This is consistent with the observed increase in the average black-hole to bulge mass ratio for massive galaxies from 0 < z < 2 [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' This large amount of historical AGN accretion relative to star formation strongly implies that AGN feedback may be responsible for quenching this galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='6 Size measurement and dynamical mass GS-9209 is an extremely compact source, which is only marginally resolved in the highest-resolution available imaging data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' The CANDELS/3DHST team RGB F210M Data Model Residual 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='5"×Springer Nature 2021 LATEX template A massive quiescent galaxy at redshift 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='658 9 [44] measured an effective radius, re = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='029 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='002′′ for GS-9209 in the HST F125W filter via S´ersic fitting, along with a S´ersic index, n = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' At z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='658, this corresponds to re = 189 ± 13 parsecs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We update this size measurement using the newly available JWST NIR- Cam F210M-band imaging, which has a FWHM of ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='07′′ (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Accounting for the AGN point-source contribution, we measure an effective radius, re = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='033 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='003′′ for the stellar component of GS-9209, along with a S´ersic index, n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' At z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='658, this corresponds to re = 215 ± 20 parsecs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' This is consistent with the CANDELS/3DHST measurement, and is ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='7 dex below the mean relationship between re and stellar mass for qui- escent galaxies at z ≃ 1 [44, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' This is interesting given that post-starburst galaxies z ≃ 1 are known to be more compact than is typical for the wider quiescent population [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We calculate a stellar-mass surface density within re of log10(Σeff/M⊙ kpc−2) = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='08, consistent with the densest stel- lar systems in the Universe [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We show the F210M data for GS-9209, along with our posterior-median model in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We estimate the dynamical mass using our size and velocity dispersion measurements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=', [40]), obtaining a value of log10(Mdyn/M⊙) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' This is ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='3 dex lower than the stellar mass we measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' As GS-9209 is only marginally resolved, even in JWST imaging data, and due to the presence of the AGN component, it is plausible that our measured re may be subject to systematic uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Deeper imaging data in the F200W or F277W bands (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=', from the JWST Advanced Deep Extragalactic Survey;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' JADES) will provide a useful check on this, particularly given the lower AGN fraction in the F277W band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Furthermore, since the pixel scale of NIRSpec is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='1′′, our velocity dispersion measurement may not accurately represent the central velocity dispersion of GS-9209, leading to an underestimated dynamical mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' It should also be noted that the stellar mass we measure is strongly dependent on our assumed IMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' A final, intriguing possibility would be a high level of rotational support in GS-9209, as has been observed for quiescent galaxies at 2 < z < 3 [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Unfor- tunately, the extremely compact nature of the source makes any attempt at resolved studies extremely challenging, even with the JWST NIRSpec integral field unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Resolved kinematics for this galaxy would be a clear use case for the High Angular Resolution Monolithic Optical and Near-infrared Integral field spectrograph (HARMONI) planned for the Extremely Large Telescope (ELT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' 3 Conclusion We report the spectroscopic confirmation of a massive quiescent galaxy, GS- 9209 at a new redshift record of z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='6582 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='002, with a stellar mass of log10(M∗/M⊙) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='61 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' This galaxy formed its stellar popula- tion over a ≃ 200 Myr period, approximately 600 − 800 Myr after the Big Bang (zform = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='2), before quenching at zquench = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' GS-9209 demonstrates unambiguously that massive galaxy formation was already well Springer Nature 2021 LATEX template 10 A massive quiescent galaxy at redshift 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='658 underway within the first billion years of cosmic history, with this object having reached log10(M∗/M⊙) > 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='3 by z = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' This galaxy also clearly demonstrates that the earliest onset of galaxy quenching was no later than ≃ 800 Myr after the Big Bang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We estimate the iron abundance and α-enhancement of GS-9209, finding [Fe/H] = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='97+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='07 and [α/Fe] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='67+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='15, suggesting the stellar mass vs iron abundance relation at z ≃ 7, when this object formed most of its stars, was ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='4 dex lower than at z ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='5 [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Whilst its spectrum is dominated by stellar emission, GS-9209 also hosts an AGN, for which we measure a black-hole mass of log10(M•/M⊙) = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='1 from the observed broad and narrow Hα emission [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We also predict a consistent value of log10(M•/M⊙) = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='1 based on the stellar velocity dispersion of GS-9209 [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Whilst large-scale star formation in GS-9209 has been quenched for almost half a billion years, the significant integrated quantity of AGN accretion implied by this large black- hole mass (≃ 4 − 5 times what would be expected given the stellar mass of this galaxy) suggests that AGN activity plausibly played a significant role in quenching star formation in this galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Based on the properties we measure, GS-9209 seems likely to be associated with the most extreme galaxy populations currently known at z > 5, such as the highest-redshift submillimetre galaxies and quasars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=', [36, 49, 50]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' GS- 9209 is also plausibly descended from an object similar to the z ≃ 8 massive galaxy candidates recently reported in the first data from the JWST CEERS programme [7], though the number density of these candidates is significantly higher than that of z > 4 quiescent galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' GS-9209 and similar objects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=', [9]) are also likely progenitors for the dense, ancient cores of the most massive galaxies in the local Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' This study, which makes use of just 5 hours of on-source integration time, demonstrates the huge potential of JWST for revolutionising our understand- ing of the high-redshift Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' It seems clear that this work will be followed rapidly by the confirmation and detailed spectroscopic exploration of large samples of z > 4 quiescent galaxies, to build up a detailed understanding of massive galaxy formation and quenching during the first billion years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' 4 Methods 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='1 Spectroscopic data reduction We reduce our NIRSpec data using the JWST Science Calibration Pipeline v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='4, using version 1017 of the JWST calibration reference data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' To improve the spectrophotometric calibration of our data, we also reduce observations of the A-type standard star 2MASS J18083474+6927286 [51], taken as part of JWST commissioning programme 1128 (PI: L¨utzgendorf) [52] using the same instrument modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We compare the resulting stellar spectrum against a spectral model for this star from the CALSPEC library [53] to construct a calibration function, which we then apply to our observations of GS-9209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Springer Nature 2021 LATEX template A massive quiescent galaxy at redshift 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='658 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='2 Photometric data reduction The majority of our photometric data are taken directly from the CANDELS GOODS South catalogue [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We supplement this with new JWST NIRCam photometric data taken as part of the Ultra Deep Field Medium-Band Survey [55] (Programme ID: 1963;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' PI: Williams).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Data are available in the F182M, F210M, F430M, F460M and F480M bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We reduce these data using the PRIMER Enhanced NIRCam Image-processing Library (PENCIL, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=', [8]), a custom version of the JWST Science Calibration Pipeline (v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='0), and using version 1011 of the JWST calibration reference data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We measure photometric fluxes for GS-9209 in large, 1′′-diameter apertures to ensure we measure the total flux in each band (the object is isolated, with no other sources within this radius, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We measure uncertainties as the standard deviation of flux values in the nearest 100 blank-sky apertures, masking out nearby objects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=', [56]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='3 Bagpipes full spectral fitting We fit the available photometry in parallel with our new spectroscopic data using the Bagpipes code [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Our model has a total of 22 free parameters, describing the stellar, dust, nebular and AGN components of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' A full list of these parameters, along with their associated priors, is given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We fit our model to the data using the MultiNest nested sampling algorithm [58–60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We use the 2016 updated version of the BC03 [61, 62] stellar population models, using the MILES stellar spectral library [63] and updated stellar evolu- tionary tracks [64, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We assume a double-power-law star-formation-history model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=', [24, 57]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We allow the logarithm of the stellar metallicity, Z∗ to vary freely from log10(Z∗/Z⊙) = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='45 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' These are the limits of the range spanned by the BC03 model grid relative to our adopted Solar metallicity value (Z⊙ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='0142 [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We mask out the narrow emission lines in our spectrum during our Bag- pipes fitting due to likely AGN contributions, whereas Bagpipes is only capable of modelling emission lines from star-forming regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We do however still include a nebular model in our Bagpipes fit to allow for the possibility of nebular continuum emission from star-forming regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We assume a stellar- birth-cloud lifetime of 10 Myr, and vary the logarithm of the ionization parameter, U, from log10(U) = −4 to −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We also allow the logarithm of the gas-phase metallicity, Zg, to vary freely from log10(Zg/Z⊙) = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='45 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Because our eventual fitted model only includes an extremely small amount of star formation within the last 10 Myr for GS-9209, this nebular component makes a negligible contribution to the fitted model spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We model attenuation of the above components by dust using the model of [66, 67], which is parameterised as a power-law deviation from the Calzetti dust attenuation law [68], and also includes a Drude profile to model the 2175˚A bump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We allow the V −band attenuation, AV to vary from 0 − 4 magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 12 A massive quiescent galaxy at redshift 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='658 Table 1 The 22 free parameters of the Bagpipes model we fit to our spectroscopic and photometric data (see Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='3), along with their associated prior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' The upper limit on τ, tobs, is the age of the Universe as a function of redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Logarithmic priors are all applied in base ten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' For parameters with Gaussian priors, the mean is µ and the standard deviation is σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Component Parameter Symbol / Unit Range Prior Hyper-parameters General Redshift z (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='6, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='7) Gaussian µ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='66 σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='01 Stellar velocity dispersion σ / km s−1 (50, 500) Logarithmic SFH Total stellar mass formed M∗ / M⊙ (1, 1013) Logarithmic Stellar metallicity Z∗ / Z⊙ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='00355, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='55) Logarithmic Double-power-law falling slope α (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='01, 1000) Logarithmic Double-power-law rising slope β (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='01, 1000) Logarithmic Double-power-law turnover time τ / Gyr (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='1, tobs) Uniform Dust V −band attenuation AV / mag (0, 4) Uniform Deviation from Calzetti slope δ (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='3) Gaussian µ = 0 σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='1 Strength of 2175˚A bump B (0, 5) Uniform Attenuation ratio for birth clouds ϵ (1, 5) Uniform AGN Power law slope (λ < 5000˚A) αλ<5000˚ A (−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='5, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='5) Gaussian µ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='5 σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='1 Power law slope (λ > 5000˚A) αλ>5000˚ A (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='5) Gaussian µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='5 σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='2 Hα broad-line flux fHα, broad / erg s−1 cm−2 (0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='5 × 10−17) Uniform Hα broad-line velocity dispersion σHα, broad / km s−1 (1000, 5000) Logarithmic Continuum flux at λ = 5100˚A f5100 / erg s−1 cm−2 ˚A−1 (0, 10−19) Uniform Nebular Ionization parameter U (10−4, 10−2) Logarithmic Gas-phase metallicity Zg / Z⊙ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='00355, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='55) Logarithmic Calibration Zero order P0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='75, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='25) Gaussian µ = 1 σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='1 First order P1 (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='25) Gaussian µ = 0 σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='1 Second order P2 (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='25) Gaussian µ = 0 σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='1 Noise White noise scaling a (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='1, 10) logarithmic Springer Nature 2021 LATEX template A massive quiescent galaxy at redshift 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='658 13 We further assume that attenuation is multiplied by an additional factor for all stars with ages below 10 Myr, and resulting nebular emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' This factor is commonly assumed to be 2, however we allow this to vary from 1 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We allow redshift to vary, using a narrow Gaussian prior with a mean of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='66 and standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We additionally convolve the spectral model with a Gaussian kernel in velocity space, to account for velocity dispersion in our target galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' The width of this kernel is allowed to vary with a logarithmic prior across a range from 50 − 500 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Separately from the above components, we also include a model for AGN continuum, broad Hα and Hβ emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Following [37], we model AGN contin- uum emission with a broken power law, with two spectral indices and a break at λrest = 5000˚A in the rest frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We vary the spectral index at λrest < 5000˚A using a Gaussian prior with a mean value of αλ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='5 (αν = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='5) and stan- dard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We also vary the spectral index at λrest > 5000˚A using a Gaussian prior with a mean value of αλ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='5 (αν = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='5) and standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We parameterise the normalisation of the AGN continuum component using f5100, the flux at rest-frame 5100˚A, which we allow to vary with a linear prior from 0 to 10−19 erg s−1 cm−2 ˚A−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We model broad Hα with a Gaussian component, varying the normalisation from 0 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='5 × 10−17 erg s−1 cm−2 using a linear prior, and the velocity dispersion from 1000 − 5000 km s−1 in the rest frame using a logarithmic prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We also include a broad Hβ component in the model, which has the same parameters as the broad Hα line, but with normalisation divided by the standard 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='86 ratio from Case B recombination theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' However, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' 2, this Hβ model peaks at around the noise level in our spectrum, and the line is therefore plausible in not being obviously detected in the observed spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We include intergalactic medium (IGM) absorption using the model of [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' To allow for imperfect spectrophotometric calibration of our spectroscopic data, we also include a second-order Chebyshev polynomial (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=', [70, 71]), which the above components of our combined model are all divided by before being compared with our spectroscopic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We finally fit an additional white noise term, which multiplies the spectroscopic uncertainties from the JWST pipeline by a factor, a, which we vary with a logarithmic prior from 1 − 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='4 Size measurement from F210M-band imaging We model the light distribution of GS-9209 in the JWST NIRCam F210M imaging data using PetroFit [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We fit these PetroFit models to our data using the MultiNest nested sampling algorithm [58–60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We use F210M in preference to the F182M band due to the smaller AGN contribution in F210M and the fact that it sits above the Balmer break, therefore being more representative of the stellar mass present rather than any ongoing star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 14 A massive quiescent galaxy at redshift 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='658 As our spectroscopic data contains strong evidence for an AGN, we fit both S´ersic and delta-function components simultaneously, convolved by an empir- ically estimated PSF, derived by stacking bright stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' In preliminary fitting, we find that the relative fluxes of these two components are entirely degen- erate with the S´ersic parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We therefore predict the AGN contribution to the flux in this band based on our full-spectral-fitting result, obtaining a value of 8 ± 1 per cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' We then impose this as a Gaussian prior on the rela- tive contributions from the S´ersic and delta function components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' The 11 free parameters of our model are the overall flux normalisation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' which we fit with a logarithmic prior,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' the effective radius,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' re,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' S´ersic index,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' ellipticity and posi- tion angle of the S´ersic component,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' the x and y centroids of both components,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' the position angle of the point spread function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' and the fraction of light in the delta-function component,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' which we fit with a Gaussian prior with a mean of 8 per cent and standard deviation of 1 per cent,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' based on our full spectral fitting result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Acknowledgements The authors would like to thank James Aird for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Carnall thanks the Leverhulme Trust for their support via a Leverhulme Early Career Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' McLure, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Dunlop, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' McLeod, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Wild, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Begley, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Donnan and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Hamadouche acknowledge the support of the Science and Technology Facilities Council.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Cullen acknowledges support from a UKRI Frontier Research Guarantee Grant (grant reference EP/X021025/1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Cimatti acknowledges support from the grant PRIN MIUR 2017 - 20173ML3WW 001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' Statement of Author Contributions ACC led the preparation of the observing proposal, reduction and analysis of the data, and preparation of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' RJM, JSD, VW, FC and AC provided advice and assistance with data reduction, analysis and interpreta- tion, as well as consulting on the preparation of the observing proposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' DJM, DM, RB and CTD reduced the JWST imaging data and prepared the empir- ical PSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' DJM, MLH and SMJ assisted with measurement of the size and morphology of GS-9209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' SW assisted with selection of GS-9209 from the CAN- DELS catalogues prior to the observing proposal being submitted.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='3847/1538-4365/abef6710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' 01426 [72] Geda, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=', 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' The Astronomical Journal 163(5), 202 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} +page_content='3847/1538-3881/ac5908' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtFIT4oBgHgl3EQf_CxR/content/2301.11413v1.pdf'} diff --git a/B9E2T4oBgHgl3EQf8wmh/content/tmp_files/2301.04222v1.pdf.txt b/B9E2T4oBgHgl3EQf8wmh/content/tmp_files/2301.04222v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..115aacd45c034de0d9f03a6d118669b3fc2e3ea3 --- /dev/null +++ b/B9E2T4oBgHgl3EQf8wmh/content/tmp_files/2301.04222v1.pdf.txt @@ -0,0 +1,2536 @@ +Geometric phases along quantum trajectories +Ludmila Viotti,1, 2 Ana Laura Gramajo,2 Paula I. Villar,3 Fernando C. Lombardo,3 and Rosario Fazio2, 4 +1Departamento de F´ısica Juan Jos´e Giambiagi, FCEyN UBA Ciudad Universitaria, Pabell´on I, 1428 Buenos Aires, Argentina +2The Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy +3Departamento de F´ısica Juan Jos´e Giambiagi, FCEyN UBA and IFIBA CONICET-UBA, +Facultad de Ciencias Exactas y Naturales, Ciudad Universitaria, Pabell´on I, 1428 Buenos Aires, Argentina +4Dipartimento di Fisica, Universit`a di Napoli ”Federico II”, Monte S. Angelo, I-80126 Napoli, Italy +(Dated: January 12, 2023) +A monitored quantum system undergoing a cyclic evolution of the parameters governing its Hamil- +tonian accumulates a geometric phase that depends on the quantum trajectory followed by the +system on its evolution. The phase value will be determined both by the unitary dynamics and +by the interaction of the system with the environment. +Consequently, the geometric phase will +acquire a stochastic character due to the occurrence of random quantum jumps. Here we study +the distribution function of geometric phases in monitored quantum systems and discuss when/if +different quantities, proposed to measure geometric phases in open quantum systems, are represen- +tative of the distribution. We also consider a monitored echo protocol and discuss in which cases +the distribution of the interference pattern extracted in the experiment is linked to the geometric +phase. Furthermore, we unveil, for the single trajectory exhibiting no quantum jumps, a topological +transition in the phase acquired after a cycle and show how this critical behavior can be observed +in an echo protocol. +For the same parameters, the density matrix does not show any singular- +ity. We illustrate all our main results by considering a paradigmatic case, a spin-1/2 immersed in +time-varying a magnetic field in presence of an external environment. The major outcomes of our +analysis are however quite general and do not depend, in their qualitative features, on the choice of +the model studied. +I. +INTRODUCTION +As Berry first stated in his seminal work [1], when a +quantum system is prepared in an energy eigenstate and +adiabatically driven in a cycle, it acquires, in addition to +the dynamical phase, a phase that depends solely on the +path traced in the ray space. Being independent of the +specific dynamics giving rise to the path, this phase is +of geometrical nature. Following Berry’s breakthrough, +consistent generalizations of the Geometric Phase (GP) +have been found for unitary evolutions which are kept +cyclic while they are not required to be adiabatic [2], in +the presence of degenerate subspaces [3], and for the case +in which both the adiabaticity and the cyclicity condi- +tions are removed [4, 5]. Further generalizations include +the definitions of GPs for mixed states [6–10] and the so- +called off-diagonal GPs [11, 12], which apply in the case +where the initial and final states are orthogonal. +GPs are profoundly linked to the theory of fiber bun- +dles and holonomies, bridging geometrical concepts like +parallel transport over curved spaces with physics [13– +15], and contributing in this way to the understanding of +quantum mechanics at the foundational level. Since their +discovery, GPs have also emerged in most diverse physical +systems [16, 17], deepening the comprehension of numer- +ous phenomena such as integer quantum Hall effect [18], +topological insulators and superconductors [19, 20], as +well as playing a pivotal role in quantum information +processing [21–23]. +The quest for implementations of geometric quantum +information processing has also spurred the search for +geometric interferometry in several different setups. The +first proposal of this kind was realized in NMR [22]. +Thereafter, Berry phases in superconducting qubits were +both studied theoretically in [24] and observed experi- +mentally for different regimes of couplings in circuit-QED +arrangements [25–30]. +In this direction, high-fidelity +quantum gates were demonstrated with trapped ions [31]. +The need to improve the performance of quantum infor- +mation processing devices against the exposure to ex- +ternal environment has led to the suggestion of non- +adiabatic geometric gates schemes [32–37]. In this con- +text, it becomes of fundamental importance to under- +stand how geometric interferometry is affected by the +presence of an external environment. Consequently, GPs +need to be generalized to deal with the systems subject +to non-unitary quantum evolution. The effect of fluctu- +ations in the classical control parameters of a quantum +cyclic evolution may average out mitigating their effect +on the accumulated Berry phase [38]. The presence of +an external bath was found to give rise to new geometric +contributions to decoherence [39, 40], as experimentally +detected in [41, 42]. Different definitions of GPs applica- +ble in the non-unitary case have been put forward. Tong +et al. [43] introduced a purification-independent formula +computed over the reduced density matrix while an aver- +age over different histories (trajectories) taking into ac- +count system-bath interaction was discussed in Carollo et +al. [44, 45] and further analyzed in [46–48]. Additional +work along these lines can be found in [49–52]. +There is, however, a different level of description of +open quantum systems which may capture features that +are washed out by simply looking at the properties of den- +sity matrices. This level is accessed, for example, when +the state of the system is continuously monitored. In this +arXiv:2301.04222v1 [quant-ph] 10 Jan 2023 + +2 +case, the quantum system is described by a wave func- +tion whose smooth evolution is interrupted by random +quantum jumps induced by the coupling with the envi- +ronment [53]. This sequence of smooth evolutions inter- +rupted by jumps is named a quantum trajectory (see [54] +for a recent review on the subject). +Goal of the present work is to describe the properties of +accumulated GP along quantum trajectories. In this ap- +proach we are inspired by the work of Gebarth et al. [55] +where the GPs induced by a sequence of weak measure- +ments stirring the system along a path in a parameter +space were analyzed. The randomness introduced by the +occurrence of jumps in a given trajectory is reflected in +the fact that the GPs inherit a stochastic nature. +By +random sampling over the trajectories, the entire distri- +bution can be reconstructed. Since the Berry phase is not +an observable, the average value does not correspond to +the phase accumulated by the average state (this is, the +density matrix). Previous works, with the notable ex- +ception of [55], either restrict the study of the dynamics +of smoothly evolving pure states with no jumps or de- +fine average quantities. Understanding the fluctuations +of GPs induced by random jumps is to a large extent un- +explored. We would like to fill this gap by studying this +distribution and whether it is related to the correspond- +ing distribution in the interference fringes in a spin-echo +experiment. Finally, we will argue that the topological +transition discussed in [55], despite the different dynam- +ical settings, is a generic feature present in adiabatically +driven monitored systems. We will show that depending +on the coupling to the external environment, the moni- +tored quantum system will show a topological transition +in the phase accumulated in a cycle and we will argue +that this transition is visible in echo dynamics. +The paper is organized as follows. In the next Section, +we will define the dynamical setting we are interested in: +A quantum system subject to a time-periodic Hamilto- +nian and coupled to an external bath. With the intention +to highlight the essence of our results, we will consider +the paradigmatic case of a two-level system that evolves +in presence of an externally varied magnetic field. The +associated density matrix is governed by the Lindblad +equation. In order to follow the dynamics of the system +along its quantum trajectories, we introduce a specific +unravelling of the Lindblad equation which relays on mi- +croscopic considerations, these aspects are introduced in +Section II. In Section III the model and its coupling to +the environment are introduced. In Section IV we de- +fine the GP that will be the founding block of all our +analysis. +For an isolated system and sufficiently slow +driving, this reduces to the Berry phase [1]. The pres- +ence of the environment induces both a smooth drift and +random jumps in the dynamics, so the evolution of the +state is generically neither adiabatic nor cyclic. To keep +the presentation self-consistent, we further include in this +same Section other definitions of GPs present in the lit- +erature. These will be employed for comparison in the +posterior Section V A, where we discuss the distribution +of the GPs accumulated along quantum trajectories and +analyze reference GP values in order to account for dif- +ferences with other definitions of GPs proposed in the +context of open quantum systems. +Due to the intrin- +sic randomness of the quantum trajectory, a monitored +echo experiment might be altered. +In Section V B we +discuss the probability distributions of the interference +fringes and detail whether/when they relate to the corre- +sponding distribution of the GPs. Our analysis of GPs in +monitored systems is completed in Section V C where we +will show that the topological transition discovered in [55] +for a specific setting is actually a generic feature in pe- +riodically driven open quantum systems. Indeed, for the +sequence of states known as no-jump trajectory, which +can be thought of as the smooth evolution generated by +a non-hermitian Hamiltonian, we find the GP displays a +complex pattern in the parameters space exhibiting sin- +gular points. +These singularities can be tracked down +to correspond to points of vanishing probability for such +a trajectory, and to reveal the border between distinct +topological sectors. The transition observed in the evo- +lution when varying the parameters is topological in the +sense that it is related to a discontinuous jump of an +integer-valued topological invariant. Section V C will be +entirely devoted to the study of this transition and ways +to detect it through an echo protocol. +A summary of +our results and concluding considerations are presented +in Section VI. The appendices give some additional in- +gredients used to compute the GP in the numerical sim- +ulations, Appendix A, a detailed analysis of the already +mentioned interference fringes distribution, Appendix B, +a brief discussion on how the distribution of GPs may de- +pend on the unravelling of the Lindblad equation (leading +to the same averaged evolution), Appendix C, and ana- +lytical treatment of the no-jump trajectory, Appendix D. +II. +FROM LINDBLAD DYNAMICS TO +QUANTUM TRAJECTORIES +Lindblad equation - +In order to make a connection +with existing literature, it is convenient to set the stage +and start from the case in which the state of an open +quantum system is described by a density matrix ρ(t). +In this case, under proper conditions, the dynamics is +governed by the Lindblad equation [56, 57] (ℏ = 1) +˙ρ = −i [H, ρ] + +� +α +[LαρL† +α − 1 +2{L† +αLα, ρ }] . +(1) +The first term in the r.h.s. of the Lindblad equation ac- +counts for the unitary evolution, while the second origi- +nates in the coupling to the environment. The strength +and the nature of this coupling are encoded in the Lind- +blad operators Lα. We will consider a Hamiltonian H +that depends periodically on time H(t + 2π/Ω) = H(t) +with T = 2π/Ω the period of a cycle in suitable param- + +3 +eter space. The Lindblad operators, if time-dependent, +should also be time-periodic Lα(t + 2π/Ω) = Lα(t). +It is useful to already at this point briefly comment on +the adiabatic limit for slow dynamics as this issue will +be central in the analysis conducted along the paper. If +the evolution is unitary, for a sufficiently large period T, +a system prepared in an eigenstate will remain in the +corresponding instantaneous eigenstate up to small cor- +rections due to Landau-Zener transitions between energy +levels. In other words, the occupancy of any given eigen- +state will not change in time. +The situation strongly +differs in presence of an environment. +In this case, a +proper adiabatic limit is not well defined, since the slow +driving limit where adiabatic dynamics sets in, is also the +regime in which the consequences of the external baths +are the most severe and the system reaches a (possibly +periodic) steady state. The adiabatic limit itself should +be reconsidered [58] in an open system, as the existence +of a continuum of energy levels makes the energy split- +tings of the system a bad reference scale for defining the +regimes. Effects due to non-adiabaticity and corrections +due to the presence of the environment seem thus to be +inextricably linked. +Monitored dynamics and quantum trajectories - +The +dynamics of the systems radically change when it is pos- +sible to continuously monitor their state. In this case, +the state of the system remains pure and consists of in- +tervals of smooth evolution interrupted at random times +by abrupt changes called quantum jumps. A sequence of +smoothly-evolving intervals together with a set of random +events is denominated a quantum trajectory. The litera- +ture on the subject is vast and we refer to the following +papers and books for a general overview [53, 54, 59, 60]. +Evolution is described in this framework as follows. If +at time t the state of the system is |ψ(t)⟩, at a later t+δt +time it will be +|ψ(t+δt)⟩ = +� +� +� +� +� +� +� +� +� +Ko|ψ(t)⟩ +√ +po(t) +with probability +po(t) +Kα|ψ(t)⟩ +√ +pα(t) +with probability +pα(t) +(2) +where o, α = 1, .. label the different operators Kα induc- +ing dynamical steps +Ko = 1 − i δt +� +H − i +2 +� +α +L† +αLα +� +Kα = +√ +δtLα (3) +and po/α(t) = ⟨ψ(t)| K† +o/αKo/α |ψ(t)⟩. Each choice in the +r.h.s. +of Eq.(2) represents evolution steps of different +characters. +The second line corresponds to the occur- +rence of a jump Kα at time t, while the first is a smooth +evolution (no jump), albeit altered from unitarity by the +fact that acquiring the information that no jumps oc- +curred modifies the evolution of the system. +The no- +jump operator Ko can also be thought of as generated +by an effective drift Hamiltonian Ho to which it relates +in the usual way Ko = 1 − i δt Ho. The full evolution +in a time interval [0, t] is therefore characterized by a se- +quence of NJ jumps of types αi occurring at times ti. We +will denote the string of these events +R(t, NJ) = {(α1, t1), . . . , (αi, ti), . . . (αNJ, tNJ)}, +(4) +with 0 ≥ ti ≥ t +∀i, the quantum trajectory. As men- +tioned above, this framework naturally emerges when the +system is continuously and indirectly monitored, so that +each trajectory can be viewed as the result of continu- +ous measurements of the environment on a given basis. +From this perspective, continuous monitoring may lead +to decoherence mitigation by the environment [61], also +post-selection and error correction schemes [62, 63] have +been proposed. +The properties of the Kraus operators Ko/α guarantee +that the probabilities to get a given outcome sum up to +one, and the time step δ t should be taken small enough +for the first order approximation to be valid, which re- +quires � +α pα ≪ 1. Averaging over every possible jump +sequence one gets back the Lindblad equation [53] in +Eq.(1), the converse implication is not valid, an infinite +number of different unravellings give rise to the same +Lindblad evolution [54]. We will address this question in +Appendix C. +III. +THE MODEL +Since we are interested in studying the impact of +an external environment on the GPs, we will con- +sider a unitary evolution over which the accumu- +lated GP, in the adiabatic limit, is the Berry phase. +To be concrete, we shall consider a spin-1/2 parti- +cle in presence of a time-dependent magnetic field +B(t) = ω ˆnB(t), whose direction is given by ˆnB = +(sin (θ) cos(Ω t), sin (θ) sin(Ω t), cos θ) with fixed polar an- +gle θ and time-varying azimuthal angle Ω t. Such unitary +evolution is generated by the Hamiltonian +H(t) = 1 +2 B(t) · σ, +(5) +with σ = (σx, σy, σz); and |0⟩ and |1⟩, the eigenstates of +σz. The instantaneous eigenstates of H(t) are denoted +|ψ−(t)⟩ and |ψ+(t)⟩ . +If the system could be kept perfectly isolated while +the direction of B(t) is adiabatically changed in a cycle +parameterized by t ∈ [0, T], with T = 2π/Ω (as shown in +Fig.1), it would acquire an adiabatic (Berry) phase φ±a = +−π(1 ∓ cos θ), where the ∓ sign depends on the energy +eigenstate in which the system was initially prepared. +Lindblad operators - For a system that evolves accord- +ing to H(t) given by Eq. (5) coupled to an environment +of harmonic oscillators a consistent time-dependent Lind- +blad equation of the form in Eq. (1) can be derived from + +4 +Figure 1. Trajectories described by the state of the system on +the Bloch sphere under different conditions. The black line +corresponds to unitary evolution in the adiabatic limit. The +purple line depicting a curly ring corresponds to general uni- +tary dynamics in which non-adiabatic corrections start to be +visible. In the presence of an environment, the quantum state +can suffer from jumps or can be smoothly driven along the +whole evolution. For a system prepared in the exited eigen- +state, the orange trajectory corresponds to a fully smooth +drift. Differently, the blue path shows a jump that projects +the state into the instantaneous ground eigenstate and is af- +terward smoothly driven. Finally, the light blue path shows a +case with several jumps, where the non-adiabatic corrections +appear in between the jumps. +microscopic considerations as long as the evolution re- +mains sufficiently slow [64, 65], with Lindblad operators +given by +L−(t) = √γ− ⟨ψ−(t)| σx |ψ+(t)⟩ |ψ−(t)⟩ ⟨ψ+(t)| +L+(t) = √γ+ ⟨ψ+(t)| σx |ψ−(t)⟩ |ψ+(t)⟩ ⟨ψ−(t)| +(6) +Ld(t) = √γd +� +i=± +⟨ψi(t)| σx |ψi(t)⟩ |ψi(t)⟩ ⟨ψi(t)| +and corresponding to decay, spontaneous excitation, and +dephasing respectively. The coupling strengths consid- +ered in this work are, in terms of the dissipation ratio Γ, +γ− = Γ ; γd = 0.32 Γ, while we consider γ+ to be negli- +gible (all the results that we will show are rather generic +and do not qualitatively depend on the chosen values). +The jumps defined by Eq. (3) from the Lindblad oper- +ators above lead, after averaging, to a consistent Lind- +blad equation for slow dynamical evolutions [66]. The +operators introduced in Eq. (6) induce transitions and +dephasing between the instantaneous eigenstates of the +Hamiltonian defined in Eq.(5). In order to keep the anal- +ysis as general as possible, we will include a further term +in the Lindbladian which requires considering a fourth +operator +Lz = √γzσz +(7) +along a fixed direction in the Bloch sphere. The particu- +lar choice of σz operator as the additional Lindblad oper- +ator is motivated by the need of introducing transitions +that do not simply involve the instantaneous eigenstates. +Any other Lindblad operator that differed from those in +Eq.(6) would lead to similar qualitative conclusions. +While the unitary evolution of the closed system will +follow the curly path indicated in purple in Fig.1, the ac- +tual dynamics will follow, with some probability, the path +indicated in blue (see Fig.1 for illustrative purposes), i.e. +it will be discontinuous and not necessary closed after a +cycle of the driving, even in the slow-driving limit. More- +over, the slower the driving, the more jumps will occur +(see light blue curve in Fig.1). The task of the next Sec- +tions is to characterize GPs under these conditions. +Smooth evolution with no jumps - A particularly inter- +esting quantum trajectory is that which is smooth along +the whole evolution. Before addressing the characteriza- +tion of GPs in indirectly monitored systems, we provide +insight into the evolution giving rise to it. +When the +records of the measurements performed on the environ- +ment reveal zero jumps, the dynamics describe a contin- +uous smooth path and is generated by an effective drift +Hamiltonian which depends both on the Hamiltonian of +the system and the Lindblad operators as described by +Eq. (3). Within the model considered in our work, the +effective drift Hamiltonian Ho governing the no-jump dy- +namics [Ko = 1 − δtHo in Eq.(3)] is given by +Ho(t) = +� +1 − i Γ +2ω f(t) +� +H(t) +(8) +with f(t) = cos2(θ) + sin2(θ) sin2(Ωt). We highlight the +fact that, due to the unitarity of σz matrix, the no-jump +evolution is completely independent of the fourth Lin- +bland operator Lz included ad-hoc and, consequently, +from the parameter γz. An illustrative example of the +trajectory generated by the above evolution, referred to +as the no-jump trajectory in what comes, is the orange +path in Fig. 1. In Appendix D we provide the analytic so- +lution for the dynamics associated to the non-Hermitian +Hamiltonian Ho(t) of Eq.(8). +While this trajectory is +unique, the number of possible (even though unevenly +probable) trajectories in which NJ > 0 jumps occur in- +creases with the number NJ of jumps, diverging as δt goes +to zero. Its uniqueness will make the no-jump trajectory +especially suitable for the analysis of some features of the +GPs, we come back to this question in Section V C. +IV. +GEOMETRIC PHASES IN OPEN SYSTEMS +- DEFINITIONS - +As mentioned in Sec. +I, the accumulation of a GP +during the dynamics of a quantum system is not neces- +sarily restricted to an adiabatic evolution. For a generic +quantum trajectory, consisting of a sequence of smoothly- +evolving intervals together with a set of random quantum +jumps R, a proper phase that deals with both aspects of +evolution can be defined. + +[1) +y +X +[0)5 +Considering the evolution in a time interval [0, T], pa- +rameterized with t, the GP associated to a trajectory in +which NJ jumps are registered at times ti, can be written +as +φ[R] = +arg ⟨ψ(0)|ψ(T)⟩ +− Im +NJ +� +i=0 +� ti+1 +ti +dt +� +ψ(t) +��� ˙ψ(t) +� +⟨ψ(t)|ψ(t)⟩ +− +� +(ti,αi)∈R +arg ⟨ψ(ti)| Kαi |ψ(ti)⟩ , +(9) +where R = R(T, NJ) for brevity, with t0 = 0 and the +convention that tNJ+1 ≡ T in the sum of integrals. The +definition of GP as given in Eq.(9) will be at the basis +of our analysis and refer to Appendix A for a deriva- +tion of this expression. As it is evident from the depen- +dence on the times and nature of the jumps, the phase +φ[R(T, NJ)] will be a stochastic variable, dependent on +the trajectory R(T, NJ). The first term in Eq. (9) is the +total relative phase between the initial and final states. +The remaining terms are of two different kinds, reflecting +the properties of the dynamics itself. The second term +features the dynamical phases accumulated along the in- +tervals of smooth evolution that take place before, be- +tween, and after jumps, and which should be subtracted +in order to access the purely geometrical object φR. The +occurrence at time ti of a jump generated by the opera- +tor Kαi introduces a contribution arg ⟨ψ(ti)| Kαi |ψ(ti)⟩ +which represents the phase difference between the state +before and after the jump. Such a term equals the GP +associated with the trajectory build-up by joining the +states by the shortest geodesic in the Hilbert space. The +expression in Eq. (9) is independent of the U(1) gauge +choice. It neither requires the trajectory to trace a close +path in the state space nor relies on adiabaticity condi- +tion. Moreover, it does not even demand unitarity as it +is well defined also if the states |ψ(ti)⟩ or |ψ(t′)⟩ are not +normalized (the norm should however be non-vanishing). +Suitable to be applied to the trajectories that emerge in +master equation unraveling, Eq. (9) has been employed +in limiting forms for addressing the definition of GPs fit- +ting non-unitary evolution. A first explored route was to +focus on the no-jump trajectory [44, 45]. This approach, +which disregards the possibility of quantum jumps by +restricting to the smooth evolution, preserves the well- +known definitions of GPs applicable to pure states and in- +cludes environmental effects through the non-hermiticity +of Ho. If no jumps are registered along the entire evolu- +tion, this is, if R(T, 0) = ∅, the GP φ0 ≡ φ[R(T, 0) = ∅] +reads +φ0 = arg ⟨ψ(0)|ψ(T)⟩ − Im +� T +0 +� +ψ(t) +��� ˙ψ(t) +� +⟨ψ(t)|ψ(t)⟩ dt +(10) +which trivially reduces to the expression for the GP +accumulated in the most general unitary evolution [5] +when this is indeed the case, and therefore the states are +instantaneously normalized, rendering the denominator +⟨ψ(t)|ψ(t)⟩ ≡ 1 ∀ t. Eq.(10) also reduces to Aharonov- +Anandan and Berry phases as the conditions required by +each definition are fulfilled, namely, for cyclic and uni- +tary while not necessarily adiabatic evolution and for +both cyclic and adiabatic evolution. +Note that phase +φ0 is ill-defined if some internal product on its argument +vanishes, this observation will become of relevance when +discussing the topological transition in Section V C. +Several other works consider the full Lindblad equation +unraveling, suggesting to define the GP of the ensemble- +averaged state ρ(t) as an average over the ensemble of +phases {φR} = φ{R} obtained by applying Eq.(10) to +each trajectory [44–46]. It has been extensively discussed +whether this is a proper definition of a GP for the density +matrix representing the state of the system as it does not +allow for a one-to-one relation between the set of density +matrices and the obtained GP values [47, 48, 67]. +Finally, a different approach introduces a generalized +GP defined directly from the reduced density matrix [43]. +The expression reads +φρ = arg +� +�� +j +� +λm(0)λm(t) ⟨ξm(0)|ξm(t)⟩ +× exp +� +− +� t +0 +dt′ � +ξm(t′) +��� ˙ξm(t′) +��� +(11) +where λk(t) and |ξk⟩ are the instantaneous eigenvalues +and eigenstates of the density matrix ρ(t) which de- +scribes the state of the system. Even though defined for +non-degenerate but otherwise general mixed states, when +computed over pure states under unitary evolution, re- +duces to the unitary expression of the GP. +All the above-mentioned proposals of GPs applicable +when dynamics are non-unitary either restrict to modi- +fied evolutions on which pure-state GP definitions would +be applicable or seek a consistently defined GP for the +reduced density matrix ρ(t), which accounts for an aver- +aged description. Stochastic processes, however, arising +from master equation unraveling, acquire independent +physical relevance in continuous monitoring schemes. As +anticipated in the introduction, the randomness intro- +duced by the occurrence of jumps in a given trajec- +tory reflects in the GPs acquiring a stochastic nature +itself. This approach, therefore, requires a study of the +environmentally-induced effects in GPs from a statistical +perspective. The probability associated with some GP +value will be related to that of individual trajectories as +P[φ] = +� +R/φ[R]=φ +P[R]. +(12) +The average phase corresponds only to the first moment +of the distribution +¯φ = +� +φP[φ] . + +6 +and in some cases may be not sufficient in characterizing +the dynamics. +For easy later reference, we provide a table summariz- +ing the GP definitions reviewed along this section +GP +Description +φa +Adiabatic Berry phase +φ[R] +GP associated to the quantum tra- +jectory R(T, NJ) +Eq.(9) +φ0 +GP +associated +to +the +no-jump +trajectory +Eq.(10) +φu +GP accumulated on general unitary +evolution ( from Eq. +(10) with +⟨ψ(t)|ψ(t)⟩ = 1) +¯φ +Average over the probability distri- +bution P[φ] +φρ +Mixed state geometric phase [43] +Eq.(11) +The next Section will be devoted to the properties of +P[φ] and how representative the different GPs applicable +to trajectories are, see Eqs. (9 - 10). As we will be show- +ing in the following, in most cases the entire probability +distribution, i.e. all higher order cumulant, is necessary +to understand the accumulation of GPs in a continuously +monitored system. We will also discuss under which cir- +cumstances and what features of P[φ] can be extracted +by geometric interferometry through a spin-echo proto- +col. +V. +RESULTS +A. +Geometric phase distribution P[φ] +We investigate in this section the distribution of the en- +semble {φR} = φ{R} of GPs obtained by employing Eq. +(9) to each individual realization (trajectory) of the evo- +lution, characterized by some set R(T, NJ). In Fig. 2 we +show two representative cases in which the correspond- +ing dynamics of a hypothetical unitary evolution would +either be faster (with small but non-zero non-adiabatic +corrections) or slow enough to be considered in the adi- +abatic regime while the environment remains the same, +characterized by the dissipation rate Γ = 10−3ω, which +leads to γ− = Γ, γd = 0.32 Γ, and negligible γ+. +We first attend the case with γz = 0, in which the en- +vironment induces jumps involving instantaneous eigen- +states only. The two situations, corresponding to the two +sets of parameters indicated before, are shown in Fig. 2, +in panels (a) and (b) respectively. In both panels, we +also plot for reference the adiabatic (Berry) result, the +no-jump and unitary GPs, and the average of the dis- +tribution. Being the Berry phase independent of Ω, it +0.0 +0.5 +1.0 +1.5 +2.0 +φ/π +0 +1 +2 +3 +4 +P[φ] +1e +1 +(a) +1.47 +1.475 +1.48 +1.485 +0 +1 +2 +3 +1e +2 +0.0 +0.5 +1.0 +1.5 +2.0 +φ/π +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +P[φ] +1e +2 +(b) +Figure 2. Probability distribution P[φ] of GPs for a magnetic +field oriented with θ = 0.34π and driven in a loop at frequen- +cies (a) Ω = 5 × 10−3ω and (b) Ω = 5 × 10−4ω. The environ- +ment is characterized by the dissipation rate Γ = 10−3ω and a +γz = 0. In both panels, the solid red line depicts the adiabatic +(Berry) phase φ+ +a , and the black dashed and dot-dashed lines +signalize the GPs φ0 and φu associated with no-jump and +general unitary evolution. The black dotted line indicates the +first moment of the distribution ¯φ. The inset in panel (a) is +a zoom in which the difference between these reference GP +values is visible. +is exactly the same for both cases, this is, φa ∼ 1.482π. +For the parameters chosen, the value φ0 computed from +Eq.(10) over the trajectory with no jumps, shows small +deviations from φa. While the values of these character- +istic GPs are similar, the entire distribution of the moni- +tored system is drastically different on each panel. In the +first case of faster driving, the period T is such that a +considerable amount of times the evolution is completed +registering no jumps, with the mean number of jumps +over the ensemble ¯NJ = 0.63. The narrow peak in the +Figure shows these cases of entire smooth evolution. In +addition, there is a small background revealing the ac- +cumulated GP along those trajectories where jumps oc- +curred. The composition of the ensemble is reflected in + +7 +the histogram by the presence of a large contribution, +corresponding to ∼ 50% of the realizations, due to the +no-jump GP-value and the remaining 50% of the counts +distributed in a broad way over the possible GP values. +This broad background distribution can be easily inter- +preted as the randomness inherited by the GP due to +the (random) time at which the jump occurred. A single +term ⟨ψ(ti)| K−i |ψ(ti)⟩ in Eq. +(9), denoting a contri- +bution to the GP from a jump at time ti, successfully +accounts for the background when considering all pos- +sible jump-times. +The peak in the distribution agrees +well with both the adiabatic and the no-jump values. +The average phase, on the other side, is a bit off due +to the small and poorly structured background, broadly +distributed over 2π. This clearly demonstrates that even +a single jump occurring at a random time leads to very +large fluctuations in the accumulated GP. In the case +with slower driving shown in panel (b) the mean num- +ber of jumps over the set of trajectories is ¯NJ = 1.77. +This means that the state of the system is much more +likely to undergo an abrupt change, or even more than +one, in each realization of the cycle. As expected, the +distribution of GPs becomes much wider, and a sharp +peak around φ0 is not visible anymore. Higher-order cu- +mulants become necessary to understand the dynamics. +The three lines, corresponding to the adiabatic, no-jump, +and average GPs do not provide clear information on the +dynamics of the monitored system. +Figure 3. Probability distribution P[φ] of GPs as a function of +the ratio Ω/ω. The field is oriented with θ = 0.34π and the en- +vironment is characterized by the dissipation rate Γ = 10−3ω +and a γz = 0 amplitude for the fourth Linbland operator. The +GP values are displayed on the y-axis, while their probability +is indicated by the intensity of the count color. The solid red +line depicts the adiabatic (Berry) phase φ+ +a , the black dashed +line indicates the GP φ0 accumulated along smooth trajecto- +ries with no jumps, and the black dotted line shows the first +moment of the distribution ¯φ. +The rate Ω/ω at which the magnetic field is rotated +has thus a direct impact on the distribution of GPs. For +larger rates, the system is exposed to the environment +for a shorter period of time, but deviations from the adi- +abatic regime become non-negligible. On the other hand, +lowering the driving frequency might result in the system +being exposed to environmental effects for too long, im- +plying strong corrections to φR from φ+ +a . Fig. 3 shows +the distribution of GP-values obtained along a range of +different Ω/ω rates which include the cases presented in +Fig.2. For high enough frequency, the distribution shows +a sharp peak around the no-jump value of the GP and al- +most no background counts. On the other hand, this no- +jump value deviates considerably from the Berry phase. +The broad background visible in panel (a) of Fig. 2 de- +velops as the frequency rate is lowered, this is, as the +relative period grows. Further on, the background turns +into a second peak, while the one in the no-jump value +decreases. For the smaller rate values, the distribution +shows the behavior depicted by panel (b) of Fig. 2, this +is, a broad single-peaked distribution. This regime shows +non-negligible environmental effects also over the GP as- +sociated with the no-jump evolution, which deviates from +the adiabatic result even though the driving is performed +slowly. We refer to Appendix D for an analytical expres- +sion for the dependence of this deviation on the different +parameters involved. The broadening exhibited by the +distribution as the frequency rate decreases, is reflected +in the increment of the distribution variance. +This is +shown in Fig. 4. +10-3 +10-2 +10-1 +Ω/ω +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +σ2� +R +� +(a) +Figure 4. Variance σ2 +{φR} of the GPs’ distribution as a func- +tion of the ratioΩ/ω. The field is oriented with θ = 0.34π +and the environment is characterized by the dissipation rate +Γ = 10−3ω and a γz = 0 (same as in Fig.3). +We conclude this section by analyzing the distribution +of GP values when γz ̸= 0. As already discussed, a non- +zero value of γz induces jumps to states that are not +instantaneous eigenstates of the Hamiltonian and thus +allows to consider of a wider class of cases. The result- +ing phenomenology depends only quantitatively on the +choice of the Lindblad operator Lz. Specifically, we take +γz = 0.1 Γ and consider, as we did before, two differ- + +2.0 +10-2 +1.5 +10-3 +1.0 +10-4 +0.5 +0.0 +10-5 +10-3 +10-2 +10-1 +P[] +m/u8 +ent values of the speed at which the system is cyclically +driven. The results are shown in Fig.5, with the qualita- +tive features of the distribution closely resembling those +obtained in the case with γz = 0. +0.0 +0.5 +1.0 +1.5 +2.0 +φ/π +0 +1 +2 +3 +4 +P[φ] +1e +1 +(a) +1.47 1.475 1.48 1.485 +0 +1 +2 +3 +1e +2 +0.0 +0.5 +1.0 +1.5 +2.0 +φ/π +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +P[φ] +1e +2 +(a) +Figure 5. Probability distribution P[φ] of the GPs for a mag- +netic field oriented with θ = 0.34π and driven in a loop at +frequencies (a) Ω = 5×10−3ω and (b) Ω = 5×10−4ω. The en- +vironment is characterized by the dissipation rate Γ = 10−3ω +and γz = 0.1 Γ. In both panels, a blue solid contour indicates +(for comparison) the γz = 0 distributions. The solid red line +depicts the adiabatic (Berry) phase φ+ +a , the black dashed and +dot-dashed lines signalize the GPs φ0 and φu associated with +no-jump and general unitary evolution. +The black dotted +line shows the first moment of the distribution ¯φ. The inset +in panel (a) zooms in to see the differences in the positions of +the lines and the peak of the distribution +Panel (a) of Fig.5 corresponds to the faster case. The +mean number of jumps ¯NJ = 0.69 is slightly above the +one obtained in the γz = 0. The additional jumps gener- +ated by Kz are not sufficient to modify the distribution +qualitatively, which continues to show a well-defined peak +(arising from the occurrence of smooth evolution with no +jumps) plus a broad small background. +In panel (b), +showing the case in which the system is driven slower, +the mean number of jumps is also slightly increased from +the γz = 0 case due to the additional presence of γz +jumps, reaching a value ¯NJ = 2.66. +The cases discussed above contain the first message of +the present work. The stochastic nature of the GP in +monitored dynamics needs to be taken into account and +it is not possible to characterize it only through a single +value. This rises the additional question of how this fact +reflects on the experimental outcomes. To address this +question, we will consider in the next Section a spin-echo +protocol and see how, when, and whether the distribution +in the interference fringes is affected by the randomness +of the process. +B. +Distribution of interference fringes in a +spin-echo protocol +If the system is prepared in an eigenstate of the Hamil- +tonian and subsequently driven in a cycle, adiabati- +cally and in absolute isolation from the environment, +then the quantum state accumulates a Berry phase that +can be measured by implementing a spin-echo proto- +col [68]. +It goes as follows. +The system is initially +prepared in a superposition state |ψ(0)⟩ which reads +(1/ +√ +2)(|ψ+(0)⟩+|ψ−(0)⟩) in terms of the ground and ex- +ited instantaneous eigenstates of H(0). Then, it is driven +for a period T, causing each eigenstate to acquire both +a dynamical and a geometric phase φ±a . A spin-flip op- +eration and a second cycle in the opposite direction lead +to a cancellation of the dynamical phases, resulting in a +purely geometric relative phase. Berry phase can thus +be extracted through state tomography [25, 27, 69] or by +realizing that the probability for the system to be back in +the initial state once the full evolution is completed, the +persistence probability, is related to the Berry phase as +|⟨ψ(0)|ψ(2 T)⟩|2 = cos2(2 φ+ +a ) [70]. The relation between +the persistent probability and the GP given above relies +on two factors: the adiabatic regime preventing the tran- +sitions between eigenstates and the exact cancellation of +the dynamical phases during the protocol. If an echo ex- +periment is performed on a system that is exposed to the +effect of the environment and continuously monitored, +the persistence probability will retain its dependence on +the dynamical evolution. Nevertheless, it is worth under- +standing to which extent it is possible to learn features +of GPs in a monitored system through an echo protocol. +For each realization of the protocol, characterized by +a sequence of jumps R(2 T, NJ), we can parametrize the +persistent probability PR through an associated angle +ϕR +PR = |⟨ψ(0)|ψ(2T)⟩|2 ≡ cos2 (2 ϕR) . +(13) +Both the persistence probability and the parameter ϕR +inherit the stochastic character of the trajectories, with +the probability of measuring a given value ϕ related to +the probability of the trajectories as + +9 +P[ϕ] = +� +R/ϕR=ϕ +P[R]. +(14) +In the limiting case in which the persistence probability +approaches its adiabatic value, ϕ will approach φ+ +a . Away +from that particular regime, ϕR is NOT equal to the GP +φR = φ[R] but, as mentioned previously, a convenient +parametrization of the spin-echo interference fringes. +The non-adiabatic and environment-induced devia- +tions from φ+ +a can be analyzed by examining the ensemble +{ϕR} = ϕ{R} that is obtained by computing Eq. (13) +for each individual realization of the protocol. This study +will also allow seeking possible relations, if any, between +the stochastic behavior of the GPs and that of experi- +mental outcomes (note that ϕR is defined modulo π/2 +and up to a sign, therefore, any relation between the dis- +tribution of GPs and the distribution of the experimental +results should take this into account). The frequency Ω +at which the magnetic field is rotated is expected, once +again, to have a direct impact on the distribution [36, 37]. +On increasing the relative value of Ω, the system will be +exposed to the disruptive influence of the environment for +shorter times, allowing to a larger extent a partial can- +cellation of the dynamical phases. At the same time, in +this regime, non-negligible deviations from the adiabatic +results will be unavoidable. On the other hand, smaller +values of Ω might result in the system being exposed to +environmental effects for too long, leading to strong de- +viations of the echo-parameter values ϕ from φ+ +a . +In analogy with what we did in section V A, we ex- +amine first the case γz = 0 and present, in Fig.6, two +representative cases in which the hypothetical unitary +evolution would either be faster or slow enough to be +considered within the adiabatic regime. These are shown +in panels (a) and (b) of Fig.6 respectively. In both pan- +els, we also display the adiabatic Berry phase φa (which +does not depend on Ω), the GP φ0 obtained in a proto- +col with no jumps, and the GP φu obtained in general +unitary evolution. The ϕ value obtained from an echo +experiment which is completed without detecting jumps +is also shown. For the parameters chosen, both panels +show very small deviations of ϕ extracted in a protocol +with no jumps from the Berry phase (see the insets in +Fig.6). It should be noted, however, that the probability +of registering this specific trajectory is different in the +two cases, as it can be seen in the differences in the full +P[ϕ] distributions. +The first striking feature that comes out is the pres- +ence of three distinct sharp peaks. The broad distribu- +tion observed in the GP values completely disappears +in the spin-echo. This behavior originates from the fact +that when γz = 0, only jumps between instantaneous +eigenstates are possible. +This particular aspect of the +unravelling leads, when combined with the properties of +the persistence probability, to a distribution of interfer- +ence fringes qualitatively different from that of the GPs. +Each of the peaks shown in panel (a) of Fig. 6 can be +1.275 +1.325 +1.375 +1.425 +1.475 +ϕ/π +0 +1 +2 +3 +4 +P[ϕ] +1e +1 +(a) +1.48173 1.48177 +0 +2 +4 +1e +3 +1.275 +1.325 +1.375 +1.425 +1.475 +ϕ/π +0 +2 +4 +6 +8 +P[ϕ] +1e +1 +(b) +1.4813 1.4817 +0 +2 +4 +6 1e +4 +Figure 6. Probability distribution P[ϕ] obtained in the echo- +protocol for a magnetic field oriented with θ = 0.34π and +driven in a loop at frequencies (a) Ω = 5×10−3ω and (b) Ω = +5×10−4ω. The environment remains the same, characterized +by the dissipation rate Γ = 10−3ω and a γz = 0. In both +panels, the solid red line depicts the adiabatic (Berry) phase +φ+ +a , and the black dashed and dash-dotted lines signalize the +GPs obtained in no-jump and unitary evolution respectively. +Furthermore, the black dash double-dotted line indicates the +ϕ value obtained in an echo protocol with no jumps. +The +insets in both panels show a range in which the result of a +smoothly performed echo experiment is distinguishable from +the Berry phase. +understood as arising from a different set of quantum tra- +jectories in the following way. For the parameters chosen +in panel (a) of Fig. 6 trajectories with at most one jump +are possible. +The three peaks correspond to protocols +with no jumps, protocols with one jump of the type L±, +and one jump of the type Ld respectively. We refer to Ap- +pendix B for a detailed justification of this identification. +Trajectories that remain smooth along the whole proto- +col induce the right peak in Fig.6 (closest to the no-jump +result, ϕ ∼ 1.475π for this choice of parameters). The +central peak, centred at the value ϕ ∼ 1.375π trivially + +10 +associated via Eq. +(13) with a persistence probability +taking the value 1/2, builds up from all those cases in +which the state of the system is, at some given time, pro- +jected into an eigenstate of H(t). In those trajectories, +all the information about the accumulated phase before +the jump is lost. As a consequence, immediately after a +jump L±, and regardless of both the previous evolution +and the time at which the jump occurred, the persistence +probability takes the exact value 1/2. The third peak, the +left one, is due to trajectories in which a jump Ld occurs. +This type of jump has the effect of introducing a π-shift +in the relative phase of the echo state, that corresponds +with the position of the left peak in Fig.6. Therefore, +the interference fringes distribution shows three peaks +out of which two encode the same information, namely, +the ϕ value of a smoothly driven protocol, while the cen- +tral peak contains almost no information. Furthermore, +the distribution is quite sharp because, for the parame- +ters chosen, the described classes of trajectories are all +detected, while more complex quantum trajectories are +highly improbable (see Appendix B). In panel (b) of Fig. +6 the two peaks located at the sides have almost van- +ished. +This reveals that when the system is driven at +lower relative frequencies, a decay jump or a spontaneous +excitation will be detected in almost every trajectory. A +similar effect is obtained if the decay rate Γ/ω increases +while keeping the ratio Ω/ω fixed. +A second aspect of the distribution P[ϕ] is which fea- +tures of GPs in open systems it captures. In panel (a) +Fig. 6, the fast-driven regime, the ϕ value obtained from +protocols with no jumps agrees well with the adiabatic +(Berry) phase, and both of these show small but visible +deviations from no-jump GP. The ϕ value is more closely +related to the adiabatic case than the actual GP accu- +mulated in smoothly drifted dynamics. For the slower +driving shown in panel (b) of Fig.6, the no-jump ϕ value +remains a good indicator of the adiabatic phase, even +though registering a smooth protocol is in this case less +probable. +Under these conditions, most of the experi- +ment realizations will contribute the central peak, which +is not related to any characteristic GP. +Inspection of Fig.6 suggests that, as in the case of the +GPs distribution, the interplay between non-adiabatic +corrections and environmentally induced jumps is bet- +ter revealed when the distribution P[ϕ] is analyzed as +a function of the rate Ω/ω. +This is shown in Fig. +7, +which includes the two paradigmatic cases of Fig. +6. +The Berry phase φa and the values φ0 and ¯φ of the GP +associated with smooth trajectories and the first moment +of the GP distribution are also given for reference. In the +non-adiabatic regime, Ω/ω ≳ 0.1 the ϕ value is most of +the time the one arising in a protocol with no jumps, and +shows appreciable but still small deviations from the adi- +abatic phase. A trajectory with a single jump might be +observed, albeit with less probability. If this is the case, +the mixing of the eigenvalues due to non-adiabatic tran- +sitions will produce slightly broad distributions around +the other two peaks, revealing the stochastic nature of +the jump times. Non-adiabatic corrections have a much +stronger impact on φ0 (for an analytical expression of +this scaling, see Appendix D), its behaviour completely +disconnects from that of the the distribution of echo pro- +tocols. +On the other side, approaching the adiabatic +regime, the three peaks get sharper. +This behavior is +accompanied by a sharp decrease in the height of the +side-peaks and an enhancement of counts on the triv- +ial, middle peak. Along the full range, there is a region +in which the interplay between environmentally-induced +and non-adiabatic effects allows for good agreement be- +tween the GP accumulated in smooth non-unitary evolu- +tion and the value of ϕ. The behavior displayed by both +the GP and the echo ”phase” in smooth non-unitary evo- +lution is further analyzed in Appendix D. Differently to +the case of the no-jump values, which display a reason- +able agreement, the inset in Fig. 7 shows that the (con- +sistently re-ranged) first moment of the GP distribution +¯φ remains, along the whole frequency range, completely +uncorrelated from both the ϕ distribution and all echo +characteristic values. +Figure 7. Probability distribution P[ϕ] of ϕ (as determined in +an echo experiment) as a function of the ratio Ω/ω. The field +is oriented with θ = 0.34π and the environment is character- +ized by the dissipation rate Γ = 10−3ω and a γz = 0 . The ϕ +values are displayed on the y-axis, while the intensity of the +count color indicates their probability. The solid red line de- +picts the adiabatic (Berry) phase φ+ +a , while the black dashed +line signalizes the no-jump GP φ0. The inset shows the prob- +ability distribution P[ϕ] accompanied by the first moment of +the GP distribution ¯φ. +The distribution changes radically when γz ̸= 0. In +what follows we discuss the case γz = 0.1 Γ with Γ = +10−3ω. We start re-considering the two representative +cases of fast and slower driving, displayed in panels (a) +and (b) of Fig. 8 respectively. The first noticeable aspect +is that, while three peaks observed in Fig. 6 (indicated +here by the blue contours) can still be detected, they are +now coexisting with a broad distribution. +As visible in panel (a) of Fig. 8, the three peaks heights + +1.475 +10-2 +1.425 +10-3 +1.375 +9 +:: +1.325 +10-4 +1.275 +10-5 +10-3 +10-2 +10-1 +P[S +m/u11 +1.275 +1.325 +1.375 +1.425 +1.475 +ϕ/π +0 +1 +2 +3 +4 +P[ϕ] +1e +1 +(a) +1.475 +1.480 +0 +2 +4 +1e +3 +1.275 +1.325 +1.375 +1.425 +1.475 +ϕ/π +0 +1 +2 +3 +4 +5 +P[ϕ] +1e +2 +(b) +1.3625 +1.3875 +0.0 +2.5 +5.0 +7.5 +1e +1 +Figure 8. Probability distribution P[ϕ] for a magnetic field +oriented with θ = 0.34π and driven in a loop at frequencies +(a) Ω = 5 × 10−3ω and (b) Ω = 5 × 10−4ω. The environment +is characterized by the dissipation rate Γ = 10−3ω, and finite +γz = 0.1 Γ. In both panels, a blue solid contour indicates the +γz = 0 distributions. The solid red line depicts the adiabatic +(Berry) phase φ+ +a , and the black dashed and dash-dotted lines +signalize the GPs obtained in no-jumps and unitary evolution +respectively. Finally, the black dash double-dotted line indi- +cates the ϕ value obtained in an echo protocol with no jumps. +The insets zoom in a range in which differences between the +reference values, panel (a), and the full magnitude of the cen- +tral peak, panel (b), are visible. +discussed previously decrease in the presence of γz. The +suppression of the peaks is accompanied by the appear- +ance of a broad background distribution covering the en- +tire range. Panel (b) of Fig. 8 attends the slow driving +situation, in which the probability to have a jump, and +even several, along each trajectory, grows. +The inclu- +sion of the Lz jump modifies the sharp-peaked distribu- +tion into a broad one, which covers the entire range of +ϕ values. In particular, while the two peaks connected +to the no-jump trajectory disappeared. +This happens +because the inclusion of this term in the Lindbladian in- +duces jumps into states other than the eigenstates of the +Hamiltonian. In this sense, we may consider the results +quite generic, not specifically dependent on the choice of +the Lindblad operator. In order to get a more complete +view of the effect of a finite γz, Fig. 9 shows the distribu- +tion of ϕ-values as a function of Ω/ω. For a non-adiabatic +evolution in which almost no jumps are detected, the be- +havior exhibited by the distribution is similar to that +observed in the γz = 0 case. When the velocity of the +driving is reduced, gradually favouring the occurrence +of jumps, the effect of introducing a finite γz value be- +comes more relevant. +The Lz jumps lead to ϕ values +that do also depend on the time at which different jumps +occurred and hence to the broad background. +Figure 9. Probability distribution P[ϕ] as a function of the +rate Ω/ω between the frequency Ω at which the magnetic +field its rotated and its amplitude ω. +The field is oriented +with θ = 0.34π and the environment is characterized by the +dissipation rate Γ = 10−3ω and γz = 0.1Γ. The ϕ values are +displayed on the y-axis, while the intensity of the count color +indicates their probability. Extra lines signalize reference GP +factors. +The solid red line indicates the adiabatic (Berry) +phase φ+ +a , while the black dashed line is the value φ0 extracted +from evolution with no jumps. +Summarizing, while the distribution of interference +fringes is, in general, quite different from that of the +phase accumulated along a single trajectory, the analysis +of a spin-echo protocol allows to extract reliable infor- +mation on both the no-jump trajectories and the adia- +batic (Berry) phase in some regimes of parameters. In +the following Section we will concentrate on the no-jump +trajectory (corresponding to the side-peaks of the per- +sistent probability in the echo-protocol) and show that +undergoes a topological transition as a function of the +coupling to the environment. + +1.475 +10-2 +1.425 +10-3 +9 1.375 +1.325 +10-4 +1.275 +10-5 +10-3 +10-2 +10-1 +P[] +2/w12 +C. +Topological transitions +As already anticipated, we conclude this analysis of +GPs in monitored systems by focusing on the no-jump +trajectory. +We will show, following in spirit the work +in Ref. [55], that the drift jump-free dynamics encode a +topological transition. We would like to emphasize that, +although the setting is very much different from that of +[55], we believe that the nature of the transition is the +same. +Our analysis is a strong hint to the conjecture +that this type of transition is rather generic for monitored +systems. +Phase diagram - +The GP φ0 given by Eq. (10) de- +pends, for every fixed θ, on the ratios Ω/ω and Γ/ω. +We recall the no-jump trajectory, and therefore the GP +associated with it, have no dependence on γz. Plotted +as a function of the above-mentioned parameters, the +GP shows discrete singularities at critical points, around +which it makes a 2π winding. Meanwhile, the probabil- +ity associated with this particular trajectory vanishes at +these points. We refer to Appendix D for details of the +analytical derivation. Fig.10 shows a color plot of the +GP in the Γ − Ω diagram at fixed values of the angle θ. +The range of the parameters is shown to highlight the +singular point and the 2π winding of the GP around it. +The white lines indicate the probability for the no-jump +trajectory, which approaches zero on reaching the singu- +larity. We will show that the collection of these singular +points delimits regions of the parameter space associated +with different topological classes of evolution. This will +be done by defining a topological invariant n ∈ Z (see be- +low) and explicitly showing it takes different values over +different regions of the parameter-rates plane. +Topological transition in the no-jump trajectory - Di- +rect inspection of the effective drift Hamiltonian shows +that if the magnetic field points in the z-direction, the +exited eigenstate |ψ+⟩ of H(t) remains fixed in a pole of +the Bloch sphere independently of the values taken by +the parameter rates Ω/ω and Γ/ω. Therefore, the GP +associated with the no-jump trajectory identically van- +ishes (mod 2π) for θ = 0 and θ = π. Without loss of +generality, the mod 2π freedom can be eliminated from +the GP by simultaneously setting φ0(θ = 0) = 0 and de- +manding continuity. In this way, φ0(θ = π) is completely +determined by the evolution and acquires a value +φ0(θ = π) = 2π n, +(15) +where n is an integer number that characterizes the de- +pendence of the GP with θ for fixed parameter values. +Being an integer, n constitutes a topological invariant be- +cause it can not be changed by smoothly deforming φ0(θ). +As a consequence, if the GP is characterized by different +values of n as a function of the various parameters, this +will impose the GP to undergo a non-smooth transfor- +mation, as the singular behavior exhibited in Fig.10. In- +deed, points in the parameter space slightly to the right +and slightly to the left of the singularity (indicated with +Figure 10. +Geometric phase associated with the no-jump +trajectory, displayed over a limited region of the parameters +plane defined by the ratios Ω/ω and Γ/ω. The value of the +GP is given by color, as indicated by the bar on the right. +The direction of the field is fixed to θ = 0.34π. +A singu- +larity is observed Ω/ω = 4.8082 × 10−3 and Γ/ω = 0.0306. +The crosses indicate points slightly to the left of the sin- +gularity (Ω/ω = 4.8 × 10−3) and slightly to the right of it +(Ω/ω = 4.8084 × 10−3), which will be shown to belong to +different topological sectors. +crosses in Fig. 10) give rise to no-jump evolutions asso- +ciated with topological invariants n = 0 and n = 1 re- +spectively, thus identifying different topological classes. +To explicitly show this, Fig. 11 compares the behavior +as a function of θ of these GPs by means of showing the +difference ∆(θ) between them. Given two points, say (1) +and (2) and labelled by crosses in Fig. 11, ∆(θ) is defined +as +∆(θ) = 1 +2π +� +φ(Γ1,Ω1) +0 +− φ(Γ2,Ω2) +0 +� +. +(16) +This difference is seen to vanish (up to some smooth small +deviations) up to θ = 0.34π, this is, until the angle of +the singularity. At this specific θ value the GP obtained +from each parameter rate abruptly deviates, so that their +difference shows a step and settles around ∆ = 1 for the +remaining range. The different topological numbers n is +reflected by the value ∆(π) = 1 for θ = π. +Over the full parameter space, the GP shows several +singularities, with locations that depend on the value of θ. +The set of singular points composes two counter-phase os- +cillating curves that define a chain of concatenated closed +regions and split the parameter-rate space into an upper +and lower region. This is shown in panel (a) of Fig. 12. +Parameters within each sector lead to the same n value. +The area below the sequence of closed regions is charac- +terized by n = −1. The points given by parameter values +Γ = 0 and Ω/ω ≪ 1, defining the adiabatic regime, be- +long to this region. The regions in between the lines are + +1e-2 +3.10 +2.0 +3.09 +10 +1.5 +3.08 +3 +3.07 +1.0 +L +0 +3.06 +X +0.5 +3.05 +3.04 +0.0 +4.8078 +4.8083 +4.8088 +m/u +1e-3 +Φ/ T13 +0.00 +0.25 +0.50 +0.75 +1.00 +θ/π +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +∆(θ) +Figure 11. +∆(θ) between GPs computed for points slightly +to the right and slightly to the left of the singularity, indicated +in Fig. 10 with x’s. The GP is, in each case, characterized +by a different value of the topological invariant n. +This is +reflected in the fact that they differ by 2π for θ = π. +topologically trivial sectors with n = 0, while the up- +per one is characterized by n = 1. It is worth pointing +out that these topological sectors are not equally proba- +ble. Besides the singular points of vanishing probability, +the probability of attaining a trajectory with no jumps +increases as Γ is reduced. This implies that the upper +topological sector is less probable than the others. +Topological transition in the echo experiment - With +the aim of seeking experimentally detectable signatures +of the topological transition, we perform a close inspec- +tion of the echo experiment that is completed without any +jump event. In Section V B, the ϕ value extracted in this +case was observed to show good agreement with the adia- +batic (Berry) phase for a wide range of frequencies. How- +ever, the close agreement of ϕ with φa will not hold for ar- +bitrarily small frequency values, and it will deviate when +the ratio Γ/Ω becomes sufficiently large. Fig. 13 shows +the ϕ value as a function of the frequency ratio. +For +easy reference and comparison, we consider an environ- +ment characterized by the dissipation rate Γ/ω = 0.0306, +which is included in the ranges exhibited by Figs. 10 to +12. +For large frequency ratio, the no-jump ϕ value shows +the behavior described in Section V B. However, ap- +proaching smaller frequencies, it shows a highly oscillat- +ing step and finally settles in the constant value ϕ ∼ +1.375π, associated with a persistence probability 1/2. +This regime will be accessed when the state at the end of +the protocol coincides, up to a global phase, with |ψ−(0)⟩, +this happens when the smooth drift suppresses the occu- +pancy of the exited eigenstate within a cycle. Full popu- +lation transfer from the excited to the ground eigenstate +taking place within the evolution cycle requires the sys- +tem to be driven at a slow frequency, smaller than that +leading to a singular point. This requirement establishes +Figure 12. Critical lines dividing the parameters’ plane into +different topological classes of the no-jump evolution. +The +classes are characterized by different n values. The critical +angle θc at which each singular point is found is indicated by +a color as described by the bar on the right. Panels (a) and +(b) display different ranges for the rates Ω/ω and Γ/ω. +a connection between the value of the echo phase and the +topological classes of evolution, as distinctive regimes of +ϕ are accessed on one and the other side of the singular +points. We refer to Appendix D for details on this point. +The limits of the range along which ϕ shows the step +and turns from ∼ φa into the central value are marked, +on Fig. 13 with two light blue dotted lines. The righter +region of the plot corresponds to evolutions characterized +by the topological number n = −1. The range between +the light-blue lines corresponds to the densely packed se- +quence of topological sectors illustrated by panel (a) of +Fig. +12. +Finally, once on the left of the last vertical +line, the evolution is associated with a value n = 1 of the +topological number. +The inset in Fig. +13 shows ϕ as a function of the +dissipation rate. In this plot, for easy reference and com- +parison, the value of the frequency rate is kept fixed at + +1e-2 +1.00 +n=1 +3.75 +3.50 +0.75 +3.25 +3 +3.00 +0.50 +L +2.75 +2.50 +0.25 +2.25 +(a) +n=-1 +0.00 +4.7 +4.8 +4.9 +m/u +1e-3 +0/元1e-2 +1.00 +3.8 +3.6 +n=1 +3.4 +0.75 +3.2 +3 +3.0 +n=0 +n=0 +0.50 +2.8 +2.6 +n=-1 +0.25 +2.4 +2.2 +(b) +0.00 +4.795 +4.800 +4.805 +4.810 +4.815 +4.820 +2/w +1e-3 +0/ T14 +Figure 13. Dependence of ϕ (black dashed line) obtained in a +protocol with no jump events, as a function of the ratio Ω/ω. +The field is oriented with θ = 0.34π and the environment is +characterized by the dissipation rate Γ = 0.0306ω, included in +the ranges displayed in Figs. 10 - 12. The adiabatic (Berry) +phase is also indicated for reference, with a red solid line. +The inset shows the ϕ value as a function of the rate Γ/ω, +with the magnetic field characterized by the same angle θ +and Ω/ω = 4.8×10−3, coinciding as well with the values used +in the previous plots. +Ω/ω = 4.8 × 10−3, also included in the ranges exhibited +by Figs. 10 to 12. Once again, the ϕ value shows good +agreement with the adiabatic phase up to some critical +Γ/Ω relation, at which it shows a decreasing step, finally +landing at ϕ ∼ 1.375π. As in the main plot, light blue +dotted lines mark the limit of the step and split the plot +into three distinctive sectors. The left of the first line +corresponds to n = −1 evolution, while the right side of +the plot, to n = 1. The space between lines, once again, +can be associated with the intermediate zone, which is a +single region (see Fig. 12 (a)) thus leading to no oscilla- +tions. +In summary, a measure of the persistent probabil- +ity in an echo protocol carries clear indications of the +topological transition. The peak structure discussed in +Section V B allows to identify the no-jump trajectory. +The subsequent analysis of this peak, as summarized in +Fig.13, is sufficient to capture the topological transition. +VI. +CONCLUSIONS +In this paper, we have studied geometric phases in +a continuously monitored quantum system. In absence +of any coupling to the environment, the cyclic time- +dependence of the Hamiltonian leads, in the adiabatic +regime, to the Berry phase, and to its consistent gener- +alization for a generic unitary evolution. The presence +of an environment induces quantum jumps so that in a +single realization of the dynamics the wave function, fol- +lowing a given quantum trajectory, accumulates a GP +that is itself a stochastic quantity. We have analyzed the +distribution of GPs by highlighting the interplay between +non-adiabatic effects and the influence of the external en- +vironment. We have shown that for slow drivings the dis- +tribution of phases is broad because of the several differ- +ent occurrences of jumps at random times. On speeding +up the driving, the number of jumps reduces and the dis- +tribution becomes peaked around the no-jump trajectory +(still deviating from the Berry phase because of the non- +adiabatic correction and the non-Hermitian drift term). +A first quantitative measure of the distribution has been +given by the variance, discussed in Fig.4. +In order to have experimental access to the GPs along +a given trajectory, we have also analyzed a spin-echo +protocol. +The structure provided by the jump opera- +tors taken together with the possibility of level transi- +tions due to non-adiabaticity and the characteristics of +the persistence probability can be set in such a way that +they lead either to the observation of broad distributions +or extremely sharp peaks. This interplay should be thus +considered in order to be explored as a tool or otherwise, +the experiment is rendered uninformative. +We have finally concentrated on the no-jump trajec- +tory, showing that it undergoes a topological transition +as a function of the dissipation strength. Interestingly, +this transition is not necessarily connected to singulari- +ties occurring in the dynamics of the density matrix. In- +deed, for the model considered herein, at the transition +point occurring in the no-jump trajectory the behavior of +the density matrix is smooth. Despite the striking differ- +ences shown between the GP and the interference fringes +of an echo experiment, traces of this transition can be +observed in the behavior of the interference fringes. +In this work, we have considered a specific model for +the jump operators corresponding to a well-defined type +of monitoring. However, it is important to understand to +which extent the properties we have discussed here de- +pend on the type of unravelling. +This question might +be of particular relevance, especially if one wants to +define topological properties associated with Markovian +systems starting from the properties of their trajectories +(there are infinite ways of unravelling the same Lindblad +dynamics). +A glimpse on this question is summarised +in Appendix C where we consider an unraveling corre- +sponding to a homodyne detection. For what concerns +the distribution the qualitative pictures we have outlined +in the body of the paper remain valid although important +quantitative differences may arise. +ACKNOWLEDGMENTS +We would like to acknowledge Alessandro Romito +for very useful discussions and critical reading of the +manuscript. The work of R.F. has been supported by the +ERC under grant agreement n.101053159 (RAVE) and by +a Google Quantum Research Award. The work of L.V., +F.L., and P.I.V. is supported by Agencia Nacional de + +1.475 +1.425 +91.375 +1.325 +10-2 +10-1 +I/w +1.275 +10-3 +10-2 +10-1 +m/u15 +Promoci´on Cient´ıfica y Tecnol´ogica (ANPCyT), Consejo +Nacional de Investigaciones Cient´ıficas y T´ecnicas (CON- +ICET), and Universidad de Buenos Aires (UBA). P.I.V. +acknowledges ICTP-Trieste Associate Program. R.F. ac- +knowledges that his research has been conducted within +the framework of the Trieste Institute for Theoretical +Quantum Technologies (TQT). +Appendix A: Pancharatnam phase along a quantum +trajectory +As stated in section II, the quantum trajectory emerg- +ing in a single monitored evolution of the system can be +understood as intervals of smooth dynamics interrupted +at random times by quantum jumps. Considered in this +way, evolution in a time interval t ∈ [0, T] is characterized +by an array of jumps of type αi occurring at times ti of +the form given by Eq. (4), and the parameter t is a con- +tinuous variable within the intervals delimited by the ti’s. +In the quantum jumps approach, the algorithm applied +in constructing the trajectories goes as follows [53]. The +time interval [0, T] is discretized into N steps of length δ t. +and the state is consistently updated at each time step +according to a randomly-decided non-hermitian operator, +as described in Eq. (2). Hence, each quantum trajectory +can also be thought of from an algorithmic point of view +as the ordered collection of states generated by the action +of a specific sequence of operators K0,α given by Eq. (3), +and is in this way a discrete set of states. +For a sequence of N discrete pure states, the suitable +GP expression is Pancharatnam phase [5, 44, 45], and is +given by +φP [ψ] = arg ⟨ψ1|ψN⟩ − arg(⟨ψ1|ψ2⟩ ... ⟨ψN−1|ψN⟩). (A1) +The Pancharatnam phase is independent of the U(1) +gauge choice and does not require the sequence to close, +rely on adiabaticity condition or demand for unitarity, +allowing for non-normalized states in the sequence, as +long as non of them perfectly vanishes. Exhibiting these +characteristics, it becomes a natural definition of GP to +be applied to monitored dynamics, in which evolution +is generated by non-hermitian operators. It equals the +unitary GP associated with the trajectory build-up from +joining consecutive states in the sequence by the shortest +geodesic in the Hilbert space. +While this definition does not imply any constraint on +the number of states in the sequence by itself, when ap- +plied in the context of quantum jumps the number N of +states is constrained from below as a consequence of the +condition reigning the time step. An evolution in time- +interval [0, T] consist of N = T/δ t ≫ 1 states. Split- +ting the sequence of states {|ψ1⟩ |ψ2⟩ ... |ψN⟩} into sets +starting and ending at those corresponding to the specific +times ti where a jump is registered, sets a bridge between +this two different descriptions of a quantum trajectory. +Each time interval [ti, ti+1], discretized in time-steps of +length δt, consist of a number of steps that depend on +the specific values of ti and ti+1. From a given jump-time +ti, any time-step in the consecutive interval can be found +as ti +ki δt, this is, by adding some amount ki ∈ N of in- +crements δt, up to some maximum value k∗ +i that satisfies +ti+1 = ti + k∗ +i δt (See Fig. 14). +0 +T +ti +|ψ(ti)⟩ +ti+1 +|ψ(ti + ki δt)⟩ +. ++ki δt +Figure 14. Illustrative diagram depicting time interval [0, T]. +Both the discretization in δt steps and the splitting at jump +times ti are indicated. The relation between times and states +is represented as well +At each given time, the outcome of a measurement +performed on the environment will be associated to the +corresponding Kraus operator acting on the system and +the state generated by its action. +Therefore, there is +a to a one-to-one correspondence between the discrete +set conforming the time interval and the array of states +forming the trajectory. The splitting at jump-times ti +can thus be mapped into the trajectory as +NJ +� +i=0 +{|ψ(ti + ki δt)⟩ ki = 0, ..., kmax +i +− 1} +(A2) +with NJ the number of jumps occurring in the trajectory +and the out-bounds indexes i = 0 and i = NJ + 1 signal- +ing the entire time-interval limits t0 = 0 and tNJ+1 = T. +Introducing such a decomposition into the formula for +Pancharatnam phase, Eq. (A1) can be re-written as +φP = arg ⟨ψ(0)|ψ(T)⟩ +− +NJ +� +i=0 +k∗ +i −1 +� +ki=1 +arg ⟨ψ(ti + ki δt)|ψ(ti + (ki + 1) δt)⟩ +− +NJ +� +i=0 +arg ⟨ψ(ti)| Kαi |ψ(ti)⟩ . +(A3) +The formula in Eq.(9) for the GP is thus associated with +a single trajectory is derived by taking the continuous +limit δt/T → 0 within the intervals of smooth evolu- +tion [44, 45]. This expression, more suitable for the exam +performed in our work, inherits all the properties of the +Pancharatnam phase from which it is obtained. +Appendix B: Interference fringes distribution +As discussed in Sec. +V B, the distribution of inter- +ference fringes from an echo experiment, which we pa- + +16 +rameterize with ϕ, shows three (sometimes sharp) peaks. +When γz = 0 only jumps between instantaneous energy +eigenstates are possible, and the three peaks emerge from +sets of trajectories of a different character as follows. +1. Smooth trajectories with no jumps generate the pil- +ing up in the no-jump value ϕ0 ∼ 1.43π +2. Trajectories in which at least one decay or spon- +taneous excitation jump occurred, projecting the +state into an eigenstate |ψ±(ti)⟩ of H(t), give rise +to the peak at ϕ ∼ 1.375π. +3. Trajectories in which only dephasing jumps took +place give rise to the peak at ϕ ∼ 1.275π. +In this appendix, we provide a detailed justification of +this observation. With the aim of providing an acces- +sible presentation of the qualitative aspects of the phe- +nomena, we will generally disregard the non-hermiticity +of the smooth evolution between jumps, thinking of those +intervals as unitary (slowly or rapidly driven) evolution. +Hence, this presentation should not be taken as a rigor- +ous quantitative analysis. +We begin with the consideration of the peak (1.) coin- +ciding with the no-jump value ϕ0 ∼ 1.34π. As presented +in Sec. II this smooth trajectory is unique and therefore +the exact same value of ϕ will be expected on every case +in which this trajectory is obtained. +We thus turn to the case in which jumps are indeed +detected, with special care on the anti-intuitive shrink- +ing of the distribution in the slower regime in which more +jumps are detected. When γz = 0 three jumps are possi- +ble within our unravelling of the Lindblad equation. Two +out of these three jumps project the state into an energy +eigenstate, namely, decay jumps and spontaneous excita- +tions. Whenever a jump of this kind takes place at some +instant of time ti, immediately after the jump the state +of the system turns into +|Ψ(ti)⟩ = ei ξ(ti)+i φ(ti) |ψ±(ti)⟩ +(B1) +with ξ(ti) the dynamical phase and φ(ti) the geometrical +phase, given by Eq. (9) accumulated up to the occur- +rence of the jump. If the protocol ends immediately after, +the persistence probability PR = | ⟨ψ(0)|ψ(2T)⟩ |2 = 1/2 +preserves no information on either the GP or the specific +characteristics of the jump. If, on the other hand, the +system continues to evolve, the possibility of obtaining +any information on a phase or the jump time will rely on +the interplay between the non-adiabatic transitions and +the existence of further jumps. If the evolution continues +from the first jump on, this will happen smoothly until ei- +ther the protocol is finished or another jump takes place. +Different regimes of Ω/ω give rise to the smooth evolution +of different natures. If the protocol is performed slowly +enough, this smooth evolution is (almost) transition-free +and |ψ(t)⟩ ∼ ei ξ(t>ti)+i φ(t>ti) |ψ±(t > ti)⟩, so the re- +sult obtained for the persistent probability remains to be +PR = 1/2. Moreover, this regime favors the occurrence +of further jumps, thus reinforcing the erasing of infor- +mation by re-projecting into eigenstates of H(t). +The +complete independence of the result on the times ti of +the jumps makes this peak (2.) extremely sharp in the +slow regime. On the other hand, if the system is driven +faster, along the smooth evolution after the jump the +state develops contributions from the other eigenstate +due to non-adiabatic effects, favoring the emergence of +relative phases and becoming +|ψ(t > ti)⟩ = A±(t > ti) |ψ±(t > ti)⟩ ++A∓(t − ti) |ψ∓(t − ti)⟩ +(B2) +with A ± (t) the amplitudes for each eigenstate. In such +a situation, the persistence probability depends on ti, +leading to the broadening observed in the central peak of +Fig.7 for faster driving, while still not trivially connected +to the GP. As anticipated in the previous paragraphs, +each jump of this kind will erase all information on the +phases and any dependence on previous jump times. The +possibility of further erasing events is mitigated in faster +protocols by the reduction of exposure to the environ- +ment. +The third peak (3.) +observed in the distribution at +ϕ ∼ 1.475π can be understood by adding dephasing +jumps to the previous discussion. A dephasing jump has +the effect of introducing a π shift in the relative phase of +the state. If the evolution afterward remains transition- +less (and no erasing jumps occur at any point), the evo- +lution resembles that of the adiabatic echo experiment +up to corrections that can be disregarded, so the per- +sistence probability takes the value P ∼ sin2(2φa) (with +cos replaced by sin due to the relative π shift). +This +situation leads to a well-defined single ϕ-value which is +independent of the time ti at which the jump took place. +Therefore, in the slow-driving range, a well-defined peak +emerges, that might however be small, as in this regime +decay jumps are likely. As the magnetic field is rotated +faster, non-adiabatic effects induce a dependence on ti on +the persistence probability. This dependence on ti is in- +herited by the ”phases” extracted, and thus responsible +for the broadening of the distribution observed in Fig. 7 +for larger Ω/ω values. +The inclusion of a jump operator ∝ σz modifies this +three-peaked distribution by leading to a broad back- +ground which is present even in the case in which it is +not the dominant process. The Kz jumps promote the +development of relative phases as they mix eigenstates of +the Hamiltonian. Even if the system has, at some time, +transitioned to an eigenstate, suffering from a σz-jump +suddenly drags it away into a superposition state. +Appendix C: Dependence on the unravelling: field +displacement +Another paradigmatic quantum trajectories scheme +arising from a different unraveling of the master equa- +tion is that of the so-called diffusive trajectories, in which + +17 +the monitored quantities produce continuously fluctuat- +ing signals instead of discontinuous jumps [71]. This is +the prototypical scheme of continuous or ideal homo- +dyne detection, which can be theoretically obtained as +a limiting case of the mentioned discrete homodyne de- +tection [59, 60, 67, 72, 73]. +The master equation Eq. +(1) is invariant under the +transformation +H(t) → H′(t) = H(t) − +√ +λ i +2 +� +α +(Kα − K† +α) +Kα → K′ +α = Kα + +√ +λ I, +(C1) +where +√ +λ ∈ R. Therefore it is possible to substitute Kα +and H(t) in Eq. +(1) by K′ +α and H′(t) without modi- +fying the averaged dynamics of the system and unravel +it using the standard direct detection (quantum jumps) +scheme applied before. When the reservoir is assumed to +be made of harmonic modes, like electromagnetic radi- +ation, adding the displacement +√ +λ to the Lindblad op- +erators corresponds to the implementation of homodyne +detection [60, 72, 74]. In this case, taking +√ +λ suitably +large leads to a measurement of the quadrature of the +system dipole Kα + K† +α. However, in order to keep the +collapse probability per step small, it would be necessary +to reduce the time step and hence increase the simula- +tion time by the same order. For this reason, we refrain +to consider finite large +√ +λ values in this section and fo- +cus on the modifications suffered by P[φ] for smaller +√ +λ +values. +In Fig. 15 we present, also for the case of this differ- +ent unravelling of the Linbland equation, the two cases +in which the driving is performed faster or slow enough +for the hypothetical unitary dynamics to be considered +adiabatic. As in the previous cases, the environment re- +mains fixed with Γ = 10−3ω and γz = 0, and we have +taken λ = 2.5 × 10−5ω < Γ. The two cases are shown in +panels (a) and (b) of Fig.15 respectively, where we also +plot the no-jump and unitary GPs, and the average of the +distribution for reference. Striking differences from the +case of direct detection arise. For this λ/ω ≪ 1, the ref- +erence values displayed remain close to those obtained in +the λ = 0, while the distributions behave differently. In +the fast-driven case displayed in panel (a), the expected +increase in jumps is reflected by the decrease of the sharp +peak piling up from no-jump trajectories. However, the +formerly broad, but still uneven background, has now +turned into a completely uniform distribution in which +each phase value (but the no-jump) is evenly probable. +The described behavior is reinforced when the system is +driven at slower frequency rates. The previously broad +while single-peaked distribution lacks, in a system mon- +itored through the operators K′ +α forming the new basis, +of any structure. +0.0 +0.5 +1.0 +1.5 +2.0 +φ/π +0 +1 +2 +3 +4 +P[φ] +1e +1 +(a) +1.47 1.475 1.48 1.485 +0 +1 +2 +3 +1e +2 +0.0 +0.5 +1.0 +1.5 +2.0 +φ/π +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +P[φ] +1e +2 +(b) +Figure 15. +Probability distribution P[φ] of GP values ob- +tained in an unravelling with K′ +α and H′(t) operators. The +magnetic field is oriented at θ = 0.34π and driven in a cycle at +frequencies (a) Ω = 5×10−3ω and (b) Ω = 5×10−4ω. The en- +vironment is characterized by the dissipation rate Γ = 10−3ω +and γz = 0. We have taken λ = 2.5 × 10−5ω. In both pan- +els, a blue solid contour recalls the distributions obtained in +the original unraveling considered in this work. Extra lines +indicate the new reference GP values. The black dashed and +dot-dashed lines signalize the GPs φ0 and φu associated with +no-jump and general unitary evolution. The black dotted line +shows the first moment of the distribution ¯φ. +Appendix D: Smooth evolution with no jumps: +Analytic approach +We provide in this Appendix some additional analyt- +ical results for the no-jumps evolution. +As previously +mentioned, this particular case can be thought of as gen- +erated by the non-hermitian Hamiltonian in Eq. (8), in +such a way that a non-normalized state +��� ˜ψ(t) +� +will follow +Schrodinger’s equation +i d +dt +��� ˜ψ(t) +� += Ho(t) +��� ˜ψ(t) +� +(D1) + +18 +where Ho(t) is not only non-hermitian but also ex- +plicitly time-dependent due to the function f(t). +The +effective +drift +Hamiltonian +shares +eigenstates +with +H(t), but the eigenvalues associated with these eigen- +states are now complex and time-dependent, given by +±ω/2 [1 − i Γ/(2ω)f(t)]. +The dynamics of the normalized state of the system +|ψ(t)⟩ = +��� ˜ψ(t) +� +�� +˜ψ(t) +��� ˜ψ(t) +� +(D2) +will be governed by the more involved, nonlinear equation +which is found by jointly differentiating Eq. (D2) and +making use of Eq. (D1). +The +not-normalized +state +can +be +expanded +into +the instantaneous eigenstates of Ho(t), was +��� ˜ψ(t) +� += +˜c+ |ψ+(t)⟩+˜c− |ψ−(t)⟩. Explicit computation of Eq. (D1) +leads to the following differential equations for the coef- +ficients ˜c±(t) +˙˜c± = +� +∓iω +2 − iΩ +2 (1 ∓ cos(θ)) ∓ Γ +4 f(t) +� +˜c±(t) ++ iΩ +2 sin(θ) ˜c∓(t), +(D3) +where the real term ∼ −Γ˜c+(t) indicates that even in +the case with no jumps, the presence of the environment +favors state transitions, as the amplitude of the excited +eigenstate is suppressed. Taking into account the nor- +malization procedure involved in turning from the not- +normalized state into the real, normalized one, this sup- +pression implies a population transfer from the excited +eigenstate into the ground state. As a consequence, any +trivial implementation of the adiabatic approximation is +prevented. A second feature observed in Eq. (D3) is that, +for the parameters chosen in this work, a good agreement +can be obtained by replacing f(t) with its mean value +f(t) ∼ 1 − sin2(θ)/2. By performing this replacement, +dynamics become easily solvable in the rotating frame. +The smooth evolution of each eigenstate of the system +is, within this approximation, given by +��ψ(±)(t) +� += N± e−iΩ/2 t +�� +±(ν + ε)e−iε/2 t ∓ (ν − ε)eiε/2 t� +|ψ±(t)⟩ +−Ω sin(θ) |ψ∓(t)⟩} , +(D4) +where both ν and ε are complex quantities given by +ν += +ω − Ω cos(θ) − i Γ/2(1 − sin2(θ)/2) and ε += +� +ν2 + Ω2 sin2(θ), N± is a normalization factor. At this +point, it should be stressed that Eq.(D4) explicitly shows +how the state +��ψ(±)(t) +� +obtained when evolving an eigen- +state will not be equal to the instantaneous eigenstate at +a later time in the general case. +Geometric phase - The GP associated with a trajec- +tory in which no jumps can be explicitly computed from +Eq. (10). While the general expression is quite involved, +it takes, for small rates Ω/ω ∼ Γ/ω of the driving fre- +quency and the dissipation rate to the amplitude of the +magnetic field, the form +φ0 ∼ − π(1 − cos(θ)) +(D5) +− π sin2(θ) +�Ω +ω + cos(θ)Ω2 +ω2 +� +− sin2(θ) +4 +�Ω +ω + cos(θ)Ω2 +ω2 +� e−4π Im(ν)/Ω − 1 +2 Im(ν)/Ω +, +where the first term in the r.h.s is the Berry phase. The +term in the second line of the equation is the main cor- +rection originating exclusively from non-adiabaticity, in +otherwise unitary evolution. The third line accounts for +the non-trivial effect of the environment in the no-jump +evolution. As Γ → 0 this term turns into a further con- +tribution due to non-adiabaticity. +Phase diagram singularities - +When computing the +accumulated GPs analyzed in Sections V A and V C we +have taken |ψ+(0)⟩ as our initial state. Thus, a vanish- +ing probability for observing this particular trajectory, +of the kind observed at the GP singular points, requires +|ψ(T)⟩ ∝ |ψ−(0)⟩. Considering the cyclic character of the +instantaneous eigenstates, this means a singular point +will take place whenever a full population transfer is +achieved exactly in a time period. It was already inferred +from the differential equations governing the evolution of +the ˜c± coefficients, that the dynamics generated by the +effective drift Hamiltonian Ho(t) favored transitions from +the excited to the ground instantaneous eigenstate. As +long as the original approximation remains accurate, the +singular points of the GP will be defined through the +equation (ν + ε) − (ν − ε)e2iπε/Ω = 0. +No-jump interference fringe - In Section V B, we have +studied the interference fringes of an echo experiment. +For this purpose, we’ve defined the convenient parameter +ϕ given by Eq. (13). Restricting to the case of a proto- +col performed without registering any jump, it was shown +that the value of ϕ displays, generally, better agreement +with the Berry phase than with the GP φ0 accumulated +by the state of the system under equal conditions, this +is, when it is smoothly driven along one period of time. +As long as the no-jump value ϕ ∼ φa, good agreement +between this “phase” and the GP will be obtained when +the second and third lines in Eq. (D5) are sufficiently +small. However, it is worth noting that the ϕ value will +not remain close to the Berry phase for arbitrarily small +driving frequency. While the protocol has shown to be +less sensitive to both non-adiabatic and environmentally +induced effects than the GP, it will account for the non- +ideal conditions. It was already shown that the environ- +ment induces population transfer from the excited to the +ground state. 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Then, the protocol +usually ends with a last spin rotatio taking the final state +back to the σz basis, where the actually compute proba- +bility is that of being in |0⟩. +[71] Different measurement schemes and physical situations +can be described recurring to symmetries of the Lindb- +land equation as a way of generating different unraveling. +Given the invariance of Eq. (1) under some joint trans- +formation Wm → W ′ +m, H → H′, the Lindblad evolution +of the averaged density matrix ρ(t) is be consequently +unchanged, while the different possible trajectories may +undergo nontrivial changes, therefore describing differ- +ent scenarios. Such a procedure can be followed to go +from direct photodetection to discrete homodyne detec- +tion schemes, in which a beam-splitter mixes the output +field with an aditional coherent field. +[72] H. M. Wiseman and G. J. Milburn, Phys. Rev. A 47, 642 +(1993). +[73] I. Percival, Quantum state diffusion (Cambridge Univer- +sity Press, 1998). +[74] N. Es’haqi-Sani, G. Manzano, R. Zambrini, and R. Fazio, +Phys. Rev. Research 2, 023101 (2020). + diff --git a/B9E2T4oBgHgl3EQf8wmh/content/tmp_files/load_file.txt b/B9E2T4oBgHgl3EQf8wmh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f5bbb27a3d652229dd1b5bea16ee9476dcc902cd --- /dev/null +++ b/B9E2T4oBgHgl3EQf8wmh/content/tmp_files/load_file.txt @@ -0,0 +1,1428 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf,len=1427 +page_content='Geometric phases along quantum trajectories Ludmila Viotti,1, 2 Ana Laura Gramajo,2 Paula I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Villar,3 Fernando C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Lombardo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='3 and Rosario Fazio2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 4 1Departamento de F´ısica Juan Jos´e Giambiagi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' FCEyN UBA Ciudad Universitaria,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Pabell´on I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 1428 Buenos Aires,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Argentina 2The Abdus Salam International Center for Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Strada Costiera 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 34151 Trieste,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Italy 3Departamento de F´ısica Juan Jos´e Giambiagi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' FCEyN UBA and IFIBA CONICET-UBA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Facultad de Ciencias Exactas y Naturales,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Ciudad Universitaria,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Pabell´on I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 1428 Buenos Aires,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Argentina 4Dipartimento di Fisica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Universit`a di Napoli ”Federico II”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Monte S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Angelo, I-80126 Napoli, Italy (Dated: January 12, 2023) A monitored quantum system undergoing a cyclic evolution of the parameters governing its Hamil- tonian accumulates a geometric phase that depends on the quantum trajectory followed by the system on its evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The phase value will be determined both by the unitary dynamics and by the interaction of the system with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Consequently, the geometric phase will acquire a stochastic character due to the occurrence of random quantum jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Here we study the distribution function of geometric phases in monitored quantum systems and discuss when/if different quantities, proposed to measure geometric phases in open quantum systems, are represen- tative of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' We also consider a monitored echo protocol and discuss in which cases the distribution of the interference pattern extracted in the experiment is linked to the geometric phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Furthermore, we unveil, for the single trajectory exhibiting no quantum jumps, a topological transition in the phase acquired after a cycle and show how this critical behavior can be observed in an echo protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' For the same parameters, the density matrix does not show any singular- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' We illustrate all our main results by considering a paradigmatic case, a spin-1/2 immersed in time-varying a magnetic field in presence of an external environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The major outcomes of our analysis are however quite general and do not depend, in their qualitative features, on the choice of the model studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' INTRODUCTION As Berry first stated in his seminal work [1], when a quantum system is prepared in an energy eigenstate and adiabatically driven in a cycle, it acquires, in addition to the dynamical phase, a phase that depends solely on the path traced in the ray space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Being independent of the specific dynamics giving rise to the path, this phase is of geometrical nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Following Berry’s breakthrough, consistent generalizations of the Geometric Phase (GP) have been found for unitary evolutions which are kept cyclic while they are not required to be adiabatic [2], in the presence of degenerate subspaces [3], and for the case in which both the adiabaticity and the cyclicity condi- tions are removed [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Further generalizations include the definitions of GPs for mixed states [6–10] and the so- called off-diagonal GPs [11, 12], which apply in the case where the initial and final states are orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' GPs are profoundly linked to the theory of fiber bun- dles and holonomies, bridging geometrical concepts like parallel transport over curved spaces with physics [13– 15], and contributing in this way to the understanding of quantum mechanics at the foundational level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Since their discovery, GPs have also emerged in most diverse physical systems [16, 17], deepening the comprehension of numer- ous phenomena such as integer quantum Hall effect [18], topological insulators and superconductors [19, 20], as well as playing a pivotal role in quantum information processing [21–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The quest for implementations of geometric quantum information processing has also spurred the search for geometric interferometry in several different setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The first proposal of this kind was realized in NMR [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Thereafter, Berry phases in superconducting qubits were both studied theoretically in [24] and observed experi- mentally for different regimes of couplings in circuit-QED arrangements [25–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In this direction, high-fidelity quantum gates were demonstrated with trapped ions [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The need to improve the performance of quantum infor- mation processing devices against the exposure to ex- ternal environment has led to the suggestion of non- adiabatic geometric gates schemes [32–37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In this con- text, it becomes of fundamental importance to under- stand how geometric interferometry is affected by the presence of an external environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Consequently, GPs need to be generalized to deal with the systems subject to non-unitary quantum evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The effect of fluctu- ations in the classical control parameters of a quantum cyclic evolution may average out mitigating their effect on the accumulated Berry phase [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The presence of an external bath was found to give rise to new geometric contributions to decoherence [39, 40], as experimentally detected in [41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Different definitions of GPs applica- ble in the non-unitary case have been put forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Tong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' [43] introduced a purification-independent formula computed over the reduced density matrix while an aver- age over different histories (trajectories) taking into ac- count system-bath interaction was discussed in Carollo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' [44, 45] and further analyzed in [46–48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Additional work along these lines can be found in [49–52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' There is, however, a different level of description of open quantum systems which may capture features that are washed out by simply looking at the properties of den- sity matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' This level is accessed, for example, when the state of the system is continuously monitored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In this arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='04222v1 [quant-ph] 10 Jan 2023 2 case, the quantum system is described by a wave func- tion whose smooth evolution is interrupted by random quantum jumps induced by the coupling with the envi- ronment [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' This sequence of smooth evolutions inter- rupted by jumps is named a quantum trajectory (see [54] for a recent review on the subject).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Goal of the present work is to describe the properties of accumulated GP along quantum trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In this ap- proach we are inspired by the work of Gebarth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' [55] where the GPs induced by a sequence of weak measure- ments stirring the system along a path in a parameter space were analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The randomness introduced by the occurrence of jumps in a given trajectory is reflected in the fact that the GPs inherit a stochastic nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' By random sampling over the trajectories, the entire distri- bution can be reconstructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Since the Berry phase is not an observable, the average value does not correspond to the phase accumulated by the average state (this is, the density matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Previous works, with the notable ex- ception of [55], either restrict the study of the dynamics of smoothly evolving pure states with no jumps or de- fine average quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Understanding the fluctuations of GPs induced by random jumps is to a large extent un- explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' We would like to fill this gap by studying this distribution and whether it is related to the correspond- ing distribution in the interference fringes in a spin-echo experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Finally, we will argue that the topological transition discussed in [55], despite the different dynam- ical settings, is a generic feature present in adiabatically driven monitored systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' We will show that depending on the coupling to the external environment, the moni- tored quantum system will show a topological transition in the phase accumulated in a cycle and we will argue that this transition is visible in echo dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In the next Section, we will define the dynamical setting we are interested in: A quantum system subject to a time-periodic Hamilto- nian and coupled to an external bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' With the intention to highlight the essence of our results, we will consider the paradigmatic case of a two-level system that evolves in presence of an externally varied magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The associated density matrix is governed by the Lindblad equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In order to follow the dynamics of the system along its quantum trajectories, we introduce a specific unravelling of the Lindblad equation which relays on mi- croscopic considerations, these aspects are introduced in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In Section III the model and its coupling to the environment are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In Section IV we de- fine the GP that will be the founding block of all our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' For an isolated system and sufficiently slow driving, this reduces to the Berry phase [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The pres- ence of the environment induces both a smooth drift and random jumps in the dynamics, so the evolution of the state is generically neither adiabatic nor cyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' To keep the presentation self-consistent, we further include in this same Section other definitions of GPs present in the lit- erature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' These will be employed for comparison in the posterior Section V A, where we discuss the distribution of the GPs accumulated along quantum trajectories and analyze reference GP values in order to account for dif- ferences with other definitions of GPs proposed in the context of open quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Due to the intrin- sic randomness of the quantum trajectory, a monitored echo experiment might be altered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In Section V B we discuss the probability distributions of the interference fringes and detail whether/when they relate to the corre- sponding distribution of the GPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Our analysis of GPs in monitored systems is completed in Section V C where we will show that the topological transition discovered in [55] for a specific setting is actually a generic feature in pe- riodically driven open quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Indeed, for the sequence of states known as no-jump trajectory, which can be thought of as the smooth evolution generated by a non-hermitian Hamiltonian, we find the GP displays a complex pattern in the parameters space exhibiting sin- gular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' These singularities can be tracked down to correspond to points of vanishing probability for such a trajectory, and to reveal the border between distinct topological sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The transition observed in the evo- lution when varying the parameters is topological in the sense that it is related to a discontinuous jump of an integer-valued topological invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Section V C will be entirely devoted to the study of this transition and ways to detect it through an echo protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' A summary of our results and concluding considerations are presented in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The appendices give some additional in- gredients used to compute the GP in the numerical sim- ulations, Appendix A, a detailed analysis of the already mentioned interference fringes distribution, Appendix B, a brief discussion on how the distribution of GPs may de- pend on the unravelling of the Lindblad equation (leading to the same averaged evolution), Appendix C, and ana- lytical treatment of the no-jump trajectory, Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' FROM LINDBLAD DYNAMICS TO QUANTUM TRAJECTORIES Lindblad equation - In order to make a connection with existing literature, it is convenient to set the stage and start from the case in which the state of an open quantum system is described by a density matrix ρ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In this case, under proper conditions, the dynamics is governed by the Lindblad equation [56, 57] (ℏ = 1) ˙ρ = −i [H, ρ] + � α [LαρL† α − 1 2{L† αLα, ρ }] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (1) The first term in the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' of the Lindblad equation ac- counts for the unitary evolution, while the second origi- nates in the coupling to the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The strength and the nature of this coupling are encoded in the Lind- blad operators Lα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' We will consider a Hamiltonian H that depends periodically on time H(t + 2π/Ω) = H(t) with T = 2π/Ω the period of a cycle in suitable param- 3 eter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The Lindblad operators, if time-dependent, should also be time-periodic Lα(t + 2π/Ω) = Lα(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' It is useful to already at this point briefly comment on the adiabatic limit for slow dynamics as this issue will be central in the analysis conducted along the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' If the evolution is unitary, for a sufficiently large period T, a system prepared in an eigenstate will remain in the corresponding instantaneous eigenstate up to small cor- rections due to Landau-Zener transitions between energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In other words, the occupancy of any given eigen- state will not change in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The situation strongly differs in presence of an environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In this case, a proper adiabatic limit is not well defined, since the slow driving limit where adiabatic dynamics sets in, is also the regime in which the consequences of the external baths are the most severe and the system reaches a (possibly periodic) steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The adiabatic limit itself should be reconsidered [58] in an open system, as the existence of a continuum of energy levels makes the energy split- tings of the system a bad reference scale for defining the regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Effects due to non-adiabaticity and corrections due to the presence of the environment seem thus to be inextricably linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Monitored dynamics and quantum trajectories - The dynamics of the systems radically change when it is pos- sible to continuously monitor their state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In this case, the state of the system remains pure and consists of in- tervals of smooth evolution interrupted at random times by abrupt changes called quantum jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' A sequence of smoothly-evolving intervals together with a set of random events is denominated a quantum trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The litera- ture on the subject is vast and we refer to the following papers and books for a general overview [53, 54, 59, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Evolution is described in this framework as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' If at time t the state of the system is |ψ(t)⟩, at a later t+δt time it will be |ψ(t+δt)⟩ = � � � � � � � � � Ko|ψ(t)⟩ √ po(t) with probability po(t) Kα|ψ(t)⟩ √ pα(t) with probability pα(t) (2) where o, α = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='. label the different operators Kα induc- ing dynamical steps Ko = 1 − i δt � H − i 2 � α L† αLα � Kα = √ δtLα (3) and po/α(t) = ⟨ψ(t)| K† o/αKo/α |ψ(t)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Each choice in the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (2) represents evolution steps of different characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The second line corresponds to the occur- rence of a jump Kα at time t, while the first is a smooth evolution (no jump), albeit altered from unitarity by the fact that acquiring the information that no jumps oc- curred modifies the evolution of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The no- jump operator Ko can also be thought of as generated by an effective drift Hamiltonian Ho to which it relates in the usual way Ko = 1 − i δt Ho.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The full evolution in a time interval [0, t] is therefore characterized by a se- quence of NJ jumps of types αi occurring at times ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' We will denote the string of these events R(t, NJ) = {(α1, t1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' , (αi, ti), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (αNJ, tNJ)}, (4) with 0 ≥ ti ≥ t ∀i, the quantum trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' As men- tioned above, this framework naturally emerges when the system is continuously and indirectly monitored, so that each trajectory can be viewed as the result of continu- ous measurements of the environment on a given basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' From this perspective, continuous monitoring may lead to decoherence mitigation by the environment [61], also post-selection and error correction schemes [62, 63] have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The properties of the Kraus operators Ko/α guarantee that the probabilities to get a given outcome sum up to one, and the time step δ t should be taken small enough for the first order approximation to be valid, which re- quires � α pα ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Averaging over every possible jump sequence one gets back the Lindblad equation [53] in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (1), the converse implication is not valid, an infinite number of different unravellings give rise to the same Lindblad evolution [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' We will address this question in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' THE MODEL Since we are interested in studying the impact of an external environment on the GPs, we will con- sider a unitary evolution over which the accumu- lated GP, in the adiabatic limit, is the Berry phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' To be concrete, we shall consider a spin-1/2 parti- cle in presence of a time-dependent magnetic field B(t) = ω ˆnB(t), whose direction is given by ˆnB = (sin (θ) cos(Ω t), sin (θ) sin(Ω t), cos θ) with fixed polar an- gle θ and time-varying azimuthal angle Ω t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Such unitary evolution is generated by the Hamiltonian H(t) = 1 2 B(t) · σ, (5) with σ = (σx, σy, σz);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' and |0⟩ and |1⟩, the eigenstates of σz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The instantaneous eigenstates of H(t) are denoted |ψ−(t)⟩ and |ψ+(t)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' If the system could be kept perfectly isolated while the direction of B(t) is adiabatically changed in a cycle parameterized by t ∈ [0, T], with T = 2π/Ω (as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='1), it would acquire an adiabatic (Berry) phase φ±a = −π(1 ∓ cos θ), where the ∓ sign depends on the energy eigenstate in which the system was initially prepared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Lindblad operators - For a system that evolves accord- ing to H(t) given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (5) coupled to an environment of harmonic oscillators a consistent time-dependent Lind- blad equation of the form in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (1) can be derived from 4 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Trajectories described by the state of the system on the Bloch sphere under different conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The black line corresponds to unitary evolution in the adiabatic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The purple line depicting a curly ring corresponds to general uni- tary dynamics in which non-adiabatic corrections start to be visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In the presence of an environment, the quantum state can suffer from jumps or can be smoothly driven along the whole evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' For a system prepared in the exited eigen- state, the orange trajectory corresponds to a fully smooth drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Differently, the blue path shows a jump that projects the state into the instantaneous ground eigenstate and is af- terward smoothly driven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Finally, the light blue path shows a case with several jumps, where the non-adiabatic corrections appear in between the jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' microscopic considerations as long as the evolution re- mains sufficiently slow [64, 65], with Lindblad operators given by L−(t) = √γ− ⟨ψ−(t)| σx |ψ+(t)⟩ |ψ−(t)⟩ ⟨ψ+(t)| L+(t) = √γ+ ⟨ψ+(t)| σx |ψ−(t)⟩ |ψ+(t)⟩ ⟨ψ−(t)| (6) Ld(t) = √γd � i=± ⟨ψi(t)| σx |ψi(t)⟩ |ψi(t)⟩ ⟨ψi(t)| and corresponding to decay, spontaneous excitation, and dephasing respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The coupling strengths consid- ered in this work are, in terms of the dissipation ratio Γ, γ− = Γ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' γd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='32 Γ, while we consider γ+ to be negli- gible (all the results that we will show are rather generic and do not qualitatively depend on the chosen values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The jumps defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (3) from the Lindblad oper- ators above lead, after averaging, to a consistent Lind- blad equation for slow dynamical evolutions [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The operators introduced in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (6) induce transitions and dephasing between the instantaneous eigenstates of the Hamiltonian defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In order to keep the anal- ysis as general as possible, we will include a further term in the Lindbladian which requires considering a fourth operator Lz = √γzσz (7) along a fixed direction in the Bloch sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The particu- lar choice of σz operator as the additional Lindblad oper- ator is motivated by the need of introducing transitions that do not simply involve the instantaneous eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Any other Lindblad operator that differed from those in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (6) would lead to similar qualitative conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' While the unitary evolution of the closed system will follow the curly path indicated in purple in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='1, the ac- tual dynamics will follow, with some probability, the path indicated in blue (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='1 for illustrative purposes), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' it will be discontinuous and not necessary closed after a cycle of the driving, even in the slow-driving limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' More- over, the slower the driving, the more jumps will occur (see light blue curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The task of the next Sec- tions is to characterize GPs under these conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Smooth evolution with no jumps - A particularly inter- esting quantum trajectory is that which is smooth along the whole evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Before addressing the characteriza- tion of GPs in indirectly monitored systems, we provide insight into the evolution giving rise to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' When the records of the measurements performed on the environ- ment reveal zero jumps, the dynamics describe a contin- uous smooth path and is generated by an effective drift Hamiltonian which depends both on the Hamiltonian of the system and the Lindblad operators as described by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Within the model considered in our work, the effective drift Hamiltonian Ho governing the no-jump dy- namics [Ko = 1 − δtHo in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (3)] is given by Ho(t) = � 1 − i Γ 2ω f(t) � H(t) (8) with f(t) = cos2(θ) + sin2(θ) sin2(Ωt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' We highlight the fact that, due to the unitarity of σz matrix, the no-jump evolution is completely independent of the fourth Lin- bland operator Lz included ad-hoc and, consequently, from the parameter γz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' An illustrative example of the trajectory generated by the above evolution, referred to as the no-jump trajectory in what comes, is the orange path in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In Appendix D we provide the analytic so- lution for the dynamics associated to the non-Hermitian Hamiltonian Ho(t) of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' While this trajectory is unique, the number of possible (even though unevenly probable) trajectories in which NJ > 0 jumps occur in- creases with the number NJ of jumps, diverging as δt goes to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Its uniqueness will make the no-jump trajectory especially suitable for the analysis of some features of the GPs, we come back to this question in Section V C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' GEOMETRIC PHASES IN OPEN SYSTEMS DEFINITIONS - As mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' I, the accumulation of a GP during the dynamics of a quantum system is not neces- sarily restricted to an adiabatic evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' For a generic quantum trajectory, consisting of a sequence of smoothly- evolving intervals together with a set of random quantum jumps R, a proper phase that deals with both aspects of evolution can be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' [1) y X [0)5 Considering the evolution in a time interval [0, T], pa- rameterized with t, the GP associated to a trajectory in which NJ jumps are registered at times ti, can be written as φ[R] = arg ⟨ψ(0)|ψ(T)⟩ − Im NJ � i=0 � ti+1 ti dt � ψ(t) ��� ˙ψ(t) � ⟨ψ(t)|ψ(t)⟩ − � (ti,αi)∈R arg ⟨ψ(ti)| Kαi |ψ(ti)⟩ , (9) where R = R(T, NJ) for brevity, with t0 = 0 and the convention that tNJ+1 ≡ T in the sum of integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The definition of GP as given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (9) will be at the basis of our analysis and refer to Appendix A for a deriva- tion of this expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' As it is evident from the depen- dence on the times and nature of the jumps, the phase φ[R(T, NJ)] will be a stochastic variable, dependent on the trajectory R(T, NJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The first term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (9) is the total relative phase between the initial and final states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The remaining terms are of two different kinds, reflecting the properties of the dynamics itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The second term features the dynamical phases accumulated along the in- tervals of smooth evolution that take place before, be- tween, and after jumps, and which should be subtracted in order to access the purely geometrical object φR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The occurrence at time ti of a jump generated by the opera- tor Kαi introduces a contribution arg ⟨ψ(ti)| Kαi |ψ(ti)⟩ which represents the phase difference between the state before and after the jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Such a term equals the GP associated with the trajectory build-up by joining the states by the shortest geodesic in the Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The expression in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (9) is independent of the U(1) gauge choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' It neither requires the trajectory to trace a close path in the state space nor relies on adiabaticity condi- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Moreover, it does not even demand unitarity as it is well defined also if the states |ψ(ti)⟩ or |ψ(t′)⟩ are not normalized (the norm should however be non-vanishing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Suitable to be applied to the trajectories that emerge in master equation unraveling, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (9) has been employed in limiting forms for addressing the definition of GPs fit- ting non-unitary evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' A first explored route was to focus on the no-jump trajectory [44, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' This approach, which disregards the possibility of quantum jumps by restricting to the smooth evolution, preserves the well- known definitions of GPs applicable to pure states and in- cludes environmental effects through the non-hermiticity of Ho.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' If no jumps are registered along the entire evolu- tion, this is, if R(T, 0) = ∅, the GP φ0 ≡ φ[R(T, 0) = ∅] reads φ0 = arg ⟨ψ(0)|ψ(T)⟩ − Im � T 0 � ψ(t) ��� ˙ψ(t) � ⟨ψ(t)|ψ(t)⟩ dt (10) which trivially reduces to the expression for the GP accumulated in the most general unitary evolution [5] when this is indeed the case, and therefore the states are instantaneously normalized, rendering the denominator ⟨ψ(t)|ψ(t)⟩ ≡ 1 ∀ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (10) also reduces to Aharonov- Anandan and Berry phases as the conditions required by each definition are fulfilled, namely, for cyclic and uni- tary while not necessarily adiabatic evolution and for both cyclic and adiabatic evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Note that phase φ0 is ill-defined if some internal product on its argument vanishes, this observation will become of relevance when discussing the topological transition in Section V C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Several other works consider the full Lindblad equation unraveling, suggesting to define the GP of the ensemble- averaged state ρ(t) as an average over the ensemble of phases {φR} = φ{R} obtained by applying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (10) to each trajectory [44–46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' It has been extensively discussed whether this is a proper definition of a GP for the density matrix representing the state of the system as it does not allow for a one-to-one relation between the set of density matrices and the obtained GP values [47, 48, 67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Finally, a different approach introduces a generalized GP defined directly from the reduced density matrix [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The expression reads φρ = arg � �� j � λm(0)λm(t) ⟨ξm(0)|ξm(t)⟩ × exp � − � t 0 dt′ � ξm(t′) ��� ˙ξm(t′) ��� (11) where λk(t) and |ξk⟩ are the instantaneous eigenvalues and eigenstates of the density matrix ρ(t) which de- scribes the state of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Even though defined for non-degenerate but otherwise general mixed states, when computed over pure states under unitary evolution, re- duces to the unitary expression of the GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' All the above-mentioned proposals of GPs applicable when dynamics are non-unitary either restrict to modi- fied evolutions on which pure-state GP definitions would be applicable or seek a consistently defined GP for the reduced density matrix ρ(t), which accounts for an aver- aged description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Stochastic processes, however, arising from master equation unraveling, acquire independent physical relevance in continuous monitoring schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' As anticipated in the introduction, the randomness intro- duced by the occurrence of jumps in a given trajec- tory reflects in the GPs acquiring a stochastic nature itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' This approach, therefore, requires a study of the environmentally-induced effects in GPs from a statistical perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The probability associated with some GP value will be related to that of individual trajectories as P[φ] = � R/φ[R]=φ P[R].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (12) The average phase corresponds only to the first moment of the distribution ¯φ = � φP[φ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 6 and in some cases may be not sufficient in characterizing the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' For easy later reference, we provide a table summariz- ing the GP definitions reviewed along this section GP Description φa Adiabatic Berry phase φ[R] GP associated to the quantum tra- jectory R(T, NJ) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (9) φ0 GP associated to the no-jump trajectory Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (10) φu GP accumulated on general unitary evolution ( from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (10) with ⟨ψ(t)|ψ(t)⟩ = 1) ¯φ Average over the probability distri- bution P[φ] φρ Mixed state geometric phase [43] Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (11) The next Section will be devoted to the properties of P[φ] and how representative the different GPs applicable to trajectories are, see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (9 - 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' As we will be show- ing in the following, in most cases the entire probability distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' all higher order cumulant, is necessary to understand the accumulation of GPs in a continuously monitored system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' We will also discuss under which cir- cumstances and what features of P[φ] can be extracted by geometric interferometry through a spin-echo proto- col.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Geometric phase distribution P[φ] We investigate in this section the distribution of the en- semble {φR} = φ{R} of GPs obtained by employing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (9) to each individual realization (trajectory) of the evo- lution, characterized by some set R(T, NJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 2 we show two representative cases in which the correspond- ing dynamics of a hypothetical unitary evolution would either be faster (with small but non-zero non-adiabatic corrections) or slow enough to be considered in the adi- abatic regime while the environment remains the same, characterized by the dissipation rate Γ = 10−3ω, which leads to γ− = Γ, γd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='32 Γ, and negligible γ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' We first attend the case with γz = 0, in which the en- vironment induces jumps involving instantaneous eigen- states only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The two situations, corresponding to the two sets of parameters indicated before, are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 2, in panels (a) and (b) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In both panels, we also plot for reference the adiabatic (Berry) result, the no-jump and unitary GPs, and the average of the dis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Being the Berry phase independent of Ω, it 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 φ/π 0 1 2 3 4 P[φ] 1e 1 (a) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='47 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='475 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='48 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='485 0 1 2 3 1e 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 φ/π 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='5 P[φ] 1e 2 (b) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Probability distribution P[φ] of GPs for a magnetic field oriented with θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='34π and driven in a loop at frequen- cies (a) Ω = 5 × 10−3ω and (b) Ω = 5 × 10−4ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The environ- ment is characterized by the dissipation rate Γ = 10−3ω and a γz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In both panels, the solid red line depicts the adiabatic (Berry) phase φ+ a , and the black dashed and dot-dashed lines signalize the GPs φ0 and φu associated with no-jump and general unitary evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The black dotted line indicates the first moment of the distribution ¯φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The inset in panel (a) is a zoom in which the difference between these reference GP values is visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' is exactly the same for both cases, this is, φa ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='482π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' For the parameters chosen, the value φ0 computed from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (10) over the trajectory with no jumps, shows small deviations from φa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' While the values of these character- istic GPs are similar, the entire distribution of the moni- tored system is drastically different on each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In the first case of faster driving, the period T is such that a considerable amount of times the evolution is completed registering no jumps, with the mean number of jumps over the ensemble ¯NJ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The narrow peak in the Figure shows these cases of entire smooth evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In addition, there is a small background revealing the ac- cumulated GP along those trajectories where jumps oc- curred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The composition of the ensemble is reflected in 7 the histogram by the presence of a large contribution, corresponding to ∼ 50% of the realizations, due to the no-jump GP-value and the remaining 50% of the counts distributed in a broad way over the possible GP values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' This broad background distribution can be easily inter- preted as the randomness inherited by the GP due to the (random) time at which the jump occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' A single term ⟨ψ(ti)| K−i |ψ(ti)⟩ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (9), denoting a contri- bution to the GP from a jump at time ti, successfully accounts for the background when considering all pos- sible jump-times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The peak in the distribution agrees well with both the adiabatic and the no-jump values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The average phase, on the other side, is a bit off due to the small and poorly structured background, broadly distributed over 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' This clearly demonstrates that even a single jump occurring at a random time leads to very large fluctuations in the accumulated GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In the case with slower driving shown in panel (b) the mean num- ber of jumps over the set of trajectories is ¯NJ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' This means that the state of the system is much more likely to undergo an abrupt change, or even more than one, in each realization of the cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' As expected, the distribution of GPs becomes much wider, and a sharp peak around φ0 is not visible anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Higher-order cu- mulants become necessary to understand the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The three lines, corresponding to the adiabatic, no-jump, and average GPs do not provide clear information on the dynamics of the monitored system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Probability distribution P[φ] of GPs as a function of the ratio Ω/ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The field is oriented with θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='34π and the en- vironment is characterized by the dissipation rate Γ = 10−3ω and a γz = 0 amplitude for the fourth Linbland operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The GP values are displayed on the y-axis, while their probability is indicated by the intensity of the count color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The solid red line depicts the adiabatic (Berry) phase φ+ a , the black dashed line indicates the GP φ0 accumulated along smooth trajecto- ries with no jumps, and the black dotted line shows the first moment of the distribution ¯φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The rate Ω/ω at which the magnetic field is rotated has thus a direct impact on the distribution of GPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' For larger rates, the system is exposed to the environment for a shorter period of time, but deviations from the adi- abatic regime become non-negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' On the other hand, lowering the driving frequency might result in the system being exposed to environmental effects for too long, im- plying strong corrections to φR from φ+ a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 3 shows the distribution of GP-values obtained along a range of different Ω/ω rates which include the cases presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' For high enough frequency, the distribution shows a sharp peak around the no-jump value of the GP and al- most no background counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' On the other hand, this no- jump value deviates considerably from the Berry phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The broad background visible in panel (a) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 2 de- velops as the frequency rate is lowered, this is, as the relative period grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Further on, the background turns into a second peak, while the one in the no-jump value decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' For the smaller rate values, the distribution shows the behavior depicted by panel (b) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 2, this is, a broad single-peaked distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' This regime shows non-negligible environmental effects also over the GP as- sociated with the no-jump evolution, which deviates from the adiabatic result even though the driving is performed slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' We refer to Appendix D for an analytical expres- sion for the dependence of this deviation on the different parameters involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The broadening exhibited by the distribution as the frequency rate decreases, is reflected in the increment of the distribution variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' This is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 10-3 10-2 10-1 Ω/ω 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='5 σ2� R � (a) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Variance σ2 {φR} of the GPs’ distribution as a func- tion of the ratioΩ/ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The field is oriented with θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='34π and the environment is characterized by the dissipation rate Γ = 10−3ω and a γz = 0 (same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' We conclude this section by analyzing the distribution of GP values when γz ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' As already discussed, a non- zero value of γz induces jumps to states that are not instantaneous eigenstates of the Hamiltonian and thus allows to consider of a wider class of cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The result- ing phenomenology depends only quantitatively on the choice of the Lindblad operator Lz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Specifically, we take γz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='1 Γ and consider, as we did before, two differ- 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 10-2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='5 10-3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 10-5 10-3 10-2 10-1 P[] m/u8 ent values of the speed at which the system is cyclically driven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='5, with the qualita- tive features of the distribution closely resembling those obtained in the case with γz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 φ/π 0 1 2 3 4 P[φ] 1e 1 (a) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='47 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='475 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='48 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='485 0 1 2 3 1e 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 φ/π 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='5 P[φ] 1e 2 (a) Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Probability distribution P[φ] of the GPs for a mag- netic field oriented with θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='34π and driven in a loop at frequencies (a) Ω = 5×10−3ω and (b) Ω = 5×10−4ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The en- vironment is characterized by the dissipation rate Γ = 10−3ω and γz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='1 Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In both panels, a blue solid contour indicates (for comparison) the γz = 0 distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The solid red line depicts the adiabatic (Berry) phase φ+ a , the black dashed and dot-dashed lines signalize the GPs φ0 and φu associated with no-jump and general unitary evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The black dotted line shows the first moment of the distribution ¯φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The inset in panel (a) zooms in to see the differences in the positions of the lines and the peak of the distribution Panel (a) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='5 corresponds to the faster case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The mean number of jumps ¯NJ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='69 is slightly above the one obtained in the γz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The additional jumps gener- ated by Kz are not sufficient to modify the distribution qualitatively, which continues to show a well-defined peak (arising from the occurrence of smooth evolution with no jumps) plus a broad small background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In panel (b), showing the case in which the system is driven slower, the mean number of jumps is also slightly increased from the γz = 0 case due to the additional presence of γz jumps, reaching a value ¯NJ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The cases discussed above contain the first message of the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The stochastic nature of the GP in monitored dynamics needs to be taken into account and it is not possible to characterize it only through a single value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' This rises the additional question of how this fact reflects on the experimental outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' To address this question, we will consider in the next Section a spin-echo protocol and see how, when, and whether the distribution in the interference fringes is affected by the randomness of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Distribution of interference fringes in a spin-echo protocol If the system is prepared in an eigenstate of the Hamil- tonian and subsequently driven in a cycle, adiabati- cally and in absolute isolation from the environment, then the quantum state accumulates a Berry phase that can be measured by implementing a spin-echo proto- col [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' It goes as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The system is initially prepared in a superposition state |ψ(0)⟩ which reads (1/ √ 2)(|ψ+(0)⟩+|ψ−(0)⟩) in terms of the ground and ex- ited instantaneous eigenstates of H(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Then, it is driven for a period T, causing each eigenstate to acquire both a dynamical and a geometric phase φ±a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' A spin-flip op- eration and a second cycle in the opposite direction lead to a cancellation of the dynamical phases, resulting in a purely geometric relative phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Berry phase can thus be extracted through state tomography [25, 27, 69] or by realizing that the probability for the system to be back in the initial state once the full evolution is completed, the persistence probability, is related to the Berry phase as |⟨ψ(0)|ψ(2 T)⟩|2 = cos2(2 φ+ a ) [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The relation between the persistent probability and the GP given above relies on two factors: the adiabatic regime preventing the tran- sitions between eigenstates and the exact cancellation of the dynamical phases during the protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' If an echo ex- periment is performed on a system that is exposed to the effect of the environment and continuously monitored, the persistence probability will retain its dependence on the dynamical evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Nevertheless, it is worth under- standing to which extent it is possible to learn features of GPs in a monitored system through an echo protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' For each realization of the protocol, characterized by a sequence of jumps R(2 T, NJ), we can parametrize the persistent probability PR through an associated angle ϕR PR = |⟨ψ(0)|ψ(2T)⟩|2 ≡ cos2 (2 ϕR) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (13) Both the persistence probability and the parameter ϕR inherit the stochastic character of the trajectories, with the probability of measuring a given value ϕ related to the probability of the trajectories as 9 P[ϕ] = � R/ϕR=ϕ P[R].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (14) In the limiting case in which the persistence probability approaches its adiabatic value, ϕ will approach φ+ a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Away from that particular regime, ϕR is NOT equal to the GP φR = φ[R] but, as mentioned previously, a convenient parametrization of the spin-echo interference fringes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The non-adiabatic and environment-induced devia- tions from φ+ a can be analyzed by examining the ensemble {ϕR} = ϕ{R} that is obtained by computing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (13) for each individual realization of the protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' This study will also allow seeking possible relations, if any, between the stochastic behavior of the GPs and that of experi- mental outcomes (note that ϕR is defined modulo π/2 and up to a sign, therefore, any relation between the dis- tribution of GPs and the distribution of the experimental results should take this into account).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The frequency Ω at which the magnetic field is rotated is expected, once again, to have a direct impact on the distribution [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' On increasing the relative value of Ω, the system will be exposed to the disruptive influence of the environment for shorter times, allowing to a larger extent a partial can- cellation of the dynamical phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' At the same time, in this regime, non-negligible deviations from the adiabatic results will be unavoidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' On the other hand, smaller values of Ω might result in the system being exposed to environmental effects for too long, leading to strong de- viations of the echo-parameter values ϕ from φ+ a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In analogy with what we did in section V A, we ex- amine first the case γz = 0 and present, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='6, two representative cases in which the hypothetical unitary evolution would either be faster or slow enough to be considered within the adiabatic regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' These are shown in panels (a) and (b) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='6 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In both pan- els, we also display the adiabatic Berry phase φa (which does not depend on Ω), the GP φ0 obtained in a proto- col with no jumps, and the GP φu obtained in general unitary evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The ϕ value obtained from an echo experiment which is completed without detecting jumps is also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' For the parameters chosen, both panels show very small deviations of ϕ extracted in a protocol with no jumps from the Berry phase (see the insets in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' It should be noted, however, that the probability of registering this specific trajectory is different in the two cases, as it can be seen in the differences in the full P[ϕ] distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The first striking feature that comes out is the pres- ence of three distinct sharp peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The broad distribu- tion observed in the GP values completely disappears in the spin-echo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' This behavior originates from the fact that when γz = 0, only jumps between instantaneous eigenstates are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' This particular aspect of the unravelling leads, when combined with the properties of the persistence probability, to a distribution of interfer- ence fringes qualitatively different from that of the GPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Each of the peaks shown in panel (a) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 6 can be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='275 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='325 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='375 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='425 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='475 ϕ/π 0 1 2 3 4 P[ϕ] 1e 1 (a) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='48173 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='48177 0 2 4 1e 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='275 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='325 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='375 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='425 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='475 ϕ/π 0 2 4 6 8 P[ϕ] 1e 1 (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='4813 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='4817 0 2 4 6 1e 4 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Probability distribution P[ϕ] obtained in the echo- protocol for a magnetic field oriented with θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='34π and driven in a loop at frequencies (a) Ω = 5×10−3ω and (b) Ω = 5×10−4ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The environment remains the same, characterized by the dissipation rate Γ = 10−3ω and a γz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In both panels, the solid red line depicts the adiabatic (Berry) phase φ+ a , and the black dashed and dash-dotted lines signalize the GPs obtained in no-jump and unitary evolution respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Furthermore, the black dash double-dotted line indicates the ϕ value obtained in an echo protocol with no jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The insets in both panels show a range in which the result of a smoothly performed echo experiment is distinguishable from the Berry phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' understood as arising from a different set of quantum tra- jectories in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' For the parameters chosen in panel (a) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 6 trajectories with at most one jump are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The three peaks correspond to protocols with no jumps, protocols with one jump of the type L±, and one jump of the type Ld respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' We refer to Ap- pendix B for a detailed justification of this identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Trajectories that remain smooth along the whole proto- col induce the right peak in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='6 (closest to the no-jump result, ϕ ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='475π for this choice of parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The central peak, centred at the value ϕ ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='375π trivially 10 associated via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (13) with a persistence probability taking the value 1/2, builds up from all those cases in which the state of the system is, at some given time, pro- jected into an eigenstate of H(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In those trajectories, all the information about the accumulated phase before the jump is lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' As a consequence, immediately after a jump L±, and regardless of both the previous evolution and the time at which the jump occurred, the persistence probability takes the exact value 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The third peak, the left one, is due to trajectories in which a jump Ld occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' This type of jump has the effect of introducing a π-shift in the relative phase of the echo state, that corresponds with the position of the left peak in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Therefore, the interference fringes distribution shows three peaks out of which two encode the same information, namely, the ϕ value of a smoothly driven protocol, while the cen- tral peak contains almost no information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Furthermore, the distribution is quite sharp because, for the parame- ters chosen, the described classes of trajectories are all detected, while more complex quantum trajectories are highly improbable (see Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In panel (b) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 6 the two peaks located at the sides have almost van- ished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' This reveals that when the system is driven at lower relative frequencies, a decay jump or a spontaneous excitation will be detected in almost every trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' A similar effect is obtained if the decay rate Γ/ω increases while keeping the ratio Ω/ω fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' A second aspect of the distribution P[ϕ] is which fea- tures of GPs in open systems it captures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In panel (a) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 6, the fast-driven regime, the ϕ value obtained from protocols with no jumps agrees well with the adiabatic (Berry) phase, and both of these show small but visible deviations from no-jump GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The ϕ value is more closely related to the adiabatic case than the actual GP accu- mulated in smoothly drifted dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' For the slower driving shown in panel (b) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='6, the no-jump ϕ value remains a good indicator of the adiabatic phase, even though registering a smooth protocol is in this case less probable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Under these conditions, most of the experi- ment realizations will contribute the central peak, which is not related to any characteristic GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Inspection of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='6 suggests that, as in the case of the GPs distribution, the interplay between non-adiabatic corrections and environmentally induced jumps is bet- ter revealed when the distribution P[ϕ] is analyzed as a function of the rate Ω/ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' This is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 7, which includes the two paradigmatic cases of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The Berry phase φa and the values φ0 and ¯φ of the GP associated with smooth trajectories and the first moment of the GP distribution are also given for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In the non-adiabatic regime, Ω/ω ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='1 the ϕ value is most of the time the one arising in a protocol with no jumps, and shows appreciable but still small deviations from the adi- abatic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' A trajectory with a single jump might be observed, albeit with less probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' If this is the case, the mixing of the eigenvalues due to non-adiabatic tran- sitions will produce slightly broad distributions around the other two peaks, revealing the stochastic nature of the jump times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Non-adiabatic corrections have a much stronger impact on φ0 (for an analytical expression of this scaling, see Appendix D), its behaviour completely disconnects from that of the the distribution of echo pro- tocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' On the other side, approaching the adiabatic regime, the three peaks get sharper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' This behavior is accompanied by a sharp decrease in the height of the side-peaks and an enhancement of counts on the triv- ial, middle peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Along the full range, there is a region in which the interplay between environmentally-induced and non-adiabatic effects allows for good agreement be- tween the GP accumulated in smooth non-unitary evolu- tion and the value of ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The behavior displayed by both the GP and the echo ”phase” in smooth non-unitary evo- lution is further analyzed in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Differently to the case of the no-jump values, which display a reason- able agreement, the inset in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 7 shows that the (con- sistently re-ranged) first moment of the GP distribution ¯φ remains, along the whole frequency range, completely uncorrelated from both the ϕ distribution and all echo characteristic values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Probability distribution P[ϕ] of ϕ (as determined in an echo experiment) as a function of the ratio Ω/ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The field is oriented with θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='34π and the environment is character- ized by the dissipation rate Γ = 10−3ω and a γz = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The ϕ values are displayed on the y-axis, while the intensity of the count color indicates their probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The solid red line de- picts the adiabatic (Berry) phase φ+ a , while the black dashed line signalizes the no-jump GP φ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The inset shows the prob- ability distribution P[ϕ] accompanied by the first moment of the GP distribution ¯φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The distribution changes radically when γz ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In what follows we discuss the case γz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='1 Γ with Γ = 10−3ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' We start re-considering the two representative cases of fast and slower driving, displayed in panels (a) and (b) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 8 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The first noticeable aspect is that, while three peaks observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 6 (indicated here by the blue contours) can still be detected, they are now coexisting with a broad distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' As visible in panel (a) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 8, the three peaks heights 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='475 10-2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='425 10-3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='375 9 :: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='325 10-4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='275 10-5 10-3 10-2 10-1 P[S m/u11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='275 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='325 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='375 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='425 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='475 ϕ/π 0 1 2 3 4 P[ϕ] 1e 1 (a) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='475 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='480 0 2 4 1e 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='275 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='325 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='375 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='425 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='475 ϕ/π 0 1 2 3 4 5 P[ϕ] 1e 2 (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='3625 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='3875 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='5 1e 1 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Probability distribution P[ϕ] for a magnetic field oriented with θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='34π and driven in a loop at frequencies (a) Ω = 5 × 10−3ω and (b) Ω = 5 × 10−4ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The environment is characterized by the dissipation rate Γ = 10−3ω, and finite γz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='1 Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In both panels, a blue solid contour indicates the γz = 0 distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The solid red line depicts the adiabatic (Berry) phase φ+ a , and the black dashed and dash-dotted lines signalize the GPs obtained in no-jumps and unitary evolution respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Finally, the black dash double-dotted line indi- cates the ϕ value obtained in an echo protocol with no jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The insets zoom in a range in which differences between the reference values, panel (a), and the full magnitude of the cen- tral peak, panel (b), are visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' discussed previously decrease in the presence of γz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The suppression of the peaks is accompanied by the appear- ance of a broad background distribution covering the en- tire range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Panel (b) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 8 attends the slow driving situation, in which the probability to have a jump, and even several, along each trajectory, grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The inclu- sion of the Lz jump modifies the sharp-peaked distribu- tion into a broad one, which covers the entire range of ϕ values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In particular, while the two peaks connected to the no-jump trajectory disappeared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' This happens because the inclusion of this term in the Lindbladian in- duces jumps into states other than the eigenstates of the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In this sense, we may consider the results quite generic, not specifically dependent on the choice of the Lindblad operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In order to get a more complete view of the effect of a finite γz, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 9 shows the distribu- tion of ϕ-values as a function of Ω/ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' For a non-adiabatic evolution in which almost no jumps are detected, the be- havior exhibited by the distribution is similar to that observed in the γz = 0 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' When the velocity of the driving is reduced, gradually favouring the occurrence of jumps, the effect of introducing a finite γz value be- comes more relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The Lz jumps lead to ϕ values that do also depend on the time at which different jumps occurred and hence to the broad background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Probability distribution P[ϕ] as a function of the rate Ω/ω between the frequency Ω at which the magnetic field its rotated and its amplitude ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The field is oriented with θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='34π and the environment is characterized by the dissipation rate Γ = 10−3ω and γz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='1Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The ϕ values are displayed on the y-axis, while the intensity of the count color indicates their probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Extra lines signalize reference GP factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The solid red line indicates the adiabatic (Berry) phase φ+ a , while the black dashed line is the value φ0 extracted from evolution with no jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Summarizing, while the distribution of interference fringes is, in general, quite different from that of the phase accumulated along a single trajectory, the analysis of a spin-echo protocol allows to extract reliable infor- mation on both the no-jump trajectories and the adia- batic (Berry) phase in some regimes of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In the following Section we will concentrate on the no-jump trajectory (corresponding to the side-peaks of the per- sistent probability in the echo-protocol) and show that undergoes a topological transition as a function of the coupling to the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='475 10-2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='425 10-3 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='375 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='325 10-4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='275 10-5 10-3 10-2 10-1 P[] 2/w12 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Topological transitions As already anticipated, we conclude this analysis of GPs in monitored systems by focusing on the no-jump trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' We will show, following in spirit the work in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' [55], that the drift jump-free dynamics encode a topological transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' We would like to emphasize that, although the setting is very much different from that of [55], we believe that the nature of the transition is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Our analysis is a strong hint to the conjecture that this type of transition is rather generic for monitored systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Phase diagram - The GP φ0 given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (10) de- pends, for every fixed θ, on the ratios Ω/ω and Γ/ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' We recall the no-jump trajectory, and therefore the GP associated with it, have no dependence on γz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Plotted as a function of the above-mentioned parameters, the GP shows discrete singularities at critical points, around which it makes a 2π winding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Meanwhile, the probabil- ity associated with this particular trajectory vanishes at these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' We refer to Appendix D for details of the analytical derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='10 shows a color plot of the GP in the Γ − Ω diagram at fixed values of the angle θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The range of the parameters is shown to highlight the singular point and the 2π winding of the GP around it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The white lines indicate the probability for the no-jump trajectory, which approaches zero on reaching the singu- larity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' We will show that the collection of these singular points delimits regions of the parameter space associated with different topological classes of evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' This will be done by defining a topological invariant n ∈ Z (see be- low) and explicitly showing it takes different values over different regions of the parameter-rates plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Topological transition in the no-jump trajectory - Di- rect inspection of the effective drift Hamiltonian shows that if the magnetic field points in the z-direction, the exited eigenstate |ψ+⟩ of H(t) remains fixed in a pole of the Bloch sphere independently of the values taken by the parameter rates Ω/ω and Γ/ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Therefore, the GP associated with the no-jump trajectory identically van- ishes (mod 2π) for θ = 0 and θ = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Without loss of generality, the mod 2π freedom can be eliminated from the GP by simultaneously setting φ0(θ = 0) = 0 and de- manding continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In this way, φ0(θ = π) is completely determined by the evolution and acquires a value φ0(θ = π) = 2π n, (15) where n is an integer number that characterizes the de- pendence of the GP with θ for fixed parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Being an integer, n constitutes a topological invariant be- cause it can not be changed by smoothly deforming φ0(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' As a consequence, if the GP is characterized by different values of n as a function of the various parameters, this will impose the GP to undergo a non-smooth transfor- mation, as the singular behavior exhibited in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In- deed, points in the parameter space slightly to the right and slightly to the left of the singularity (indicated with Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Geometric phase associated with the no-jump trajectory, displayed over a limited region of the parameters plane defined by the ratios Ω/ω and Γ/ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The value of the GP is given by color, as indicated by the bar on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The direction of the field is fixed to θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='34π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' A singu- larity is observed Ω/ω = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='8082 × 10−3 and Γ/ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The crosses indicate points slightly to the left of the sin- gularity (Ω/ω = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='8 × 10−3) and slightly to the right of it (Ω/ω = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='8084 × 10−3), which will be shown to belong to different topological sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' crosses in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 10) give rise to no-jump evolutions asso- ciated with topological invariants n = 0 and n = 1 re- spectively, thus identifying different topological classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' To explicitly show this, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 11 compares the behavior as a function of θ of these GPs by means of showing the difference ∆(θ) between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Given two points, say (1) and (2) and labelled by crosses in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 11, ∆(θ) is defined as ∆(θ) = 1 2π � φ(Γ1,Ω1) 0 − φ(Γ2,Ω2) 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (16) This difference is seen to vanish (up to some smooth small deviations) up to θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='34π, this is, until the angle of the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' At this specific θ value the GP obtained from each parameter rate abruptly deviates, so that their difference shows a step and settles around ∆ = 1 for the remaining range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The different topological numbers n is reflected by the value ∆(π) = 1 for θ = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Over the full parameter space, the GP shows several singularities, with locations that depend on the value of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The set of singular points composes two counter-phase os- cillating curves that define a chain of concatenated closed regions and split the parameter-rate space into an upper and lower region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' This is shown in panel (a) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Parameters within each sector lead to the same n value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The area below the sequence of closed regions is charac- terized by n = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The points given by parameter values Γ = 0 and Ω/ω ≪ 1, defining the adiabatic regime, be- long to this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The regions in between the lines are 1e-2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='09 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='08 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 L 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='06 X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='8078 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='8083 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='8088 m/u 1e-3 Φ/ T13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='00 θ/π 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 ∆(θ) Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' ∆(θ) between GPs computed for points slightly to the right and slightly to the left of the singularity, indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 10 with x’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The GP is, in each case, characterized by a different value of the topological invariant n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' This is reflected in the fact that they differ by 2π for θ = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' topologically trivial sectors with n = 0, while the up- per one is characterized by n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' It is worth pointing out that these topological sectors are not equally proba- ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Besides the singular points of vanishing probability, the probability of attaining a trajectory with no jumps increases as Γ is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' This implies that the upper topological sector is less probable than the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Topological transition in the echo experiment - With the aim of seeking experimentally detectable signatures of the topological transition, we perform a close inspec- tion of the echo experiment that is completed without any jump event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In Section V B, the ϕ value extracted in this case was observed to show good agreement with the adia- batic (Berry) phase for a wide range of frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' How- ever, the close agreement of ϕ with φa will not hold for ar- bitrarily small frequency values, and it will deviate when the ratio Γ/Ω becomes sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 13 shows the ϕ value as a function of the frequency ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' For easy reference and comparison, we consider an environ- ment characterized by the dissipation rate Γ/ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0306, which is included in the ranges exhibited by Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 10 to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' For large frequency ratio, the no-jump ϕ value shows the behavior described in Section V B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' However, ap- proaching smaller frequencies, it shows a highly oscillat- ing step and finally settles in the constant value ϕ ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='375π, associated with a persistence probability 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' This regime will be accessed when the state at the end of the protocol coincides, up to a global phase, with |ψ−(0)⟩, this happens when the smooth drift suppresses the occu- pancy of the exited eigenstate within a cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Full popu- lation transfer from the excited to the ground eigenstate taking place within the evolution cycle requires the sys- tem to be driven at a slow frequency, smaller than that leading to a singular point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' This requirement establishes Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Critical lines dividing the parameters’ plane into different topological classes of the no-jump evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The classes are characterized by different n values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The critical angle θc at which each singular point is found is indicated by a color as described by the bar on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Panels (a) and (b) display different ranges for the rates Ω/ω and Γ/ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' a connection between the value of the echo phase and the topological classes of evolution, as distinctive regimes of ϕ are accessed on one and the other side of the singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' We refer to Appendix D for details on this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The limits of the range along which ϕ shows the step and turns from ∼ φa into the central value are marked, on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 13 with two light blue dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The righter region of the plot corresponds to evolutions characterized by the topological number n = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The range between the light-blue lines corresponds to the densely packed se- quence of topological sectors illustrated by panel (a) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Finally, once on the left of the last vertical line, the evolution is associated with a value n = 1 of the topological number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The inset in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 13 shows ϕ as a function of the dissipation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In this plot, for easy reference and com- parison, the value of the frequency rate is kept fixed at 1e-2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='00 n=1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='75 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='75 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='25 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='50 L 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='25 (a) n=-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='00 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='9 m/u 1e-3 0/元1e-2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='6 n=1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='75 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='2 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 n=0 n=0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='6 n=-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='2 (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='00 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='795 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='800 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='805 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='810 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='815 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='820 2/w 1e-3 0/ T14 Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Dependence of ϕ (black dashed line) obtained in a protocol with no jump events, as a function of the ratio Ω/ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The field is oriented with θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='34π and the environment is characterized by the dissipation rate Γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0306ω, included in the ranges displayed in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 10 - 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The adiabatic (Berry) phase is also indicated for reference, with a red solid line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The inset shows the ϕ value as a function of the rate Γ/ω, with the magnetic field characterized by the same angle θ and Ω/ω = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='8×10−3, coinciding as well with the values used in the previous plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Ω/ω = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='8 × 10−3, also included in the ranges exhibited by Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 10 to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Once again, the ϕ value shows good agreement with the adiabatic phase up to some critical Γ/Ω relation, at which it shows a decreasing step, finally landing at ϕ ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='375π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' As in the main plot, light blue dotted lines mark the limit of the step and split the plot into three distinctive sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The left of the first line corresponds to n = −1 evolution, while the right side of the plot, to n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The space between lines, once again, can be associated with the intermediate zone, which is a single region (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 12 (a)) thus leading to no oscilla- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In summary, a measure of the persistent probabil- ity in an echo protocol carries clear indications of the topological transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The peak structure discussed in Section V B allows to identify the no-jump trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The subsequent analysis of this peak, as summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='13, is sufficient to capture the topological transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' CONCLUSIONS In this paper, we have studied geometric phases in a continuously monitored quantum system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In absence of any coupling to the environment, the cyclic time- dependence of the Hamiltonian leads, in the adiabatic regime, to the Berry phase, and to its consistent gener- alization for a generic unitary evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The presence of an environment induces quantum jumps so that in a single realization of the dynamics the wave function, fol- lowing a given quantum trajectory, accumulates a GP that is itself a stochastic quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' We have analyzed the distribution of GPs by highlighting the interplay between non-adiabatic effects and the influence of the external en- vironment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' We have shown that for slow drivings the dis- tribution of phases is broad because of the several differ- ent occurrences of jumps at random times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' On speeding up the driving, the number of jumps reduces and the dis- tribution becomes peaked around the no-jump trajectory (still deviating from the Berry phase because of the non- adiabatic correction and the non-Hermitian drift term).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' A first quantitative measure of the distribution has been given by the variance, discussed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In order to have experimental access to the GPs along a given trajectory, we have also analyzed a spin-echo protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The structure provided by the jump opera- tors taken together with the possibility of level transi- tions due to non-adiabaticity and the characteristics of the persistence probability can be set in such a way that they lead either to the observation of broad distributions or extremely sharp peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' This interplay should be thus considered in order to be explored as a tool or otherwise, the experiment is rendered uninformative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' We have finally concentrated on the no-jump trajec- tory, showing that it undergoes a topological transition as a function of the dissipation strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Interestingly, this transition is not necessarily connected to singulari- ties occurring in the dynamics of the density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In- deed, for the model considered herein, at the transition point occurring in the no-jump trajectory the behavior of the density matrix is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Despite the striking differ- ences shown between the GP and the interference fringes of an echo experiment, traces of this transition can be observed in the behavior of the interference fringes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In this work, we have considered a specific model for the jump operators corresponding to a well-defined type of monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' However, it is important to understand to which extent the properties we have discussed here de- pend on the type of unravelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' This question might be of particular relevance, especially if one wants to define topological properties associated with Markovian systems starting from the properties of their trajectories (there are infinite ways of unravelling the same Lindblad dynamics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' A glimpse on this question is summarised in Appendix C where we consider an unraveling corre- sponding to a homodyne detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' For what concerns the distribution the qualitative pictures we have outlined in the body of the paper remain valid although important quantitative differences may arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' ACKNOWLEDGMENTS We would like to acknowledge Alessandro Romito for very useful discussions and critical reading of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The work of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' has been supported by the ERC under grant agreement n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='101053159 (RAVE) and by a Google Quantum Research Award.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The work of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=', F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=', and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' is supported by Agencia Nacional de 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='475 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='425 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='375 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='325 10-2 10-1 I/w 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='275 10-3 10-2 10-1 m/u15 Promoci´on Cient´ıfica y Tecnol´ogica (ANPCyT), Consejo Nacional de Investigaciones Cient´ıficas y T´ecnicas (CON- ICET), and Universidad de Buenos Aires (UBA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' acknowledges ICTP-Trieste Associate Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' ac- knowledges that his research has been conducted within the framework of the Trieste Institute for Theoretical Quantum Technologies (TQT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Appendix A: Pancharatnam phase along a quantum trajectory As stated in section II, the quantum trajectory emerg- ing in a single monitored evolution of the system can be understood as intervals of smooth dynamics interrupted at random times by quantum jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Considered in this way, evolution in a time interval t ∈ [0, T] is characterized by an array of jumps of type αi occurring at times ti of the form given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (4), and the parameter t is a con- tinuous variable within the intervals delimited by the ti’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In the quantum jumps approach, the algorithm applied in constructing the trajectories goes as follows [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The time interval [0, T] is discretized into N steps of length δ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' and the state is consistently updated at each time step according to a randomly-decided non-hermitian operator, as described in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Hence, each quantum trajectory can also be thought of from an algorithmic point of view as the ordered collection of states generated by the action of a specific sequence of operators K0,α given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (3), and is in this way a discrete set of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' For a sequence of N discrete pure states, the suitable GP expression is Pancharatnam phase [5, 44, 45], and is given by φP [ψ] = arg ⟨ψ1|ψN⟩ − arg(⟨ψ1|ψ2⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' ⟨ψN−1|ψN⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (A1) The Pancharatnam phase is independent of the U(1) gauge choice and does not require the sequence to close, rely on adiabaticity condition or demand for unitarity, allowing for non-normalized states in the sequence, as long as non of them perfectly vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Exhibiting these characteristics, it becomes a natural definition of GP to be applied to monitored dynamics, in which evolution is generated by non-hermitian operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' It equals the unitary GP associated with the trajectory build-up from joining consecutive states in the sequence by the shortest geodesic in the Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' While this definition does not imply any constraint on the number of states in the sequence by itself, when ap- plied in the context of quantum jumps the number N of states is constrained from below as a consequence of the condition reigning the time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' An evolution in time- interval [0, T] consist of N = T/δ t ≫ 1 states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Split- ting the sequence of states {|ψ1⟩ |ψ2⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' |ψN⟩} into sets starting and ending at those corresponding to the specific times ti where a jump is registered, sets a bridge between this two different descriptions of a quantum trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Each time interval [ti, ti+1], discretized in time-steps of length δt, consist of a number of steps that depend on the specific values of ti and ti+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' From a given jump-time ti, any time-step in the consecutive interval can be found as ti +ki δt, this is, by adding some amount ki ∈ N of in- crements δt, up to some maximum value k∗ i that satisfies ti+1 = ti + k∗ i δt (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 0 T ti |ψ(ti)⟩ ti+1 |ψ(ti + ki δt)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' +ki δt Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Illustrative diagram depicting time interval [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Both the discretization in δt steps and the splitting at jump times ti are indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The relation between times and states is represented as well At each given time, the outcome of a measurement performed on the environment will be associated to the corresponding Kraus operator acting on the system and the state generated by its action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Therefore, there is a to a one-to-one correspondence between the discrete set conforming the time interval and the array of states forming the trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The splitting at jump-times ti can thus be mapped into the trajectory as NJ � i=0 {|ψ(ti + ki δt)⟩ ki = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=', kmax i − 1} (A2) with NJ the number of jumps occurring in the trajectory and the out-bounds indexes i = 0 and i = NJ + 1 signal- ing the entire time-interval limits t0 = 0 and tNJ+1 = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Introducing such a decomposition into the formula for Pancharatnam phase, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (A1) can be re-written as φP = arg ⟨ψ(0)|ψ(T)⟩ − NJ � i=0 k∗ i −1 � ki=1 arg ⟨ψ(ti + ki δt)|ψ(ti + (ki + 1) δt)⟩ − NJ � i=0 arg ⟨ψ(ti)| Kαi |ψ(ti)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (A3) The formula in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (9) for the GP is thus associated with a single trajectory is derived by taking the continuous limit δt/T → 0 within the intervals of smooth evolu- tion [44, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' This expression, more suitable for the exam performed in our work, inherits all the properties of the Pancharatnam phase from which it is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Appendix B: Interference fringes distribution As discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' V B, the distribution of inter- ference fringes from an echo experiment, which we pa- 16 rameterize with ϕ, shows three (sometimes sharp) peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' When γz = 0 only jumps between instantaneous energy eigenstates are possible, and the three peaks emerge from sets of trajectories of a different character as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Smooth trajectories with no jumps generate the pil- ing up in the no-jump value ϕ0 ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='43π 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Trajectories in which at least one decay or spon- taneous excitation jump occurred, projecting the state into an eigenstate |ψ±(ti)⟩ of H(t), give rise to the peak at ϕ ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='375π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Trajectories in which only dephasing jumps took place give rise to the peak at ϕ ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='275π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In this appendix, we provide a detailed justification of this observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' With the aim of providing an acces- sible presentation of the qualitative aspects of the phe- nomena, we will generally disregard the non-hermiticity of the smooth evolution between jumps, thinking of those intervals as unitary (slowly or rapidly driven) evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Hence, this presentation should not be taken as a rigor- ous quantitative analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' We begin with the consideration of the peak (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=') coin- ciding with the no-jump value ϕ0 ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='34π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' As presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' II this smooth trajectory is unique and therefore the exact same value of ϕ will be expected on every case in which this trajectory is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' We thus turn to the case in which jumps are indeed detected, with special care on the anti-intuitive shrink- ing of the distribution in the slower regime in which more jumps are detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' When γz = 0 three jumps are possi- ble within our unravelling of the Lindblad equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Two out of these three jumps project the state into an energy eigenstate, namely, decay jumps and spontaneous excita- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Whenever a jump of this kind takes place at some instant of time ti, immediately after the jump the state of the system turns into |Ψ(ti)⟩ = ei ξ(ti)+i φ(ti) |ψ±(ti)⟩ (B1) with ξ(ti) the dynamical phase and φ(ti) the geometrical phase, given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (9) accumulated up to the occur- rence of the jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' If the protocol ends immediately after, the persistence probability PR = | ⟨ψ(0)|ψ(2T)⟩ |2 = 1/2 preserves no information on either the GP or the specific characteristics of the jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' If, on the other hand, the system continues to evolve, the possibility of obtaining any information on a phase or the jump time will rely on the interplay between the non-adiabatic transitions and the existence of further jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' If the evolution continues from the first jump on, this will happen smoothly until ei- ther the protocol is finished or another jump takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Different regimes of Ω/ω give rise to the smooth evolution of different natures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' If the protocol is performed slowly enough, this smooth evolution is (almost) transition-free and |ψ(t)⟩ ∼ ei ξ(t>ti)+i φ(t>ti) |ψ±(t > ti)⟩, so the re- sult obtained for the persistent probability remains to be PR = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Moreover, this regime favors the occurrence of further jumps, thus reinforcing the erasing of infor- mation by re-projecting into eigenstates of H(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The complete independence of the result on the times ti of the jumps makes this peak (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=') extremely sharp in the slow regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' On the other hand, if the system is driven faster, along the smooth evolution after the jump the state develops contributions from the other eigenstate due to non-adiabatic effects, favoring the emergence of relative phases and becoming |ψ(t > ti)⟩ = A±(t > ti) |ψ±(t > ti)⟩ +A∓(t − ti) |ψ∓(t − ti)⟩ (B2) with A ± (t) the amplitudes for each eigenstate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In such a situation, the persistence probability depends on ti, leading to the broadening observed in the central peak of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='7 for faster driving, while still not trivially connected to the GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' As anticipated in the previous paragraphs, each jump of this kind will erase all information on the phases and any dependence on previous jump times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The possibility of further erasing events is mitigated in faster protocols by the reduction of exposure to the environ- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The third peak (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=') observed in the distribution at ϕ ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='475π can be understood by adding dephasing jumps to the previous discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' A dephasing jump has the effect of introducing a π shift in the relative phase of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' If the evolution afterward remains transition- less (and no erasing jumps occur at any point), the evo- lution resembles that of the adiabatic echo experiment up to corrections that can be disregarded, so the per- sistence probability takes the value P ∼ sin2(2φa) (with cos replaced by sin due to the relative π shift).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' This situation leads to a well-defined single ϕ-value which is independent of the time ti at which the jump took place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Therefore, in the slow-driving range, a well-defined peak emerges, that might however be small, as in this regime decay jumps are likely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' As the magnetic field is rotated faster, non-adiabatic effects induce a dependence on ti on the persistence probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' This dependence on ti is in- herited by the ”phases” extracted, and thus responsible for the broadening of the distribution observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 7 for larger Ω/ω values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The inclusion of a jump operator ∝ σz modifies this three-peaked distribution by leading to a broad back- ground which is present even in the case in which it is not the dominant process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The Kz jumps promote the development of relative phases as they mix eigenstates of the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Even if the system has, at some time, transitioned to an eigenstate, suffering from a σz-jump suddenly drags it away into a superposition state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Appendix C: Dependence on the unravelling: field displacement Another paradigmatic quantum trajectories scheme arising from a different unraveling of the master equa- tion is that of the so-called diffusive trajectories, in which 17 the monitored quantities produce continuously fluctuat- ing signals instead of discontinuous jumps [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' This is the prototypical scheme of continuous or ideal homo- dyne detection, which can be theoretically obtained as a limiting case of the mentioned discrete homodyne de- tection [59, 60, 67, 72, 73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The master equation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (1) is invariant under the transformation H(t) → H′(t) = H(t) − √ λ i 2 � α (Kα − K† α) Kα → K′ α = Kα + √ λ I, (C1) where √ λ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Therefore it is possible to substitute Kα and H(t) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (1) by K′ α and H′(t) without modi- fying the averaged dynamics of the system and unravel it using the standard direct detection (quantum jumps) scheme applied before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' When the reservoir is assumed to be made of harmonic modes, like electromagnetic radi- ation, adding the displacement √ λ to the Lindblad op- erators corresponds to the implementation of homodyne detection [60, 72, 74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In this case, taking √ λ suitably large leads to a measurement of the quadrature of the system dipole Kα + K† α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' However, in order to keep the collapse probability per step small, it would be necessary to reduce the time step and hence increase the simula- tion time by the same order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' For this reason, we refrain to consider finite large √ λ values in this section and fo- cus on the modifications suffered by P[φ] for smaller √ λ values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 15 we present, also for the case of this differ- ent unravelling of the Linbland equation, the two cases in which the driving is performed faster or slow enough for the hypothetical unitary dynamics to be considered adiabatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' As in the previous cases, the environment re- mains fixed with Γ = 10−3ω and γz = 0, and we have taken λ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='5 × 10−5ω < Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The two cases are shown in panels (a) and (b) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='15 respectively, where we also plot the no-jump and unitary GPs, and the average of the distribution for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Striking differences from the case of direct detection arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' For this λ/ω ≪ 1, the ref- erence values displayed remain close to those obtained in the λ = 0, while the distributions behave differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In the fast-driven case displayed in panel (a), the expected increase in jumps is reflected by the decrease of the sharp peak piling up from no-jump trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' However, the formerly broad, but still uneven background, has now turned into a completely uniform distribution in which each phase value (but the no-jump) is evenly probable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The described behavior is reinforced when the system is driven at slower frequency rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The previously broad while single-peaked distribution lacks, in a system mon- itored through the operators K′ α forming the new basis, of any structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 φ/π 0 1 2 3 4 P[φ] 1e 1 (a) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='47 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='475 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='48 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='485 0 1 2 3 1e 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 φ/π 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='5 P[φ] 1e 2 (b) Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Probability distribution P[φ] of GP values ob- tained in an unravelling with K′ α and H′(t) operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The magnetic field is oriented at θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='34π and driven in a cycle at frequencies (a) Ω = 5×10−3ω and (b) Ω = 5×10−4ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The en- vironment is characterized by the dissipation rate Γ = 10−3ω and γz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' We have taken λ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='5 × 10−5ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In both pan- els, a blue solid contour recalls the distributions obtained in the original unraveling considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Extra lines indicate the new reference GP values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The black dashed and dot-dashed lines signalize the GPs φ0 and φu associated with no-jump and general unitary evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The black dotted line shows the first moment of the distribution ¯φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Appendix D: Smooth evolution with no jumps: Analytic approach We provide in this Appendix some additional analyt- ical results for the no-jumps evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' As previously mentioned, this particular case can be thought of as gen- erated by the non-hermitian Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (8), in such a way that a non-normalized state ��� ˜ψ(t) � will follow Schrodinger’s equation i d dt ��� ˜ψ(t) � = Ho(t) ��� ˜ψ(t) � (D1) 18 where Ho(t) is not only non-hermitian but also ex- plicitly time-dependent due to the function f(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The effective drift Hamiltonian shares eigenstates with H(t), but the eigenvalues associated with these eigen- states are now complex and time-dependent, given by ±ω/2 [1 − i Γ/(2ω)f(t)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The dynamics of the normalized state of the system |ψ(t)⟩ = ��� ˜ψ(t) � �� ˜ψ(t) ��� ˜ψ(t) � (D2) will be governed by the more involved, nonlinear equation which is found by jointly differentiating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (D2) and making use of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (D1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The not-normalized state can be expanded into the instantaneous eigenstates of Ho(t), was ��� ˜ψ(t) � = ˜c+ |ψ+(t)⟩+˜c− |ψ−(t)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Explicit computation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (D1) leads to the following differential equations for the coef- ficients ˜c±(t) ˙˜c± = � ∓iω 2 − iΩ 2 (1 ∓ cos(θ)) ∓ Γ 4 f(t) � ˜c±(t) + iΩ 2 sin(θ) ˜c∓(t), (D3) where the real term ∼ −Γ˜c+(t) indicates that even in the case with no jumps, the presence of the environment favors state transitions, as the amplitude of the excited eigenstate is suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Taking into account the nor- malization procedure involved in turning from the not- normalized state into the real, normalized one, this sup- pression implies a population transfer from the excited eigenstate into the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' As a consequence, any trivial implementation of the adiabatic approximation is prevented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' A second feature observed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (D3) is that, for the parameters chosen in this work, a good agreement can be obtained by replacing f(t) with its mean value f(t) ∼ 1 − sin2(θ)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' By performing this replacement, dynamics become easily solvable in the rotating frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The smooth evolution of each eigenstate of the system is, within this approximation, given by ��ψ(±)(t) � = N± e−iΩ/2 t �� ±(ν + ε)e−iε/2 t ∓ (ν − ε)eiε/2 t� |ψ±(t)⟩ −Ω sin(θ) |ψ∓(t)⟩} , (D4) where both ν and ε are complex quantities given by ν = ω − Ω cos(θ) − i Γ/2(1 − sin2(θ)/2) and ε = � ν2 + Ω2 sin2(θ), N± is a normalization factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' At this point, it should be stressed that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (D4) explicitly shows how the state ��ψ(±)(t) � obtained when evolving an eigen- state will not be equal to the instantaneous eigenstate at a later time in the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Geometric phase - The GP associated with a trajec- tory in which no jumps can be explicitly computed from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' While the general expression is quite involved, it takes, for small rates Ω/ω ∼ Γ/ω of the driving fre- quency and the dissipation rate to the amplitude of the magnetic field, the form φ0 ∼ − π(1 − cos(θ)) (D5) − π sin2(θ) �Ω ω + cos(θ)Ω2 ω2 � − sin2(θ) 4 �Ω ω + cos(θ)Ω2 ω2 � e−4π Im(ν)/Ω − 1 2 Im(ν)/Ω , where the first term in the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='s is the Berry phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The term in the second line of the equation is the main cor- rection originating exclusively from non-adiabaticity, in otherwise unitary evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The third line accounts for the non-trivial effect of the environment in the no-jump evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' As Γ → 0 this term turns into a further con- tribution due to non-adiabaticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Phase diagram singularities - When computing the accumulated GPs analyzed in Sections V A and V C we have taken |ψ+(0)⟩ as our initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Thus, a vanish- ing probability for observing this particular trajectory, of the kind observed at the GP singular points, requires |ψ(T)⟩ ∝ |ψ−(0)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Considering the cyclic character of the instantaneous eigenstates, this means a singular point will take place whenever a full population transfer is achieved exactly in a time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' It was already inferred from the differential equations governing the evolution of the ˜c± coefficients, that the dynamics generated by the effective drift Hamiltonian Ho(t) favored transitions from the excited to the ground instantaneous eigenstate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' As long as the original approximation remains accurate, the singular points of the GP will be defined through the equation (ν + ε) − (ν − ε)e2iπε/Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' No-jump interference fringe - In Section V B, we have studied the interference fringes of an echo experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' For this purpose, we’ve defined the convenient parameter ϕ given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Restricting to the case of a proto- col performed without registering any jump, it was shown that the value of ϕ displays, generally, better agreement with the Berry phase than with the GP φ0 accumulated by the state of the system under equal conditions, this is, when it is smoothly driven along one period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' As long as the no-jump value ϕ ∼ φa, good agreement between this “phase” and the GP will be obtained when the second and third lines in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (D5) are sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' However, it is worth noting that the ϕ value will not remain close to the Berry phase for arbitrarily small driving frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' While the protocol has shown to be less sensitive to both non-adiabatic and environmentally induced effects than the GP, it will account for the non- ideal conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' It was already shown that the environ- ment induces population transfer from the excited to the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' The asymmetry between the smooth evo- lution of each eigenstate should be expected to prevent, at some point, the cancellation of the dynamical evolu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Figure 13 illustrates this situation, by showing the 19 ϕ value as a function of the rate between the driving fre- quency and the field amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' While for larger rates ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='01 the phase reproduces the behavior discussed in Section V B, this situation does not hold if the rate is lowered enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' At some critical value, the parameter extracted from the echo protocol starts deviating from the adiabatic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' A rather singular situation arises when the state at the end of the protocol coincides, up to a global phase, with |ψ−(0)⟩, so that the persistence probability turns P = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' As a consequence, the ϕ ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content='375π value ob- served in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' 13, trivially associated with P ∼ 1/2 by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (13) is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' In this case, the three peaks ob- served in the distribution P[ϕ] (see Section V B) merge into a single, central peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' This regime is accessed when full population transfer occurs within a cycle and the system reaches a steady state ∼ |ψ−(t)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' As we have discussed above, the parameters leading to full popula- tion transfer at the exact time of a cycle correspond to singular points of the GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Then, full population trans- fer within a cycle implies 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with a last spin rotatio taking the final state back to the σz basis, where the actually compute proba- bility is that of being in |0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' [71] Different measurement schemes and physical situations can be described recurring to symmetries of the Lindb- land equation as a way of generating different unraveling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Given the invariance of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' (1) under some joint trans- formation Wm → W ′ m, H → H′, the Lindblad evolution of the averaged density matrix ρ(t) is be consequently unchanged, while the different possible trajectories may undergo nontrivial changes, therefore describing differ- ent scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Such a procedure can be followed to go from direct photodetection to discrete homodyne detec- tion schemes, in which a beam-splitter mixes the output field with an aditional coherent field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' [72] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Wiseman and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Milburn, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' A 47, 642 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' [73] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Percival, Quantum state diffusion (Cambridge Univer- sity Press, 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' [74] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Es’haqi-Sani, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Manzano, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Zambrini, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Fazio, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} +page_content=' Research 2, 023101 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E2T4oBgHgl3EQf8wmh/content/2301.04222v1.pdf'} diff --git a/BtA0T4oBgHgl3EQfAP8-/content/tmp_files/2301.01959v1.pdf.txt b/BtA0T4oBgHgl3EQfAP8-/content/tmp_files/2301.01959v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a50b9ce1b0b2a0f8e4f431f66c4c6431f3864f7d --- /dev/null +++ b/BtA0T4oBgHgl3EQfAP8-/content/tmp_files/2301.01959v1.pdf.txt @@ -0,0 +1,2371 @@ +1 + +Application of Machine Learning to Sporadic +Experimental Data for Understanding Epitaxial Strain +Relaxation + +Jin Young Oh1, Dongwon Shin2,*, and Woo Seok Choi1,* +1Department of Physics, Sungkyunkwan University, Suwon 16419, Republic of Korea +2Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, +USA +Corresponding author e-mail : shind@ornl.gov, choiws@skku.edu + + +2 + +ABSTRACT +Understanding epitaxial strain relaxation is one of the key challenges in functional thin films +with strong structure-property relation. Herein, we employ an emerging data analytics approach +to quantitatively evaluate the underlying relationships between critical thickness (hc) of strain +relaxation and various physical and chemical features, despite the sporadic experimental data +points available. First, we have collected and refined reported hc of perovskite oxide thin +film/substrate system to construct a consistent sub-dataset which captures a common trend +among the varying experimental details. Then, we employ correlation analyses and feature +engineering to find the most relevant feature set which include Poisson’s ratio and lattice +mismatch. With the insight offered by correlation analyses and feature engineering, machine +learning (ML) models have been trained to deduce a decent accuracy, which has been further +validated experimentally. The demonstrated framework is expected to be efficiently extended +to the other classes of thin films in understanding hc. +KEYWORDS: epitaxial strain, perovskite oxide, pulsed laser deposition, machine learning + + + + +3 + +1. Introduction +Epitaxial strain and its relaxation mechanism in transition metal oxide thin films and +heterostructures are critical for understanding and tailoring the strain-induced emergent +functional properties.1-3 The physical and chemical properties of perovskite oxide thin films +are strongly affected by the microscopic lattice structure via sensitive structure-property +relation, primarily altered by epitaxial strain and its relaxation. The in-plane lattice constant of +the thin film follows that of the substrate up to a specific thickness, defined as the critical +thickness (hc) typically in the range of a few tens of nanometers because of the epitaxial strain +imposed by the substrate.4, 5 For films of thickness above hc, epitaxial strain relaxation occurs +and the in-plane lattice constant returns to the original bulk value concomitantly with the +introduction of dislocations.6-8 By studying and assessing hc in various perovskite oxide thin +films and heterostructures, the fundamental correlation between hc and epitaxial strain would +lead to a better understanding of the strain relaxation mechanism in general. +The People-Bean (PB) model is one of the most comprehensive and successful approaches +for predicting hc.9-11 It is a phenomenological model that considers the energies of strain and +dislocation within the thin film.12 It compares the strain energy density, 2G +1 + v +1 − v hε2 (where G +is the shear modulus, ν is the Poisson ratio, which is the ratio between the out-of-plane (εoop) +and in-plane lattice mismatch (εip), h is the thickness, and ε is the lattice mismatch; G, ν, and h +are the intrinsic values of the thin film), with the dislocation energy density, +Gb2 +8π√2afilm ln( +h +b ) +(where afilm is the in-plane lattice constant of the film in the bulk phase, and b is the Burger’s +vector, which is proportional to afilm). When the strain energy density exceeds the dislocation +energy density at hc = +b (1 − v) +40π (1 + v) +1 +ε2 ln( +hc +b ), misfit dislocations start to be created with epitaxial +strain relaxation. The PB model has been used to successfully predict the hc of various + +4 + +perovskite oxide thin film systems, including LaAlO3 (LAO) and PbTiO3 thin films on SrTiO3 +(STO) substrates and BaTiO3 thin films on Scandate substrates.13, 14 +The data analytics approach is an emerging tool in materials science and condensed matter +physics with practical problem-solving abilities. For example, it can be applied to constructing +a magnetic phase diagram by predicting the Néel temperatures of cubic lattices and ferroelectric +phase diagram from experimental Raman spectra.15, 16 The approach was also employed to +characterize structural dynamics in glassy liquids and predict the yield strength of high- +temperature Cr alloy.17, 18 Specifically for the case of perovskite oxides, machine learning (ML) +was used to predict thermodynamic stabilities,19 lattice constants,20 thermal expansion,21 and +the synthesizability of new compounds.22 +We propose that the approach can be further applied to efficiently identify the correlation +between various input features and the hc of perovskite oxide thin films by considering several +different factors that allow going beyond the PB model. Despite the effective predictability of +hc, the PB model also has limitations in being universally applied. For example, the PB model +often fails to predict hc in systems with unconventional strain-relaxation characteristics, such +as ferroelastic thin films or low mismatched systems.23, 24 If adequately applied, the data +analytics approach will include the concerted effect from various parameters, such as synthesis +methods, growth conditions, and type of materials, in determining the epitaxial strain relaxation. +Additionally, a quantitative ranking of the relevant parameters in terms of their importance in +determining hc will be possible through correlation analyses. Finally, new augmented +functional forms based on combined features can be created to reach high correlations, +providing insights in understanding the epitaxial strain relaxation. +In this study, we perform data analytics using correlation analyses and feature engineering +to train ML models to understand the epitaxial strain relaxation of perovskite oxide epitaxial + +5 + +thin films. We explain the data analytics process adopted in the current study in Section. 2. We +discuss the challenges and limitations in applying the data analytics to actual experimental data, +which are inconsistent and sporadic. In Section. 3, we present the data analytics results and +discuss the epitaxial strain relaxation mechanism in terms of the PB model. We conclude our +study in Section. 4, and briefly explain experimental process for validating the data analytics +in Section. 5. + +2. Data Analytics Process: Challenges and Suggested Resolutions +The data analytics process, illustrated in Figure 1, consists of four steps: (1) Dataset +construction, (2) correlation analyses, (3) feature set compilation, and (4) ML model training. +Below, we list challenges and resolutions in each step. +2.1. Dataset Construction +We collected 82 experimentally reported hc for the perovskite oxide thin films, as shown in +Table S1. Due to the sporadic nature of the data points, we encountered practical challenges in +introducing consistent features that comprehensively capture the various experimental +conditions. For example, our dataset contains data from six growth methods and 11 different +substrates. For the growth of the same La0.7Sr0.3MnO3/LAO (001) system, magnetron +sputtering and pulsed laser deposition (PLD) result in drastically different hc of 2.5 and 12 nm, +respectively,25, 26 highlighting the influence of the growth method on hc. On the other hand, +substrates with a significant lattice mismatch or orthorhombic crystal structure would further +complicate the analyses. Hence, we compiled a relatively small yet highly consistent dataset +by grouping only data points with a similar pedigree (Figure 1a). Our final dataset comprises +23 data points, as shown in Table S2. We have selected the results for the thin films grown on +STO, LAO, and (LaAlO3)0.3(Sr2TaAlO6)0.7 (LSAT) (001) substrates by PLD.26-42 Despite small + +6 + +size of the dataset, the following approach was found to be efficient in assessing the epitaxial +strain relaxation and predicting hc. +2.2. Correlation Analyses +Correlation analyses let us quantitatively examine the contribution of individual features +quantitatively and develop physical conclusions (Figure 1b). The analyses identify key +physical/chemical features to determine hc based on two distinct correlation coefficients. +Maximal information coefficient (MIC) quantifies nonlinear correlations, and Pearson +correlation coefficient (PCC) describes linear correlations with either positive or negative +correlations.43 +2.3. Feature Set Compilation +Feature engineering, a process of adjusting features is necessary for achieving realistic and +reliable data analytics results.44, 45 With the information of correlation scores, we adopted +various physical hypotheses describing the relation between hc and epitaxial strain relaxation +for the feature set compilation. The optimum feature set found via this process will be used for +the ML training. This study classifies features into three categories: ionic, phase, and PB model +features (Figure 1b and Table 1). Growth-related parameters, such as thermal expansion +coefficient and growth temperature might be considered as important features in determining +hc. However, our correlation analyses showed small correlation scores for those parameters, +implying that the growth procedure does not strongly affect hc. The application of a physical +hypothesis for each feature set (Table 2) is justified as follows. In set A, we speculated the +ionic properties, including atomic weight, electron affinity, electronegativity, ionization energy, +ionic radius, and oxidation state of individual ions in perovskite structures for the thin film and +substrate, might influence hc. Considerations related to the oxidation states of constituent ions +were applied to all ionic features. In set B, the general phase features related to epitaxial strain, + +7 + +including afilm, ν, and ε, were essential in determining hc. In set C, the PB model features were +selected to examine the validity of the PB model, including PB factor, XPB = afilm +1 − v +1 + v +1 +ε2; strain +energy density factor, ES = G +1 + v +1 − v ε2; and dislocation energy density factor, ED = G afilm +ln(afilm). These features were directly adopted from the PB model, but the scale constants were +eliminated to reduce them into the simplest numerical form. We also omitted h from the original +PB formulas to remove the self-recurring thickness effect. These combined features were +expected to provide a concerted approach in understanding hc. Set D includes both PB model +features and phase features simultaneously, which lets us examine any synergetic effect +between the features. +2.4. ML Model Training +ML model training was performed by using an open-source data analytics frontend, +Advanced data SCiEnce toolkit for Non-Data Scientist (ASCENDS) (Figure 1d).46 ML models +were trained for a given dataset and feature sets while changing detailed conditions, such as +the type of algorithm and scaler, which are intrinsic training parameters. We also tuned the +hyperparameter corresponding to the scaler used. Four algorithms, i.e., nearest neighbor (NN) +regression, kernel ridge (KR) regression, Bayesian ridge (BR) regression, and support vector +machine (SVM), were adopted to train the models.46 The NN47, KR48, 49, BR50, 51, and SVM52 +regression models were utilized as four representative ML models. NN model employs the +results of the k-nearest neighbors' average values for the given data points. The function only +takes a portion of the pertinent dataset because it can only be approximated locally. KR is one +of the non-parametric forms of ridge regression that combines the kernel technique and ridge +regression. It develops a linear model in the implicit feature space caused by the appropriate +kernel and data. KR simplifies the computation of inner products in a high-dimensional space + +8 + +by employing the kernel approach. It correlates to a non-linear function in the original space +for the non-linear kernels. BR model is a linear-based model, which assumes a relationship +between the input and output variables by fitting a linear equation. Instead of employing point +estimates, BR formulates a linear relationship using probability distributors. SVM can handle +both classification and regression issues. SVM creates a set of hyperplanes in high-dimensional +space to classify the data points for a classification task. SVM is more versatile for regression +problems by enclosing the function in the ε-insensitive region (ε-tube). To balance model +complexity and prediction error in the SVM regression, this tube reformulates the regression +problem to identify the function that deviates from the acquired targets throughout all training +data the least. ASCENDS saves metadata regarding each training model so that deviation of +accuracy (R2) can be calculated by using ten times of trial. The ML model with the highest +accuracy was trained using features of high correlation. We have further validated the ML +model by comparing its estimated hc value with the directly obtained experimental hc value +from an example not available in the literature. + +3. Results and Discussion +The result of correlation analyses (Figure 1b) for the relevant features are shown in Figure +2. Despite the differences between MIC and PCC, XPB and ES commonly show high correlation +scores. The MIC (absolute PCC) scores are 0.687 (0.950) and 0.687 (0.629) for XPB and ES, +respectively. In contrast, the correlation scores of ED are 0.435 and 0.025 for MIC and PCC, +respectively, which is significantly lower than those of XPB and ES. This suggests that XPB and +ES are more important than ED, among the PB features. Because XPB and ES are different from +ED in that they contain both ν and ε, it can be further inferred that a combination of ν and ε is +critical in constructing the most efficient feature. The result is more intriguing because + +9 + +individual ν or ε alone do not exhibit particularly high correlation scores, yet the combined +features of XPB and ES become the most physically relevant. Note that the overall correlation +scores of ionic features in set A (Figure S2) are lower than the ones of set C, and set D with +XPB, ES. +To compile the features (Figure 1c), we compare the training results of each feature set from +feature engineering (Table 2). The results show that the feature sets C and D, including the PB +model features, are highly reliable. Figure 3a exhibits the R2 of the ML model for the feature +sets presented in Table 2. For set A, all algorithms produced R2 < 0.8. The low R2 values and +large deviations imply that the feature set does not represent a valid physical situation. This +result is not surprising because the ionic features do not consider any interaction between the +film and the substrate. Only the BR algorithm results in a reasonable accuracy for set B, +suggesting that set B does not contain enough critical features for predicting hc. Notably, it can +be inferred that individual v and ε are insufficient to construct a valid prediction model. On the +other hand, set C exhibits consistently high R2 values, indicating good model training. Set D +also shows high R2 values similar to set C. From the results of sets C and D, it is evident that +the PB features are crucial in determining hc. Figure 3b presents an example of the ML training +results obtained using set C and the BR algorithm, with the highest R2 value of 0.87. The +diagonal grey region has a slope of 1, indicating the correspondence of the predicted and actual +experimental values of hc. The prediction is encouraging, especially considering experimental +uncertainty and sporadic data points, and confirms feature engineering has successfully +deduced reliable feature sets. +With the insight from the results of correlation analyses and feature set compilation, we +predicted with the trained ML surrogate models (Figure 1d), as summarized in Figure 4. The +hc of the STO/LSAT system is predicted by various ML models based on different feature sets + +10 + +and algorithms. Figure 4a shows the hc values obtained by the ML models from sets A, B, C, +and D (vertical bars) with the BR algorithm. The predictions using feature sets A (42.7 ± 1.5 +nm) and B (36.5 ± 6.2 nm) largely underestimates hc. On the other hand, using feature sets C +(78.8 ± 1.0 nm) and D (77.2 ± 4.2 nm) the predictions lie just beneath the PB model calculation +result (86.4 nm, red horizontal dashed line). We further compare the algorithm-dependent +results of using sets C (Figure 4b) and D (Figure 4c). Set C produces more consistent results +with less variation among different algorithms. This reiterates that the individual features of v +and ε included in set D might obscure the effective model construction. Their augmented form +is essential in understanding the epitaxial strain relaxation. +For more realistic validation of our ML model, we compared our results to the experimental +result. The X-ray reflectivity (XRR) results of four samples with thicknesses of 37.2, 72.0, 88.5, +and 117.0 nm are shown in Figure S1. X-ray diffraction reciprocal space map (XRD-RSM) +measurements were taken for the samples to investigate the epitaxial strain relaxation. As the +epitaxial strain relaxes with increasing thickness, additional Bragg peaks (i.e., relaxed regions) +emerge, breaking the mirror symmetry of the original Bragg peak (i.e., strained region). The +upper panels in Figure S1c show the RSMs around the substrate LSAT (103) and the film STO +(103) peaks. The regions marked by white boxes are magnified in the lower panels to assess +the strain relaxation of the STO thin film in further detail. The peaks are symmetric for the 37.2 +and 72.0 nm films, but they become progressively asymmetric for the 88.5 and 117.0 nm films, +suggesting that the strain relaxation occurs between 72.0 and 88.5 nm. This strain relaxation +behavior was further quantitatively examined using a bi-Gaussian fitting of the STO (103) peak +(Figure 5 and S2). Bi-Gaussian function, which has a distinct standard deviation for the left +(W1) and right half (W2) of the peak, is an effective tool for quantifying the asymmetry that +originates from strain relaxation. The line profiles through the black lines in the lower panels + +11 + +of Figure S1c and their bi-Gaussian fittings (red lines) are plotted in Figure S3a. As shown in +Figure 5, (W1 − W2)/W2, the normalized difference between the width on the left and right side +increases dramatically as the film thickness ≥ 88.5 nm, the thickness near which the strain +relaxation begins. Therefore, the hc of the STO/LSAT system was experimentally determined +to be 72.0 – 88.5 nm and it is consistent with our ML model result. +Whereas the original PB model show decent prediction of hc as expected (Fig. S4a), the data +analytics approach provides hidden insight of the epitaxial strain relaxation of perovskite oxide +thin film system. Particularly, we note that the strong correlations between hc and various +features are not evident when hc values from the literature are directly plotted (Figure S4). For +example, Figures S4b-d show hc values plotted as functions of ES, v, and ε. This emphasizes +the merit of applying data analytics which quantitatively characterize the augmented features +of XPB and ES to be essential in understanding the epitaxial strain relaxation. As discussed +previously, both XPB and ES include the parameters v and ε, yet feature sets including pure v +and ε, do not result in particularly high precision in predicting hc. Physically, this might imply +that independent information on either the value of the in-plane lattice structure, ε, or the +relationship between the in-plane and out-of-plane lattice structure, v, does not provide +meaningful understanding of the epitaxial strain relaxation. Again, the augmented features of +XPB and ES are critical, as the elastic modulation of the thin film should be interpreted as a +three-dimensional phenomenon. + +4. Conclusion +We demonstrate the feasibility of establishing a streamlined data analytics workflow to +efficiently evaluate and introduce relevant features that capture the physics of strain relaxation +in epitaxial thin films. First, we collected various experimental hc data which are sporadic in + +12 + +nature. Second, we augmented physical/chemical features for detailed correlation analyses. +Third, we refined the dataset into consistent sub-dataset by applying the result of correlation +analyses and prevailing physical conditions, which inevitably reduced our dataset. Despite the +small number of sporadic data, our carefully chosen conditions were proven to be highly +consistent. Consequently, the data analytics process presented in the current study based on the +PB model provides an obvious first step for understanding hc by quantitatively identifying key +features (i.e., the Poisson’s ratio v and the lattice mismatch ε) that affect the epitaxial strain +relaxation. We experimentally validated the predicted hc of STO thin films grown on LSAT +(001) substrates, showing a good agreement. This study introduces challenges in ML approach +using sporadic experimental dataset and proposes its systematic solutions. By doing so, we +emphasize that using refined dataset within the context of modern data analytics can help +achieving a better understanding. In particular, the quantitative analyses of ML successfully +provide us with the unique physical insight about three-dimensional nature of epitaxial strain +relaxation mechanism intuitively by focusing on the key features. Initiating a framework for +understanding the epitaxial strain relaxation would inspire the community to consistently +collect/compile the dataset. Furthermore, we anticipate that the demonstrated data analytics +approach can be further applied beyond the example used in the present study. + +5. Experimental Section +We experimentally fabricated epitaxial STO thin films on LSAT substrates and determined +the actual range of hc to validate the data analytics approach (Figure S1). The system was +selected because it was not available in the literature, so pure prediction is possible. STO thin +films were fabricated on LSAT (001) substrate using PLD. We grew the film at 750 °C and +100 mTorr of O2 partial pressure, using a KrF excimer laser (248 nm; IPEX-868, + +13 + +Lightmachinery) with 1.5 J cm-2 of fluence and 5 Hz of repetition rate. The thicknesses of the +STO thin films were determined by XRR (PANalytical X’Pert and a Rigaku Smartlab XRD), +as the films had atomically sharp surfaces and interfaces. + +Acknowledgements +This work was supported by the Basic Science Research Programs through the National +Research Foundation of Korea (NRF) (NRF-2021R1A2C201134012). + +ORCID +Woo Seok Choi https://orcid.org/0000-0002-2872-6191 + +References +1. B. Kim, P. Liu, J. M. 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Neural Comput. 1992; 4:415-47. +51. M. E. Tipping. Sparse bayesian learning and the relevance vector machine. J. Mach. Learn. +Res. 2001; 1:211–44. +52. R. Khanna and M. Awad, "Efficient Learning Machines: Theories, Concepts, and +Applications for Engineers and System Designers." Apress, (2015). + + + + + + + + + + + + +20 + + +FIGURE 1. Schematic workflow of modern data analytics for predicting hc. a) Experimental +hc values of perovskite oxide thin films on STO, LAO, and LSAT (001) substrates fabricated +by PLD are collected for the dataset construction. b) Quantitative correlation analyses provide +insight into epitaxial strain relaxation by highlighting the underlying correlation. c) Various +physical features and feature sets were examined and constructed to create an ideal feature for +data analytics. d) Model prediction was applied to predict hc and compare it with the +experimental value. + + +Prediction +STO,LAO,LSAT (001) +100 +F R=0.87 +of hc +Substrates +Actual h. (nm) +80 +(a) +Pulsed Laser +Dataset +60 +Deposition(PLD +Construction +40 +PerovskiteOxide +20 +Thin Films +0 +20 +40 +60 +80 +100 +Predicted hc (nm) +IBulkState +1.0 +(b) +Positive +(d) +Misfit +Negative +MLModel +Dislocation +Correlation +Target: +Analysis +Training +he ++Strained +State +0.4 +Substrate +0.2 +00 +XpB +afilm +ED +Features +lonic,Phase-baseo +(c) +PBModel-inspired +Identifying +Feature Set +Features +Compilation +KeyFeatures1.0 +1.0 +(a) +(b) +Positive +Negative +0.8 +0.8 +0.6 +0.6 +MIC +PCC +0.4 +0.4 +0.2 +0.2 +0.0 +0.0 +XpB +Es +afilm +Ep +XpB +Es +a film +Ep +Features +Features21 + +FIGURE 2. Correlation analyses in MIC and PCC for the features in set D. Correlation +coefficients in a) MIC and b) PCC. For the PCC, red (blue) bars represent positive (negative) +correlations. + +FIGURE 3. Accuracy (R2) of each feature set sorted by different algorithms. a) The graph +shows the R2 values of the model training using four algorithms, NN, KR, BR, and SVM. b) +The ML training result of set C using the BR algorithm with the highest R2 value. Each point +represents experimental data with hc values with their predicted values. The thick gray line +represents a slope of 1. BR, Bayesian ridge; KR, kernel ridge; NN, nearest neighbor; SVM, +support vector machine. + + +(a) +1.0 +NN +KR +BR +SVM +(b) 100 +R2 = 0.87 +0.9 +0.8 +80 +0.7 +Accuracy (R2) +0.6 +60 +0.5 +0.4 +40 +0.3 +0.2 +20 +0.1 +0.0 +0 +A +B +C +D +0 +20 +40 +60 +80 +100 +Feature set +Predicted h. (nm)(a) +100 +(b) +100 +(c) +100 +PB calculation +Set C +Set D +80 +80 +80 +h。 (nm) +60 +60 +60 +40 +40 +40 +20 +20 +20 +0 +A +B +c +D +NN +KR +BR +SVM +NN +KR +BR +SVM +Featureset +Algorithm +Algorithm22 + +FIGURE 4. Comparison between the ML estimated, PB model calculated, and experimental +values of hc for the STO/LSAT system. a) hc values with error bars (standard deviation) +obtained by the ML models using sets A, B, C, and D (purple vertical bars) with the best- +performing BR algorithm. hc values with error bars (standard deviation) obtained by the ML +models using b) set C and c) set D based on different algorithms (colored bars). Direct PB +model calculation (red horizontal dashed line) are also shown for comparison. + +FIGURE 5. Asymmetrical line profiles of the STO film peak and epitaxial strain relaxation. +Black solid circles represent experimental (W1 − W2)/W2 values plotted as a function of the +film thickness, quantitatively characterizing the peak asymmetry, and hence, the epitaxial strain + +Set B +Set A +People-Bean Model +60 +Set +Set C +50 +Fully relaxed +(W,-W2)/W2 (%) +40 +30 +Partially relaxed +20 +Fully strained +10 +Experimental h。 +30 +40 +50 +60 +70 +80 +90 +100 110 120 130 +thickness (nm)23 + +relaxation. The vertical bars represent the ML predicted hc values from various feature sets, +where their thicknesses correspond to the error bars. The hc values of Set C and Set D lie well +between experimental hc region indicated by an arrow. + + +Ionic features +Atomic weight +Electron affinity +Electronegativity +Ionization energy +Ionic radius +Oxidation states +Phase features +Definition +Symbol +Lattice constant of film +afilm +afilm +Lattice mismatch +afilm − asub +asub + +ε +Poisson ratio +− 𝜀oop +𝜀ip +v +PB model features +Definition +Symbol +PB factor +afilm 1 – v +1 + v 1 +ε2 +XPB +Strain energy density factor +G 1 + v +1 – v ε2 +ES +Dislocation energy density factor +G afilm ln(afilm) +ED + +24 + +TABLE 1. Description of features used in data analytics. Definitions and symbols of each +individual and combined feature used for data analytics. The features were classified into three +types: ionic features, phase features, and PB model features. + +Feature set +Features +A +Ionic features +B +afilm, ν, ε +C +XPB, Es, ED +D +XPB, Es, ED, afilm, ν, ε + +TABLE 2. Corresponding features of each feature set. The table lists the four feature sets using +features that correspond under different physical hypotheses: ionic features set A, phase +features set B, PB features set C, combined features set D. + +SUPPORTING INFORMATION +Additional supporting information may be found in the online version of the article at the +publisher’s website. + + + + + + +25 + +Supplemental Materials + + +Application of Machine Learning to Sporadic +Experimental Data for Understanding Epitaxial Strain +Relaxation + +Jin Young Oh1, Dongwon Shin2,*, and Woo Seok Choi1,* +1Department of Physics, Sungkyunkwan University, Suwon 16419, Republic of Korea +2Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, +USA + +Corresponding author e-mail : shind@ornl.gov, choiws@skku.edu + + + + + + + + + + +26 + + +FIGURE S1. Characterization of the strain state of STO thin films on LSAT substrate. (a) +XRR, (b) θ-2θ, and (c) XRD-RSM data for epitaxial STO thin films on LSAT (001) substrates +with representative thicknesses. The asterisk in (b) indicates the LSAT (002) substrate peak +and the peak around 46.1° marks the STO (002) peak. + + + +1 +2 +3 +44 +45 +46 +47 +48 +49 +(b) +37.2 nm +72.0 +88.5 +117.0 +Intensity (arb.units) +2q (degrees) +(a) +Intensity (arb. units) +2q (degrees) +* +0.256 +0.260 +0.760 +0.765 +qz (Å-1) +qx (Å-1) +0.256 +0.260 +0.256 +0.260 +0.256 +0.260 +0.25 +0.26 +0.76 +0.78 +STO +(103) +qz (Å-1) +37.2 +LSAT +(103) +(c) +0.25 +0.26 +72.0 +0.25 +0.26 +88.5 +0.25 +0.26 +117.0 + +27 + + +FIGURE S2. Correlation analyses in MIC and PCC for the features in set A. Correlation +coefficients for set A, ionic feature set. Overall correlation scores are relatively lower than set +C, and Set D. + + + + + + + +IRBF +AWBF +IRAF +AWBS +OSBF +IRAS +OSAF +AWAF +IEAF +ENBF +IRBS +EAAS +AWAS +ENAS +OSAS +IEAS +EABF +EAAF +ENAF +ENBS +IEBS +EABS +IEBF +OSBS +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +AWBF +ENBF +AWAF +IRAF +ENAF +IEAF +IRBF +EABF +IEBF +EAAF +OSAF +OSBF +AWAS +EAAS +EABS +ENAS +ENBS +IEAS +OSAS +IRAS +AWBS +IRBS +IEBS +OSBS +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 + Positive + Negative +(b) +Features +PCC +(a) +MIC +Features + +28 + + +FIGURE S3. Detailed information of the asymmetric plot. (a) Line profiles crossing the STO +(103) peak and bi-Gaussian fitting. (b) The table summarizes the (W1 − W2)/W2 values. + + +Thickness (nm) 37.2 +72.0 +88.5 +117.0 +W1 (10-4⋅Å-1) +6.61 +4.87 +5.42 +6.33 +W2 (10-4⋅Å-1) +5.87 +4.31 +4.60 +3.99 +(W1 − W2)/W2 +12.61% +12.99% +17.82% +58.64% +(b) + +29 + +FIGURE S4. Correlation between hc and various features. A plot of actual hc with respect to +various features, including (a) hc from PB model, (b) Es (c) v, and (d) ε. + + + + + + + + +-3 +-2 +-1 +0 +1 +2 +3 +0 +20 +40 +60 +80 +100 +0.25 +0.30 +0.35 +0.40 +0 +20 +40 +60 +80 +100 +0 +400 +800 +1200 +1600 +0 +20 +40 +60 +80 +100 +0 +20 +40 +60 +80 +100 +120 +0 +20 +40 +60 +80 +100 + + +hc (nm) +e (%) +(a) +hc (nm) +n +hc (nm) +ES (GPa) +(d) +(b) +(c) +hc (nm) +hc from PB model (nm) + +30 + +Author/year +DOI link +Thin film +Substrate hc +H.L.Ju 1998 +10.1063/1.367864 +La0.66Sr0.33MnO3 +LaAlO3 +10 +H.L.Ju 1998 +10.1063/1.367864 +La0.66Sr0.33MnO3 +LaAlO3 +30 +A.Tebano 2006 +10.1103/PhysRevB.74.245116 +La0.7Sr0.3MnO3 +LaAlO3 +12 +S.H.Lim 2007 +10.1002/adfm.200700055 +BiFeO3 +SrTiO3 +50 +Y.H.Chu 2007 +10.1063/1.2750524 +BiFeO3 +SrTiO3 +30 +S.Geprags 2011 +http://mediatum.ub.tum.de/?id=1091602 BiFeO3 +SrTiO3 +22 +J.L.Maurice 2003 +10.1080/14786430310001603436 +La0.66Sr0.33MnO3 +SrTiO3 +100 +P.M.Vaghefi 2017 +10.1088/1361-6463/aa80bf +La0.7Sr0.3MnO3 +SrTiO3 +100 +L. 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+10.1103/PhysRevLett.98.217602 +PbZr0.2Ti0.8O3 +SrTiO3 +40 +S.Venkatesan 2009 +10.1103/PhysRevB.78.104112 +TbMnO3 +SrTiO3 +5 +Y.Dai 2016 +10.1063/1.4962853 +Sr0.63Ba0.37TiO3 +DyScO3 +40 +Y.Dai 2016 +10.1063/1.4962853 +Sr0.875Ba0.125TiO3 +DyScO3 +18 +Y.Dai 2016 +10.1063/1.4962853 +SrTiO3 +DyScO3 +11 +Y.Dai 2016 +10.1063/1.4962853 +SrTiO3 +TbScO3 +9 +Y.Dai 2016 +10.1063/1.4962853 +SrTiO3 +GdScO3 +5 +W.S.Choi 2012 +10.1021/nl302562f +LaCoO3 +LaAlO3 +26 +S.Zhong 2006 +10.1557/jmr.2006.0193 +PbZr0.2Ti0.8O3 +SrTiO3 +12 +A.Herpers 2014 +10.1063/1.4900817 +Pr0.48Ca0.52MnO3 +SrTiO3 +1.5 +D.Fuchs 2002 +10.1063/1.1461897 +LaAlO3 +LSAT +3 +D.Fuchs 2002 +10.1063/1.1461897 +SrTiO3 +LSAT +30 +D.Fuchs 2002 +10.1063/1.1461897 +La0.4Sr0.6CoO3 +LSAT +90 +X.Wang 2012 +10.1080/14786435.2012.657709 +BaTiO3 +SrTiO3 +2 +X.Wang 2012 +10.1080/14786435.2012.657709 +BaTiO3 +SrTiO3 +4 +S.Jan 2016 +10.14279/depositonce-4997 +NaNbO3 +NdGaO3 +20 +S.Jan 2016 +10.14279/depositonce-4997 +NaNbO3 +DyScO3 +27 +T.Wang 2013 +10.1063/1.4833248 +SrTiO3 +LSAT +180 +C.M.Foster 1998 +10.1063/1.360121 +PbTiO3 +SrTiO3 +150 +C.M.Foster 1998 +10.1063/1.360121 +PbTiO3 +MgO +10 +C.M.Foster 1998 +10.1063/1.360121 +PbTiO3 +LaAlO3 +10 +S.H.Oh 2004 +10.1063/1.1690484 +SrRuO3 +SrTiO3 +10 + +31 + + +B.S.Kwak 1992 +10.1103/PhysRevLett.68.3733 +PbTiO3 +KTaO3 +34 +B.S.Kwak 1992 +10.1103/PhysRevLett.68.3733 +PbTiO3 +KTaO3 +250 +K.Hirai 2013 +10.1063/1.4817505 +SrFeO2.5 +DyScO3 +50 +R.A.Rao 1999 +10.1063/1.122749 +La0.8Ca0.2MnO3 +SrTiO3 +25 +R.A.Rao 1999 +10.1063/1.122749 +La0.8Ca0.2MnO3 +LaAlO3 +5 +L.Peng 2003 +10.1063/1.1631055 +SrTiO3 +LaAlO3 +50 +L.Peng 2003 +10.1063/1.1631055 +SrTiO3 +MgO +50 +J.Zhang 2001 +10.1088/0953-8984/23/33/334211 +La0.05Ba0.95MnO3 +SrTiO3 +20 +G.Catalan 2005 +10.1103/PhysRevB.72.020102 +Ba0.5Sr0.5TiO3 +MgO +100 +G.Catalan 2005 +10.1103/PhysRevB.72.020102 +Ba0.5Sr0.5TiO3 +MgO +200 +G.Gao 2007 +10.1063/1.2429903 +La0.7Ca0.3MnO3 +SrTiO3 +30 +D.H.Kim 2008 +10.1063/1.2830799 +BiFeO3 +SrTiO3 +90 +T.L.Meyer 2015 +10.1063/1.4937170 +La1.85Sr0.15CuO4 +SrTiO3 +15 +T.L.Meyer 2015 +10.1063/1.4937170 +La1.85Sr0.15CuO4 +LaAlO3 +80 +T.L.Meyer 2015 +10.1063/1.4937170 +La1.85Sr0.15CuO4 +LaSrAlO4 35 +V.V.Mehta 2015 +10.1103/PhysRevB.91.144418 +LaCoO3 +SrTiO3 +15 +V.V.Mehta 2015 +10.1103/PhysRevB.91.144418 +LaCoO3 +SrTiO3 +73 +V.V.Mehta 2015 +10.1103/PhysRevB.91.144418 +LaCoO3 +LSAT +15 +V.V.Mehta 2015 +10.1103/PhysRevB.91.144418 +LaCoO3 +LSAT +73 +V.V.Mehta 2015 +10.1103/PhysRevB.91.144418 +LaCoO3 +LaAlO3 +8 +Y.B.Xu 2016 +10.1038/srep35172 +PbTiO3 +LaAlO3 +45 +A.I.Khan 2014 +10.1063/1.4885551 +PbZr0.2Ti0.8O3 +SrTiO3 +40 +E.Breckenfeld 2013 10.1039/c3tc31653j +Sr1.04TiO3 +NdGaO3 +60 +E.Breckenfeld 2013 10.1039/c3tc31653j +Sr0.96TiO3 +NdGaO3 +300 +S.Gariglio 2007 +10.1063/1.2740171 +PbZr0.2Ti0.8O3 +SrTiO3 +25 +V.Pena 2006 +10.1016/j.jpcs.2005.10.022 +La0.67Ca0.33MnO3 +LaSrAlO4 5 +Xuan Shen 2015 +10.1063/1.4906430 +SrZr0.68Ti0.32O3 +Ge +11.4 +H.P.Sun 2004 +10.1063/1.1728300 +BaTiO3 +SrTiO3 +50 +H.P.Sun 2004 +10.1063/1.1728300 +BaTiO3 +SrTiO3 +20 +K.Daoudi 2010 +10.1016/j.jallcom.2010.07.035 +La0.7Sr0.3CoO3 +SrTiO3 +100 +M.D. Biegalski 2008 10.1063/1.3037216 +SrTiO3 +DyScO3 +200 +J.Santiso 2016 +10.1021/acsami.6b02896 +La0.7Sr0.3MnO3 +LaAlO3 +30 +A.Petraru 2007 +10.1063/1.2745277 +BaTiO3 +SrTiO3 +10 +K.Saito 2006 +10.1143/JJAP.45.7311 +BiFeO3 +SrTiO3 +90 +Z.Fu 2017 +10.1063/1.4975342 +BiFeO3 +LSAT +15 +H.Terauchi 1992 +10.1143/JPSJ.61.2194 +BaTiO3 +SrTiO3 +80 +Y.C.Liang 2005 +10.1016/j.tsf.2005.07.187 +La0.7Ba0.3MnO3 +SrTiO3 +35 +Y.C.Liang 2005 +10.1016/j.tsf.2005.07.187 +La0.7Ba0.3MnO3 +SrTiO3 +34 +S.Stemmer 1995 +10.1002/pssa.2211470115 +PbTiO3 +SrTiO3 +50 +S.Stemmer 1995 +10.1002/pssa.2211470115 +PbTiO3 +SrTiO3 +100 +S.Stemmer 1995 +10.1002/pssa.2211470115 +PbTiO3 +SrTiO3 +15 +A.Duk 2013 +10.1063/1.4794405 +NaNbO3 +TbScO3 +73 +Y.Wu 2011 +10.1063/1.3567297 +(BiScO3)0.36(PbTiO3)0.64 SrTiO3 +15 +F.Sandiumenge 2016 10.1002/admi.201600106 +La0.7Sr0.3MnO3 +LaAlO3 +73 +F.Sandiumenge 2016 10.1002/admi.201600106 +La0.7Sr0.3MnO3 +LaAlO3 +8 +M.Kuroda 2018 +10.1063/1.5007332 +SmFeO3 +LaAlO3 +45 + +32 + + +TABLE S1. Original dataset. The dataset includes 82 data on hc of perovskite oxide thin films +from the literature. hc values are expressed in unit of nm. The highlighted data are the ones +used for ML training. + + + +33 + +hc +10 +30 +12 +50 +30 +22 +100 +100 +100 +10 +12 +5 +10 +3.2 +5 +10 +10 +10 +15 +15 +10 +42 +80 +Dislocation energy +density factor +-28.6545 +-28.6545 +-28.6545 +-14.6732 +-14.6732 +-14.6732 +-28.6545 +-28.6545 +-28.6545 +-38.6099 +-25.2942 +-25.2942 +-25.2942 +-25.2942 +-25.2942 +-25.2942 +-25.2942 +-25.2942 +-25.2942 +-14.6732 +-25.2942 +-17.993 +-17.993 +Strain energy +density factor +814.844 +814.844 +915.442 +147.363 +147.363 +147.363 +95.6127 +95.6127 +95.6127 +1550.54 +595.836 +595.836 +595.836 +595.836 +595.836 +595.836 +595.836 +595.836 +595.836 +147.363 +595.836 +80.4425 +80.4425 +PB factor +0.03709 +0.03709 +0.03302 +0.10749 +0.10749 +0.10749 +0.31612 +0.31612 +0.31612 +0.02567 +0.04623 +0.04623 +0.04623 +0.04623 +0.04623 +0.04623 +0.04623 +0.04623 +0.04623 +0.10749 +0.04623 +0.23306 +0.23306 +Poisson ratio +0.35 +0.35 +0.4 +0.3 +0.3 +0.3 +0.35 +0.35 +0.35 +0.26 +0.27 +0.27 +0.27 +0.27 +0.27 +0.27 +0.27 +0.27 +0.27 +0.3 +0.27 +0.26 +0.26 +Lattice mismatch +2.24274 +2.24274 +2.24274 +1.40845 +1.40845 +1.40845 +-0.76825 +-0.76825 +-0.76825 +-2.94494 +2.22791 +2.22791 +2.22791 +2.22791 +2.22791 +2.22791 +2.22791 +2.22791 +2.22791 +1.40845 +2.22791 +-0.98242 +-0.98242 +Lattice constant +0.3875 +0.3875 +0.3875 +0.396 +0.396 +0.396 +0.3875 +0.3875 +0.3875 +0.379 +0.3992 +0.3992 +0.3992 +0.3992 +0.3992 +0.3992 +0.3992 +0.3992 +0.3992 +0.396 +0.3992 +0.383 +0.383 +OSBS +3 +3 +3 +4 +4 +4 +4 +4 +4 +4 +4 +4 +4 +4 +4 +4 +4 +4 +4 +4 +4 +3.41 +3.41 +IRBS +0.535 +0.535 +0.535 +0.605 +0.605 +0.605 +0.605 +0.605 +0.605 +0.605 +0.605 +0.605 +0.605 +0.605 +0.605 +0.605 +0.605 +0.605 +0.605 +0.605 +0.605 +0.594 +0.594 +IEBS +28.45 +28.45 +28.45 +43.27 +43.27 +43.27 +43.27 +43.27 +43.27 +43.27 +43.27 +43.27 +43.27 +43.27 +43.27 +43.27 +43.27 +43.27 +43.27 +43.27 +43.27 +29.9 +29.9 +ENBS +1.61 +1.61 +1.61 +1.54 +1.54 +1.54 +1.54 +1.54 +1.54 +1.54 +1.54 +1.54 +1.54 +1.54 +1.54 +1.54 +1.54 +1.54 +1.54 +1.54 +1.54 +1.5649 +1.5649 +EABS +0.4328 +0.4328 +0.4328 +0.084 +0.084 +0.084 +0.084 +0.084 +0.084 +0.084 +0.084 +0.084 +0.084 +0.084 +0.084 +0.084 +0.084 +0.084 +0.084 +0.084 +0.084 +0.3869 +0.3869 +AWBS +26.98 +26.98 +26.98 +47.87 +47.87 +47.87 +47.87 +47.87 +47.87 +47.87 +47.87 +47.87 +47.87 +47.87 +47.87 +47.87 +47.87 +47.87 +47.87 +47.87 +47.87 +90.09 +90.09 +OSAS +3 +3 +3 +2 +2 +2 +2 +2 +2 +2 +2 +2 +2 +2 +2 +2 +2 +2 +2 +2 +2 +2.18 +2.18 +IRAS +1.36 +1.36 +1.36 +1.44 +1.44 +1.44 +1.44 +1.44 +1.44 +1.44 +1.44 +1.44 +1.44 +1.44 +1.44 +1.44 +1.44 +1.44 +1.44 +1.44 +1.44 +1.54 +1.54 +IEAS +19.18 +19.18 +19.18 +11.03 +11.03 +11.03 +11.03 +11.03 +11.03 +11.03 +11.03 +11.03 +11.03 +11.03 +11.03 +11.03 +11.03 +11.03 +11.03 +11.03 +11.03 +12.5 +12.5 +ENAS +1.1 +1.1 +1.1 +0.95 +0.95 +0.95 +0.95 +0.95 +0.95 +0.95 +0.95 +0.95 +0.95 +0.95 +0.95 +0.95 +0.95 +0.95 +0.95 +0.95 +0.95 +0.977 +0.977 +EAAS +0.47 +0.47 +0.47 +0.05206 +0.05206 +0.05206 +0.05206 +0.05206 +0.05206 +0.05206 +0.05206 +0.05206 +0.05206 +0.05206 +0.05206 +0.05206 +0.05206 +0.05206 +0.05206 +0.05206 +0.05206 +0.12724 +0.12724 +AWAS +138.9 +138.9 +138.9 +87.62 +87.62 +87.62 +87.62 +87.62 +87.62 +87.62 +87.62 +87.62 +87.62 +87.62 +87.62 +87.62 +87.62 +87.62 +87.62 +87.62 +87.62 +96.85 +96.85 + +34 + +OSBF +3.3 +3.3 +3.3 +3 +3 +3 +3.3 +3.3 +3.3 +3 +4 +4 +4 +4 +4 +4 +4 +4 +4 +3 +4 +4 +4 +IRBF +0.5577 +0.5577 +0.565 +0.55 +0.55 +0.55 +0.5577 +0.565 +0.565 +0.535 +0.605 +0.605 +0.605 +0.605 +0.605 +0.605 +0.605 +0.605 +0.605 +0.55 +0.605 +0.53 +0.53 +IEBF +39.1182 +39.1182 +38.929 +30.65 +30.65 +30.65 +39.1182 +38.929 +38.929 +28.45 +43.27 +43.27 +43.27 +43.27 +43.27 +43.27 +43.27 +43.27 +43.27 +30.65 +43.27 +51.3 +51.3 +ENBF +1.55 +1.55 +1.55 +1.96 +1.96 +1.96 +1.55 +1.55 +1.55 +1.61 +1.54 +1.54 +1.54 +1.54 +1.54 +1.54 +1.54 +1.54 +1.54 +1.96 +1.54 +1.88 +1.88 +EABF +0 +0 +0 +0.151 +0.151 +0.151 +0 +0 +0 +0.4328 +0.084 +0.084 +0.084 +0.084 +0.084 +0.084 +0.084 +0.084 +0.084 +0.151 +0.084 +0.6633 +0.6633 +AWBF +54.94 +54.94 +54.94 +55.85 +55.85 +55.85 +54.94 +54.94 +54.94 +26.98 +47.87 +47.87 +47.87 +47.87 +47.87 +47.87 +47.87 +47.87 +47.87 +55.85 +47.87 +58.93 +58.93 +OSAF +2.64 +2.64 +2.7 +3 +3 +3 +2.64 +2.7 +2.7 +3 +2 +2 +2 +2 +2 +2 +2 +2 +2 +3 +2 +2 +2 +IRAF +1.3728 +1.3728 +1.384 +1.17 +1.17 +1.17 +1.3728 +1.384 +1.384 +1.36 +1.61 +1.61 +1.61 +1.61 +1.61 +1.61 +1.61 +1.61 +1.61 +1.17 +1.61 +1.44 +1.44 +IEAF +16.2987 +16.2987 +16.735 +25.56 +25.56 +25.56 +16.2987 +16.735 +16.735 +19.18 +10 +10 +10 +10 +10 +10 +10 +10 +10 +25.56 +10 +11.03 +11.03 +ENAF +1.0395 +1.0395 +1.055 +2.02 +2.02 +2.02 +1.0395 +1.055 +1.055 +1.1 +0.89 +0.89 +0.89 +0.89 +0.89 +0.89 +0.89 +0.89 +0.89 +2.02 +0.89 +0.95 +0.95 +EAAF +0.32738 +0.32738 +0.34462 +0.9424 +0.9424 +0.9424 +0.32738 +0.34462 +0.34462 +0.47 +0.1446 +0.1446 +0.1446 +0.1446 +0.1446 +0.1446 +0.1446 +0.1446 +0.1446 +0.9424 +0.1446 +0.05206 +0.05206 +AWAF +120.589 +120.589 +123.516 +209 +209 +209 +120.589 +123.516 +123.516 +138.9 +137.3 +137.3 +137.3 +137.3 +137.3 +137.3 +137.3 +137.3 +137.3 +209 +137.3 +87.62 +87.62 +Substrate +LaAlO3 +LaAlO3 +LaAlO3 +SrTiO3 +SrTiO3 +SrTiO3 +SrTiO3 +SrTiO3 +SrTiO3 +SrTiO3 +SrTiO3 +SrTiO3 +SrTiO3 +SrTiO3 +SrTiO3 +SrTiO3 +SrTiO3 +SrTiO3 +SrTiO3 +SrTiO3 +SrTiO3 +LSAT +LSAT +Thin film +La0.66Sr0.33Mn +O3 +La0.66Sr0.33Mn +O3 +La0.7Sr0.3MnO +3 +BiFeO3 +BiFeO3 +BiFeO3 +La0.66Sr0.33Mn +O3 +La0.7Sr0.3MnO +3 +La0.7Sr0.3MnO +3 +LaAlO3 +BaTiO3 +BaTiO3 +BaTiO3 +BaTiO3 +BaTiO3 +BaTiO3 +BaTiO3 +BaTiO3 +BaTiO3 +BiFeO3 +BaTiO3 +SrCoO3 +SrCoO3 +Author/year +H.L.Ju 1998 +H.L.Ju 1998 +A.Tebano 2006 +S.H.Lim 2007 +Y.H.Chu 2007 +S.Geprags 2011 +J.L.Maurice 2003 +P.M.Vaghefi 2017 +L. Ranno 2002 +L.Qiao 2011 +T.Suzuki 1999 +A.Visinoiu 2002 +R.Guo 2016 +J.Zhu 2006 +M.Fujimoto 2002 +M.Fujimoto 2002 +G.H.Lee 2001 +H.I.Seo 2020 +H.I.Seo 2020 +H.J.Lee 2016 +Y.S.Kim 2005 +D.Zhang 2019 +D.Zhang 2019 + +TABLE S2. Final dataset. The refined dataset with 23 data on hc of perovskite oxide thin films +used for ML training. Ionic features, phase features, and PB model features are also shown. +Lattice constant and PB factor are expressed in unit of nm. Strain energy density factor and +dislocation energy density factor are expressed in units of GPa and GPa∙nm respectively. Ionic +features are presented using the following notation: AW (atomic weight), EA, (electron affinity), +EN (electronegativity), IE (ionization energy), IR (ionic radius), OS (oxidation states), of A or +B cation in film (F) or substrate (S) material of perovskite ABO3. + diff --git a/BtA0T4oBgHgl3EQfAP8-/content/tmp_files/load_file.txt b/BtA0T4oBgHgl3EQfAP8-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..56fc7bd08a1f5a514637852095aa56d496a307a1 --- /dev/null +++ b/BtA0T4oBgHgl3EQfAP8-/content/tmp_files/load_file.txt @@ -0,0 +1,2225 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf,len=2224 +page_content='1 Application of Machine Learning to Sporadic Experimental Data for Understanding Epitaxial Strain Relaxation Jin Young Oh1, Dongwon Shin2,*, and Woo Seok Choi1,* 1Department of Physics, Sungkyunkwan University, Suwon 16419, Republic of Korea 2Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Corresponding author e-mail : shind@ornl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='gov, choiws@skku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='edu 2 ABSTRACT Understanding epitaxial strain relaxation is one of the key challenges in functional thin films with strong structure-property relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Herein, we employ an emerging data analytics approach to quantitatively evaluate the underlying relationships between critical thickness (hc) of strain relaxation and various physical and chemical features, despite the sporadic experimental data points available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' First, we have collected and refined reported hc of perovskite oxide thin film/substrate system to construct a consistent sub-dataset which captures a common trend among the varying experimental details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Then, we employ correlation analyses and feature engineering to find the most relevant feature set which include Poisson’s ratio and lattice mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' With the insight offered by correlation analyses and feature engineering, machine learning (ML) models have been trained to deduce a decent accuracy, which has been further validated experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' The demonstrated framework is expected to be efficiently extended to the other classes of thin films in understanding hc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' KEYWORDS: epitaxial strain, perovskite oxide, pulsed laser deposition, machine learning 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Introduction Epitaxial strain and its relaxation mechanism in transition metal oxide thin films and heterostructures are critical for understanding and tailoring the strain-induced emergent functional properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='1-3 The physical and chemical properties of perovskite oxide thin films are strongly affected by the microscopic lattice structure via sensitive structure-property relation, primarily altered by epitaxial strain and its relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' The in-plane lattice constant of the thin film follows that of the substrate up to a specific thickness, defined as the critical thickness (hc) typically in the range of a few tens of nanometers because of the epitaxial strain imposed by the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='4, 5 For films of thickness above hc, epitaxial strain relaxation occurs and the in-plane lattice constant returns to the original bulk value concomitantly with the introduction of dislocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='6-8 By studying and assessing hc in various perovskite oxide thin films and heterostructures, the fundamental correlation between hc and epitaxial strain would lead to a better understanding of the strain relaxation mechanism in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' The People-Bean (PB) model is one of the most comprehensive and successful approaches for predicting hc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='9-11 It is a phenomenological model that considers the energies of strain and dislocation within the thin film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='12 It compares the strain energy density, 2G 1 + v 1 − v hε2 (where G is the shear modulus, ν is the Poisson ratio, which is the ratio between the out-of-plane (εoop) and in-plane lattice mismatch (εip), h is the thickness, and ε is the lattice mismatch;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' G, ν, and h are the intrinsic values of the thin film), with the dislocation energy density, Gb2 8π√2afilm ln( h b ) (where afilm is the in-plane lattice constant of the film in the bulk phase, and b is the Burger’s vector, which is proportional to afilm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' When the strain energy density exceeds the dislocation energy density at hc = b (1 − v) 40π (1 + v) 1 ε2 ln( hc b ), misfit dislocations start to be created with epitaxial strain relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' The PB model has been used to successfully predict the hc of various 4 perovskite oxide thin film systems, including LaAlO3 (LAO) and PbTiO3 thin films on SrTiO3 (STO) substrates and BaTiO3 thin films on Scandate substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='13, 14 The data analytics approach is an emerging tool in materials science and condensed matter physics with practical problem-solving abilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' For example, it can be applied to constructing a magnetic phase diagram by predicting the Néel temperatures of cubic lattices and ferroelectric phase diagram from experimental Raman spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='15, 16 The approach was also employed to characterize structural dynamics in glassy liquids and predict the yield strength of high- temperature Cr alloy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='17, 18 Specifically for the case of perovskite oxides, machine learning (ML) was used to predict thermodynamic stabilities,19 lattice constants,20 thermal expansion,21 and the synthesizability of new compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='22 We propose that the approach can be further applied to efficiently identify the correlation between various input features and the hc of perovskite oxide thin films by considering several different factors that allow going beyond the PB model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Despite the effective predictability of hc, the PB model also has limitations in being universally applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' For example, the PB model often fails to predict hc in systems with unconventional strain-relaxation characteristics, such as ferroelastic thin films or low mismatched systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='23, 24 If adequately applied, the data analytics approach will include the concerted effect from various parameters, such as synthesis methods, growth conditions, and type of materials, in determining the epitaxial strain relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Additionally, a quantitative ranking of the relevant parameters in terms of their importance in determining hc will be possible through correlation analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Finally, new augmented functional forms based on combined features can be created to reach high correlations, providing insights in understanding the epitaxial strain relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' In this study, we perform data analytics using correlation analyses and feature engineering to train ML models to understand the epitaxial strain relaxation of perovskite oxide epitaxial 5 thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' We explain the data analytics process adopted in the current study in Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' We discuss the challenges and limitations in applying the data analytics to actual experimental data, which are inconsistent and sporadic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' In Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' 3, we present the data analytics results and discuss the epitaxial strain relaxation mechanism in terms of the PB model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' We conclude our study in Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' 4, and briefly explain experimental process for validating the data analytics in Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Data Analytics Process: Challenges and Suggested Resolutions The data analytics process, illustrated in Figure 1, consists of four steps: (1) Dataset construction, (2) correlation analyses, (3) feature set compilation, and (4) ML model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Below, we list challenges and resolutions in each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Dataset Construction We collected 82 experimentally reported hc for the perovskite oxide thin films, as shown in Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Due to the sporadic nature of the data points, we encountered practical challenges in introducing consistent features that comprehensively capture the various experimental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' For example, our dataset contains data from six growth methods and 11 different substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' For the growth of the same La0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='7Sr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='3MnO3/LAO (001) system, magnetron sputtering and pulsed laser deposition (PLD) result in drastically different hc of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='5 and 12 nm, respectively,25, 26 highlighting the influence of the growth method on hc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' On the other hand, substrates with a significant lattice mismatch or orthorhombic crystal structure would further complicate the analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Hence, we compiled a relatively small yet highly consistent dataset by grouping only data points with a similar pedigree (Figure 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Our final dataset comprises 23 data points, as shown in Table S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' We have selected the results for the thin films grown on STO, LAO, and (LaAlO3)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='3(Sr2TaAlO6)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='7 (LSAT) (001) substrates by PLD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='26-42 Despite small 6 size of the dataset, the following approach was found to be efficient in assessing the epitaxial strain relaxation and predicting hc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Correlation Analyses Correlation analyses let us quantitatively examine the contribution of individual features quantitatively and develop physical conclusions (Figure 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' The analyses identify key physical/chemical features to determine hc based on two distinct correlation coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Maximal information coefficient (MIC) quantifies nonlinear correlations, and Pearson correlation coefficient (PCC) describes linear correlations with either positive or negative correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='43 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Feature Set Compilation Feature engineering, a process of adjusting features is necessary for achieving realistic and reliable data analytics results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='44, 45 With the information of correlation scores, we adopted various physical hypotheses describing the relation between hc and epitaxial strain relaxation for the feature set compilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' The optimum feature set found via this process will be used for the ML training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' This study classifies features into three categories: ionic, phase, and PB model features (Figure 1b and Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Growth-related parameters, such as thermal expansion coefficient and growth temperature might be considered as important features in determining hc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' However, our correlation analyses showed small correlation scores for those parameters, implying that the growth procedure does not strongly affect hc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' The application of a physical hypothesis for each feature set (Table 2) is justified as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' In set A, we speculated the ionic properties, including atomic weight, electron affinity, electronegativity, ionization energy, ionic radius, and oxidation state of individual ions in perovskite structures for the thin film and substrate, might influence hc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Considerations related to the oxidation states of constituent ions were applied to all ionic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' In set B, the general phase features related to epitaxial strain, 7 including afilm, ν, and ε, were essential in determining hc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' In set C, the PB model features were selected to examine the validity of the PB model, including PB factor, XPB = afilm 1 − v 1 + v 1 ε2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' strain energy density factor, ES = G 1 + v 1 − v ε2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' and dislocation energy density factor, ED = G afilm ln(afilm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' These features were directly adopted from the PB model, but the scale constants were eliminated to reduce them into the simplest numerical form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' We also omitted h from the original PB formulas to remove the self-recurring thickness effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' These combined features were expected to provide a concerted approach in understanding hc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Set D includes both PB model features and phase features simultaneously, which lets us examine any synergetic effect between the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' ML Model Training ML model training was performed by using an open-source data analytics frontend, Advanced data SCiEnce toolkit for Non-Data Scientist (ASCENDS) (Figure 1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='46 ML models were trained for a given dataset and feature sets while changing detailed conditions, such as the type of algorithm and scaler, which are intrinsic training parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' We also tuned the hyperparameter corresponding to the scaler used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Four algorithms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=', nearest neighbor (NN) regression, kernel ridge (KR) regression, Bayesian ridge (BR) regression, and support vector machine (SVM), were adopted to train the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='46 The NN47, KR48, 49, BR50, 51, and SVM52 regression models were utilized as four representative ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=" NN model employs the results of the k-nearest neighbors' average values for the given data points." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' The function only takes a portion of the pertinent dataset because it can only be approximated locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' KR is one of the non-parametric forms of ridge regression that combines the kernel technique and ridge regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' It develops a linear model in the implicit feature space caused by the appropriate kernel and data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' KR simplifies the computation of inner products in a high-dimensional space 8 by employing the kernel approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' It correlates to a non-linear function in the original space for the non-linear kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' BR model is a linear-based model, which assumes a relationship between the input and output variables by fitting a linear equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Instead of employing point estimates, BR formulates a linear relationship using probability distributors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' SVM can handle both classification and regression issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' SVM creates a set of hyperplanes in high-dimensional space to classify the data points for a classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' SVM is more versatile for regression problems by enclosing the function in the ε-insensitive region (ε-tube).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' To balance model complexity and prediction error in the SVM regression, this tube reformulates the regression problem to identify the function that deviates from the acquired targets throughout all training data the least.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' ASCENDS saves metadata regarding each training model so that deviation of accuracy (R2) can be calculated by using ten times of trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' The ML model with the highest accuracy was trained using features of high correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' We have further validated the ML model by comparing its estimated hc value with the directly obtained experimental hc value from an example not available in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Results and Discussion The result of correlation analyses (Figure 1b) for the relevant features are shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Despite the differences between MIC and PCC, XPB and ES commonly show high correlation scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' The MIC (absolute PCC) scores are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='687 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='950) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='687 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='629) for XPB and ES, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' In contrast, the correlation scores of ED are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='435 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='025 for MIC and PCC, respectively, which is significantly lower than those of XPB and ES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' This suggests that XPB and ES are more important than ED, among the PB features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Because XPB and ES are different from ED in that they contain both ν and ε, it can be further inferred that a combination of ν and ε is critical in constructing the most efficient feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' The result is more intriguing because 9 individual ν or ε alone do not exhibit particularly high correlation scores, yet the combined features of XPB and ES become the most physically relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Note that the overall correlation scores of ionic features in set A (Figure S2) are lower than the ones of set C, and set D with XPB, ES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' To compile the features (Figure 1c), we compare the training results of each feature set from feature engineering (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' The results show that the feature sets C and D, including the PB model features, are highly reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Figure 3a exhibits the R2 of the ML model for the feature sets presented in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' For set A, all algorithms produced R2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' The low R2 values and large deviations imply that the feature set does not represent a valid physical situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' This result is not surprising because the ionic features do not consider any interaction between the film and the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Only the BR algorithm results in a reasonable accuracy for set B, suggesting that set B does not contain enough critical features for predicting hc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Notably, it can be inferred that individual v and ε are insufficient to construct a valid prediction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' On the other hand, set C exhibits consistently high R2 values, indicating good model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Set D also shows high R2 values similar to set C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' From the results of sets C and D, it is evident that the PB features are crucial in determining hc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Figure 3b presents an example of the ML training results obtained using set C and the BR algorithm, with the highest R2 value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' The diagonal grey region has a slope of 1, indicating the correspondence of the predicted and actual experimental values of hc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' The prediction is encouraging, especially considering experimental uncertainty and sporadic data points, and confirms feature engineering has successfully deduced reliable feature sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' With the insight from the results of correlation analyses and feature set compilation, we predicted with the trained ML surrogate models (Figure 1d), as summarized in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' The hc of the STO/LSAT system is predicted by various ML models based on different feature sets 10 and algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Figure 4a shows the hc values obtained by the ML models from sets A, B, C, and D (vertical bars) with the BR algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' The predictions using feature sets A (42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='5 nm) and B (36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='5 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='2 nm) largely underestimates hc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' On the other hand, using feature sets C (78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='0 nm) and D (77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='2 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='2 nm) the predictions lie just beneath the PB model calculation result (86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='4 nm, red horizontal dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' We further compare the algorithm-dependent results of using sets C (Figure 4b) and D (Figure 4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Set C produces more consistent results with less variation among different algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' This reiterates that the individual features of v and ε included in set D might obscure the effective model construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Their augmented form is essential in understanding the epitaxial strain relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' For more realistic validation of our ML model, we compared our results to the experimental result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' The X-ray reflectivity (XRR) results of four samples with thicknesses of 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='2, 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='0, 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='5, and 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='0 nm are shown in Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' X-ray diffraction reciprocal space map (XRD-RSM) measurements were taken for the samples to investigate the epitaxial strain relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' As the epitaxial strain relaxes with increasing thickness, additional Bragg peaks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=', relaxed regions) emerge, breaking the mirror symmetry of the original Bragg peak (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=', strained region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' The upper panels in Figure S1c show the RSMs around the substrate LSAT (103) and the film STO (103) peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' The regions marked by white boxes are magnified in the lower panels to assess the strain relaxation of the STO thin film in further detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' The peaks are symmetric for the 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='2 and 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='0 nm films, but they become progressively asymmetric for the 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='5 and 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='0 nm films, suggesting that the strain relaxation occurs between 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='0 and 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='5 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' This strain relaxation behavior was further quantitatively examined using a bi-Gaussian fitting of the STO (103) peak (Figure 5 and S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Bi-Gaussian function, which has a distinct standard deviation for the left (W1) and right half (W2) of the peak, is an effective tool for quantifying the asymmetry that originates from strain relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' The line profiles through the black lines in the lower panels 11 of Figure S1c and their bi-Gaussian fittings (red lines) are plotted in Figure S3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' As shown in Figure 5, (W1 − W2)/W2, the normalized difference between the width on the left and right side increases dramatically as the film thickness ≥ 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='5 nm, the thickness near which the strain relaxation begins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Therefore, the hc of the STO/LSAT system was experimentally determined to be 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='0 – 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='5 nm and it is consistent with our ML model result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Whereas the original PB model show decent prediction of hc as expected (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' S4a), the data analytics approach provides hidden insight of the epitaxial strain relaxation of perovskite oxide thin film system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Particularly, we note that the strong correlations between hc and various features are not evident when hc values from the literature are directly plotted (Figure S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' For example, Figures S4b-d show hc values plotted as functions of ES, v, and ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' This emphasizes the merit of applying data analytics which quantitatively characterize the augmented features of XPB and ES to be essential in understanding the epitaxial strain relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' As discussed previously, both XPB and ES include the parameters v and ε, yet feature sets including pure v and ε, do not result in particularly high precision in predicting hc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Physically, this might imply that independent information on either the value of the in-plane lattice structure, ε, or the relationship between the in-plane and out-of-plane lattice structure, v, does not provide meaningful understanding of the epitaxial strain relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Again, the augmented features of XPB and ES are critical, as the elastic modulation of the thin film should be interpreted as a three-dimensional phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Conclusion We demonstrate the feasibility of establishing a streamlined data analytics workflow to efficiently evaluate and introduce relevant features that capture the physics of strain relaxation in epitaxial thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' First, we collected various experimental hc data which are sporadic in 12 nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Second, we augmented physical/chemical features for detailed correlation analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Third, we refined the dataset into consistent sub-dataset by applying the result of correlation analyses and prevailing physical conditions, which inevitably reduced our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Despite the small number of sporadic data, our carefully chosen conditions were proven to be highly consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Consequently, the data analytics process presented in the current study based on the PB model provides an obvious first step for understanding hc by quantitatively identifying key features (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=', the Poisson’s ratio v and the lattice mismatch ε) that affect the epitaxial strain relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' We experimentally validated the predicted hc of STO thin films grown on LSAT (001) substrates, showing a good agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' This study introduces challenges in ML approach using sporadic experimental dataset and proposes its systematic solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' By doing so, we emphasize that using refined dataset within the context of modern data analytics can help achieving a better understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' In particular, the quantitative analyses of ML successfully provide us with the unique physical insight about three-dimensional nature of epitaxial strain relaxation mechanism intuitively by focusing on the key features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Initiating a framework for understanding the epitaxial strain relaxation would inspire the community to consistently collect/compile the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Furthermore, we anticipate that the demonstrated data analytics approach can be further applied beyond the example used in the present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Experimental Section We experimentally fabricated epitaxial STO thin films on LSAT substrates and determined the actual range of hc to validate the data analytics approach (Figure S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' The system was selected because it was not available in the literature, so pure prediction is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' STO thin films were fabricated on LSAT (001) substrate using PLD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' We grew the film at 750 °C and 100 mTorr of O2 partial pressure, using a KrF excimer laser (248 nm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' IPEX-868, 13 Lightmachinery) with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='5 J cm-2 of fluence and 5 Hz of repetition rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' The thicknesses of the STO thin films were determined by XRR (PANalytical X’Pert and a Rigaku Smartlab XRD), as the films had atomically sharp surfaces and interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Acknowledgements This work was supported by the Basic Science Research Programs through the National Research Foundation of Korea (NRF) (NRF-2021R1A2C201134012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' ORCID Woo Seok Choi https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='org/0000-0002-2872-6191 References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Kim, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Liu, J.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Salamanca-Riba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' The Effects of Multiphase Formation on Strain Relaxation and Magnetization in Multiferroic BiFeO3 Thin Films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Mater.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Strain Relaxation in the Epitaxy of La2/3Sr1/3MnO3 Grown by Pulsed- Laser Deposition on SrTiO3(001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Philos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' 83(28):3201-24.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Feng, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Tailoring Self-Polarization of BaTiO3 Thin Films by Interface Engineering and Flexoelectric Effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Interfaces.' metadata={'source': 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+page_content=' Tikhonov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Goncharsky, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Stepanov, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Yagola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Numerical methods for the solution of ill-posed problems: Springer Science & Business Media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Mackay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Bayesian Interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Neural Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' 4:415-47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Tipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Sparse bayesian learning and the relevance vector machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Mach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' 1:211–44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Khanna and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Awad, "Efficient Learning Machines: Theories, Concepts, and Applications for Engineers and System Designers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='" Apress, (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' 20 FIGURE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Schematic workflow of modern data analytics for predicting hc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' a) Experimental hc values of perovskite oxide thin films on STO, LAO, and LSAT (001) substrates fabricated by PLD are collected for the dataset construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' b) Quantitative correlation analyses provide insight into epitaxial strain relaxation by highlighting the underlying correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' c) Various physical features and feature sets were examined and constructed to create an ideal feature for data analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' d) Model prediction was applied to predict hc and compare it with the experimental value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Prediction STO,LAO,LSAT (001) 100 F R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='87 of hc Substrates Actual h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' (nm) 80 (a) Pulsed Laser Dataset 60 Deposition(PLD Construction 40 PerovskiteOxide 20 Thin Films 0 20 40 60 80 100 Predicted hc (nm) IBulkState 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='0 (b) Positive (d) Misfit Negative MLModel Dislocation Correlation Target: Analysis Training he +Strained State 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='4 Substrate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='2 00 XpB afilm ED Features lonic,Phase-baseo (c) PBModel-inspired Identifying Feature Set Features Compilation KeyFeatures1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='0 (a) (b) Positive Negative 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='6 MIC PCC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='0 XpB Es afilm Ep XpB Es a film Ep Features Features21 FIGURE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Correlation analyses in MIC and PCC for the features in set D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Correlation coefficients in a) MIC and b) PCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' For the PCC, red (blue) bars represent positive (negative) correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' FIGURE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Accuracy (R2) of each feature set sorted by different algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' a) The graph shows the R2 values of the model training using four algorithms, NN, KR, BR, and SVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' b) The ML training result of set C using the BR algorithm with the highest R2 value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Each point represents experimental data with hc values with their predicted values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' The thick gray line represents a slope of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' BR, Bayesian ridge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' KR, kernel ridge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' NN, nearest neighbor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' SVM, support vector machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' (a) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='0 NN KR BR SVM (b) 100 R2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='8 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='7 Accuracy (R2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='6 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='4 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='2 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='0 0 A B C D 0 20 40 60 80 100 Feature set Predicted h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' (nm)(a) 100 (b) 100 (c) 100 PB calculation Set C Set D 80 80 80 h。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' (nm) 60 60 60 40 40 40 20 20 20 0 A B c D NN KR BR SVM NN KR BR SVM Featureset Algorithm Algorithm22 FIGURE 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Comparison between the ML estimated, PB model calculated, and experimental values of hc for the STO/LSAT system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' a) hc values with error bars (standard deviation) obtained by the ML models using sets A, B, C, and D (purple vertical bars) with the best- performing BR algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' hc values with error bars (standard deviation) obtained by the ML models using b) set C and c) set D based on different algorithms (colored bars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Direct PB model calculation (red horizontal dashed line) are also shown for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' FIGURE 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Asymmetrical line profiles of the STO film peak and epitaxial strain relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Black solid circles represent experimental (W1 − W2)/W2 values plotted as a function of the film thickness, quantitatively characterizing the peak asymmetry, and hence, the epitaxial strain Set B Set A People-Bean Model 60 Set Set C 50 Fully relaxed (W,-W2)/W2 (%) 40 30 Partially relaxed 20 Fully strained 10 Experimental h。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' 30 40 50 60 70 80 90 100 110 120 130 thickness (nm)23 relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' The vertical bars represent the ML predicted hc values from various feature sets, where their thicknesses correspond to the error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' The hc values of Set C and Set D lie well between experimental hc region indicated by an arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Ionic features Atomic weight Electron affinity Electronegativity Ionization energy Ionic radius Oxidation states Phase features Definition Symbol Lattice constant of film afilm afilm Lattice mismatch afilm − asub asub ε Poisson ratio − 𝜀oop 𝜀ip v PB model features Definition Symbol PB factor afilm 1 – v 1 + v 1 ε2 XPB Strain energy density factor G 1 + v 1 – v ε2 ES Dislocation energy density factor G afilm ln(afilm) ED 24 TABLE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Description of features used in data analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Definitions and symbols of each individual and combined feature used for data analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' The features were classified into three types: ionic features, phase features, and PB model features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Feature set Features A Ionic features B afilm, ν, ε C XPB, Es, ED D XPB, Es, ED, afilm, ν, ε TABLE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Corresponding features of each feature set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' The table lists the four feature sets using features that correspond under different physical hypotheses: ionic features set A, phase features set B, PB features set C, combined features set D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' SUPPORTING INFORMATION Additional supporting information may be found in the online version of the article at the publisher’s website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' 25 Supplemental Materials Application of Machine Learning to Sporadic Experimental Data for Understanding Epitaxial Strain Relaxation Jin Young Oh1, Dongwon Shin2,*, and Woo Seok Choi1,* 1Department of Physics, Sungkyunkwan University, Suwon 16419, Republic of Korea 2Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Corresponding author e-mail : shind@ornl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='gov, choiws@skku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='edu 26 FIGURE S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Characterization of the strain state of STO thin films on LSAT substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' (a) XRR, (b) θ-2θ, and (c) XRD-RSM data for epitaxial STO thin films on LSAT (001) substrates with representative thicknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' The asterisk in (b) indicates the LSAT (002) substrate peak and the peak around 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='1° marks the STO (002) peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' 1 2 3 44 45 46 47 48 49 (b) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='2 nm 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='0 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='5 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='0 Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='units) 2q (degrees) (a) Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' units) 2q (degrees) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='256 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='256 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='260 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='78 STO (103) qz (Å-1) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='2 LSAT (103) (c) 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='26 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='0 27 FIGURE S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Correlation analyses in MIC and PCC for the features in set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Correlation coefficients for set A, ionic feature set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Overall correlation scores are relatively lower than set C, and Set D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' IRBF AWBF IRAF AWBS OSBF IRAS OSAF AWAF IEAF ENBF IRBS EAAS AWAS ENAS OSAS IEAS EABF EAAF ENAF ENBS IEBS EABS IEBF OSBS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='0 AWBF ENBF AWAF IRAF ENAF IEAF IRBF EABF IEBF EAAF OSAF OSBF AWAS EAAS EABS ENAS ENBS IEAS OSAS IRAS AWBS IRBS IEBS OSBS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='0 Positive Negative (b) Features PCC (a) MIC Features 28 FIGURE S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Detailed information of the asymmetric plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' (a) Line profiles crossing the STO (103) peak and bi-Gaussian fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' (b) The table summarizes the (W1 − W2)/W2 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Thickness (nm) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='2 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='0 88.' metadata={'source': 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+page_content='3MnO 3 BiFeO3 BiFeO3 BiFeO3 La0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='66Sr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='33Mn O3 La0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='7Sr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='3MnO 3 La0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='7Sr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='3MnO 3 LaAlO3 BaTiO3 BaTiO3 BaTiO3 BaTiO3 BaTiO3 BaTiO3 BaTiO3 BaTiO3 BaTiO3 BiFeO3 BaTiO3 SrCoO3 SrCoO3 Author/year H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='Ju 1998 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='Ju 1998 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='Tebano 2006 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='Lim 2007 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='Chu 2007 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='Geprags 2011 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='Maurice 2003 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='Vaghefi 2017 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Ranno 2002 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='Qiao 2011 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='Suzuki 1999 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='Visinoiu 2002 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='Guo 2016 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='Zhu 2006 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='Fujimoto 2002 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='Fujimoto 2002 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='Lee 2001 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='Seo 2020 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='Seo 2020 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='Lee 2016 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='Kim 2005 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='Zhang 2019 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content='Zhang 2019 TABLE S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Final dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' The refined dataset with 23 data on hc of perovskite oxide thin films used for ML training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Ionic features, phase features, and PB model features are also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Lattice constant and PB factor are expressed in unit of nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Strain energy density factor and dislocation energy density factor are expressed in units of GPa and GPa∙nm respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} +page_content=' Ionic features are presented using the following notation: AW (atomic weight), EA, (electron affinity), EN (electronegativity), IE (ionization energy), IR (ionic radius), OS (oxidation states), of A or B cation in film (F) or substrate (S) material of perovskite ABO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtA0T4oBgHgl3EQfAP8-/content/2301.01959v1.pdf'} diff --git a/DdAyT4oBgHgl3EQfefik/content/tmp_files/2301.00323v1.pdf.txt b/DdAyT4oBgHgl3EQfefik/content/tmp_files/2301.00323v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e091c3a18b942b007438f9aaccd086e53b052554 --- /dev/null +++ b/DdAyT4oBgHgl3EQfefik/content/tmp_files/2301.00323v1.pdf.txt @@ -0,0 +1,675 @@ +Magic angle in thermal conductivity of twisted bilayer graphene +Yajuan Cheng,1 Zheyong Fan,2 Tao Zhang,1 Masahiro Nomura,3 +Sebastian Volz,3 Guimei Zhu,4, † Baowen Li,5, ∗ and Shiyun Xiong6, ‡ +1School of Physics and Materials Science, Guangzhou University, Guangzhou 510006, China +2College of Physical Science and Technology, Bohai University, Jinzhou 121013, P. R. China +3Laboratory for Integrated Micro Mechatronic Systems (LIMMS/CNRS-IIS), +The University of Tokyo, Tokyo 153-8505, Japan +4School of Microelectronics, Southern University of Science and Technology, Shenzhen 518055, China +5Department of Materials Science and Engineering, Department of Physics, +Southern University of Science and Technology, Shenzhen 518055, China +Paul M. Rady Department of Mechanical Engineering and Department of Physics, +University of Colorado, Boulder, Colorado 80305-0427, USA +6Guangzhou Key Laboratory of Low-Dimensional Materials and Energy Storage Devices, +School of Materials and Energy, Guangdong University of Technology, Guangzhou 510006, China +In this Letter, we report a magic angle of 1.08◦ in the thermal conductivity of twisted bilayer +graphene (TBLG). Within the supercell of a moir´e lattice, there exist sites with different staking +modes between the two graphene layers, which serve as scatterers for the phonons that reduce +the thermal conductivity of TBLG compared to that of untwisted bilayer graphene. Our detailed +study reveals that the thermal magic angle arises from the competition between the weakened +spatial dependence of vibrational amplitude and stress on one hand, and the increased number of +scattering sites on the other hand. The rapid decrease of spatial mismatch for atomic vibrational +amplitudes and local stresses at small angles significantly weakens the scattering strength of a single +scatterer. In contrast, the reduced crystal period with twist angle dramatically increases scatterer +density. The combination of the two effects eventually leads to the apparition of this irregularity +in heat conduction. Our work not only enriches the research of twisted graphene, but also reveals +the underlying physics of the thermal magic angle, which should impact heat conduction in two- +dimensional materials in general. +Twisted bilayer graphene (TBLG) exhibits a moir´e +pattern with a larger second lattice periodicity. When +the twist angle between the two graphene layers reaches +1.08◦, which called magic angle [1–3], band hybridization +and avoided crossings emerge and result in the formation +of flat bands near the Dirac point. This unusual behav- +ior leads to many novel phenomena that are not prevalent +either in a single-layer or in a bilayer graphene. Among +many others are electronic correlation, superconductiv- +ity, spontaneous ferromagnetism, quantized anomalous +Hall states, and topologically protected states. +This +magic angle has attracted substantial research interests +in recent years since its theoretical prediction and exper- +imental observation [4–13]. +In this work, we report an abnormal behavior of ther- +mal property in the vicinity of 1.08◦, where the thermal +conductivity shows a local dip. Our systematic investi- +gations with homogeneous non-equilibrium molecular dy- +namic (HNEMD) simulations reveal that the magic arises +from the competition between the reduced spatial depen- +dence of atomic vibrational amplitude and stress on one +side, and the increased density of scattering sites on the +other side. The former weakens the scattering effect of +a single site while the latter strengthens the scattering +rates. +The Moir´e lattice formed in TBLG has a lattice pa- +rameter that decreases with increasing twist angle θ as +amoire = +a +2 sin(θ/2), a = 2.452 ˚A being the lattice param- +eter of monolayer graphene. The unit cell of the Moir´e +lattice has an hexagonal shape as shown by the black +lines in the top panel of Fig. 1. Inside the unit cell of the +Moir´e lattice, the stacking between the two layers is not +uniform. Typical stacking modes include AA stack, AB +stack, and SP stack, which are schematically illustrated +in the bottom panel of Fig. 1. +To construct an orthogonal simulation box, we adopted +a larger unit cell with a rectangle shape as indicated by +the blue dashed lines in Fig. 1. The number of atoms +in such a rectangular cell is twice that in the hexagonal +unit cell. To simulate the thermal transport properties +of TBLG, we performed HNEMD simulations [14, 15] +implemented in the graphics processing units molecular +dynamics (GPUMD) package [16]. The intralayer C-C +interaction is described by the optimized Tersoff poten- +tial [17] while the interlayer C-C interaction is governed +by the interlayer potential (ILP) [18]. The ILP poten- +tial has been proven to provide a better description for +the interlayer weak Van der Waals interactions than the +commonly used Lennard-Jones (LJ) potential as demon- +strated by Lebedeva et al [18]. +The cutoff of the ILP +potential is chosen as 15 ˚A, beyond which atomic inter- +action energies are negligible. During the simulations, a +time step of 1.0 fs is adopted for the integration of New- +ton’s equation. The details of HNEMD formulation as +well as the decomposition of the TC into the in-plane +and out-of-plane modes can be found in the supplemen- +arXiv:2301.00323v1 [cond-mat.mes-hall] 1 Jan 2023 + +2 +FIG. 1. Top panel: the Moir´e lattice formed in TBLG. The +relative position of AA, AB, and SP stacks are illustrated by +the circles and the unit cell of the Moir´e lattice is indicated +by the black parallelogram. +Bottom panel: the atomic ar- +rangements of AA, AB, and SP stacks. Red and blue atoms +correspond to the atoms in the bottom and top layers, respec- +tively. +tary material. +Fig. 2(a) illustrates the twisted angle dependent ther- +mal conductivity (TC) at 300 K. In general, the TC of +TBLG is reduced compared to the untwisted one. Be- +low 1.08◦, the TC decreases rapidly with the increase of +twist angle. While beyond 10◦, the TC slightly increases, +which is in agreement with other simulations [19–21] and +experiments [22, 23]. Surprisingly, we observe an abnor- +mal TC valley in the vicinity of 1.08◦. Beyond that the +TC increases rapidly and reach a local maximum when +the angle reaches around 3◦. After that the TC decreases. +It is well known that 1.08◦ corresponds to the magic +angle for electron transport, at which superconductivity +emerges at low temperatures [2, 3]. +Thus we call this +angle - thermal magic angle. +What is the underlying +mechanism of this coincidence? +Our study shows that the thermal magic angle also +emerges at other temperatures as shown in Fig. 2(c) and +(d). Moreover, to verify whether the thermal magic an- +gle is related to the adopted interlayer potential, we also +calculated the TC of TBLG with the LJ potential. The +results emphasize that an abnormal TC increase start- +ing from 1.56◦ is also preserved (Fig. S2). Therefore, +we believe that the observed thermal magic angle is an +FIG. 2. (a) Total, in-plane, and out-of-plane thermal conduc- +tivity of TBLG as a function of twist angle from 0◦ to 30◦ +at 300 K. (b) Total thermal conductivity of TBLG with twist +angle below 5◦. (c) Total thermal conductivity of TBLG ver- +sus with twist angle at different temperature: 300 K, 400 K, +and 500 K. (d) Normalized thermal conductivity with respect +to the value of untwisted structure as a function of twist angle +at 300 K, 400 K, and 500 K. +intrinsic property of TBLG. +For two-dimensional materials, the in-plane and out- +of-plane modes contribute much differently to the total +TC [24]. To examine such a difference and the effect of +the twist angle on the in-plane and out-of-plane modes, +we decompose the total TC into the contributions of in- +plane and out-of-plane modes at 300 K following the +method proposed in Ref. +15. +The corresponding re- +sults are reported in Fig. 2(a). Similarly to other two- +dimensional materials, the out-of-plane modes contribute +more to the total TC than the in-plane ones. Interest- +ingly, the in-plane TC is more sensitive to the twist angle +below 1.08◦ while remaining almost constant beyond 3◦. +As a result, the in-plane TC around 1.08◦ corresponds +to the global minimum. In contrast, the out-of-plane TC +rapidly decreases from 3◦ to 10◦. +Moreover, the total +TC increase beyond 15◦ is fully contributed by the out- +of-plane modes. +In the following, we shall illustrate that the magic ther- + +Stack +x +AA stack +AB/BA stack +SP stack2.4 +(a) +(b) +total +2.4 +2.2 +- out-of-plane +K) +2.0 +in-plane +W/m +2.3 +1.8 +1.6 +(103 +2.2 +1.2 +2.1 +1.0 +0.8 +0 +5 +10 15 20 25 30 +0.0 +2.5 +5.0 + (degree) +0 (degree)2.5F +(c) +1.05 +300 +(p)) +300 K +K +400 K +400 K +500 K +500 K +1.00: +K +2.0 +0.95 +0.90 +1.5 +0.85 +1.0 +0.80 + 10 15 20 25 30 +0.0 +2.5 +5.0 +0 +5 +0 (degree) +θ (degree)3 +FIG. 3. Spatial distribution of in-plan atomic vibrational amplitude at (a) θ = 1.08◦, (b) θ = 1.56◦, and (c) θ = 5.0◦ (unit: +˚A). The bright yellow regions correspond to AA stacks in (a) and (b). Spatial distribution of atomic (normal) stress along the +x direction (σxx) of the bottom graphene layer for (d) 1.08◦, (e) 1.56◦, and (f) 5◦ at 300 K (unit: eV). The red disks denote +the AA stack centers. The averaged stress for all systems is shifted to zero. (g) the atomic in-plane vibrational amplitude +distribution at different angles. (h) the distribution of atomic normal stress for all atoms at indicated angles. (i) Twist angle +dependent S-factor in TBLG calculated by Eq. 1. The constant C is chosen as 6×10−6 ˚AeV/nm2 to make the value of S-factor +in the same range of std(σ). +mal property is due to the competition between the re- +duced mismatch of local properties (atomic vibrational +amplitude and stress) and increased scattering center +density. +In TBLG, the Moir´e unit cell is relatively large and the +stacking is not uniform inside the unit cell. It changes +from the stable AB stack to the unstable AA and SP +stacks continuously (Fig. 1). Such a non-uniform stack +can scatter phonons strongly, thus reducing the TC. On +the other hand, it can also lead to spatial-dependent vi- +brational and mechanical properties, such as the atomic +vibrational amplitude and the atomic stress. +We will +show that the thermal magic angle arises from the inter- +play between the increased number of scattering centers +on one side, and the weakened spatial-dependent atomic +vibrational amplitude and stress on the other side. +At zero temperature, the AA stack has a larger equi- +librium interlayer distance (3.65 ˚A) compared to the one +of the AB stack (3.38 ˚A) (Fig. S3). For small twist an- +gles, the AA and AB stack centers are far away from each +other, and the interlayer distances around AA and AB +centers highlight distinct differences (Fig. +S4). +With +the increase of θ, the reduced AA and AB distances +make their interlayer interactions around AA and AB +regions more balanced. Therefore the interlayer distance +becomes less spatial dependent (Fig. S4). The spatial +dependent interlayer distance (especially at small angles) +affect the atomic vibrations in different regions. To ex- +amine those, we analyze the atomic trajectories for each +case. +Due to the fluctuations of TBLG in the out-of- +plane direction (z), we only count the atomic displace- +ment along the in-plane directions (x and y). +Figs. 3(a)-(c) report the time-averaged atomic in-plane +vibration amplitude Rxy at 1.08◦, 1.56◦ and 5◦ (300 K). + +0.085 +0.12 +(b) +140 +(a) +(c) +200 +140 +0.051 +120 +0.08 +0.11 120 +160 +100 +0.05 +100 +0.075 +0.10 +80 +80 +0.049 +0.07 +60 +0.09 +60 +80 +40 +40 +0.065 +0.048 +40 +0.08 +20 +20 +10.06 +0 +0.047 +0.07 +0 +40 +¥120160 200 240 +40 +80 +120 +160 +2040 +08 +60 +80 100 120 140 +x (A) +x (A) +x (A) +0.15 +140 +0.15 +0.15 +(e) +(f) +(d) +140 +200 +120 +0.1 +0.1 +0.1 +120 +160 +100 +100 +0.05 +0.05 +0.05 +80 +120 +80 +0 +0 +0 +60 +60 +80 +-0.05 +-0.05 +-0.05 +40 +40 +-0.1 +-0.1 +40 +-0.1 +20 +20 +-0.15 +0 +-0.15 +-0.15 +0 +0 +40 +80120 160 200 240 +40 +80 +120 +160 +20 +6080 100 120 140 +x (A) +x (A) +x (A) +(eV-1) +40 +(h) +I S-factor +(z-wu) I +g) +0.10 +10 +103 +- 0° +0° +0.5° +0.5° +0.08 +1.08° +1.08° + S-factor +AA stack density +102 +1.56° +1.56° + std(R) +()ps +0.06 +3° +3° + std() +5° +5° +101 +Distribution +0.04 +10-2 +10 +0.02 +100 +0 +0.00 +10-3 +0.05 +0.1 +0.15 0.2 +0.3 +-0.3-0.2-0.1 +0.0 +¥0.1 +0.20.3 +0 +2 +4 +6 +8 +10 +Atomic displacement Rxx (A) +Atomic in-plane stress (eV) +Twist angle (degree)4 +At small angles, the spatial distribution of Rxy clearly +represents the Moir´e patterns, where Rxy is maximized +at the AA stack regions. The mismatch magnitude of +the vibrational amplitudes at different regions reduces +rapidly as the twist angle increases. In Fig. 3(g), the +distribution range of Rxy is large at small angles but re- +duces rapidly with the increase of the twist angle. Start- +ing from 3◦, Rxy becomes a normal distribution with a +small standard deviation, which is similar to the one of +the untwisted structure. The averaged Rxy at θ ≥ 3◦ is +reaching 0.05˚A, which is slightly larger than the corre- +sponding value of the untwisted structure (Fig. S7). +Similarly to Rxy, we find that the in-plane atomic +stress is also spatial dependent because the rotation +changes the periodicity of graphene along a specific spa- +tial direction, leading to different period lengths along the +same direction for the top and bottom layers. The lattice +mismatch can cause the atoms from the top and bottom +layers to attract or repel each other along a specific in- +plane direction. Since the attraction and repulsion occur +between the two layers, the stresses of the top and bot- +tom layers are directed in opposite directions with similar +magnitudes at the same place (Fig. S8). +Fig. +3(d)-(f) illustrates the time-averaged normal +stress along the x direction (i.e., σxx) of the bottom +layer graphene for θ = 1.08◦, 1.56◦ and 5◦, respec- +tively, for temperature at 300 K. Different from The +maximum/minimum of vibrational amplitudes that are +located at the AA or AB stack, the maximum and min- +imum of the stresses are located between the two neigh- +boring AA regions. +Due to the symmetry of the TBLG, the maxi- +mum/minimum of the normal stresses σxx and σyy are +located at two edges of the triangle formed by AA cen- +ters (solid lines in Fig. 3(d), while another edge of the +triangle (dashed line) possesses the maximum/minimum +shear stress τxy as shown in Fig. S11. At small twist +angles, the atoms at different regions carry fairly differ- +ent stresses. Such a local stress difference reduces rapidly +with the increase of the twist angle and becomes negligi- +ble eventually. +To check the distribution of atomic stresses, +we +counted the distributions of σxx and σyy as shown in +Fig. 3(h). It clearly shows that the stress is distributed +in a broad range in the small twist angle region. For ex- +ample, σxx and σyy are ranging from -0.2 to 0.2 eV when +θ = 0.5◦. +With the increase of θ, the stress distribu- +tion range reduces and it becomes a normal distribution +with reduced stand deviations starting from 5◦, which is +similar to the case of the untwisted structure (Fig. S10). +The structure-induced spatial inhomogeneity in TBLG +can strongly scatter phonons, which induces the reduc- +tion of TC in bilayer graphene with a twist. Larger vibra- +tional amplitudes result in stronger anharmonicity while +local stresses can lead to the mismatch of phonon fre- +quencies. The total phonon scattering strength depends +both on the scattering strength of each individual scat- +terer and the density of scatters. In TBLG, the scatter- +ing strength of a single scatterer can be characterized by +the standard deviation of the corresponding distribution +function, namely std(Rxy) and std(σ). Since the number +of stress and vibrational amplitude maxima/minima is +directly related to the AA stack numbers, we use the AA +stack density ρAA to represent the density of scatterers. +The variation of ρAA, std(Rxy) and std(σ) as a func- +tion of θ are illustrated in Fig. S12. At small angles, +ρAA increases dramatically with θ while both std(Rxy) +and std(σ) decreases rapidly. +The increased ρAA pro- +duces more scattering sites while the reduced std(Rxy) +and std(σ) weakens the scattering strength of a single +scattering site. As a result, the two effects compete with +each other and eventually lead to the abnormal TC in- +crease after 1.08◦. Here we can define a S-factor: +S = +C +std(Rxy)std(σ)ρAA +(1) +to quantify the effect of the scattering, where C is a +constant. The denominator std(Rxy)std(σ)ρAA charac- +terizes the total scattering strength, i.e., the product of +scattering strength of a single scatterer and site density +of scatterers. +Fig. 3(i) illustrates the variation of S-factor with twist +angle at 300 K. Interestingly, the twist angle dependent +S-factor features the same trend as TC. Starting from +0.5◦, the S-factor decreases first and reaches to a local +minimum around 1.08◦. +With further increase of the +twist angle, the S-factor increases and reaches to a lo- +cal maximum around 3◦. After that, it reduces with the +increase of twist angle. The similar trend of S-factor and +TC with the variation of twist angle quantitatively con- +firms the thermal magic arises from the interplay between +the reduced scattering strength of a single scatterer and +the increased scattering site density. +With the further increase of the twist angle after 3◦, +the weakening effect of the scattering strength of a single +scattering site is negligible and the TC is governed by the +increased scattering site density again, which leads to the +TC decrease after 3◦. We note that beyond 10◦, the TC +is slightly enhanced with the increase of the twist angle, +which is related to a coherent phonon transport effect +similar to case in superlattices [25–27]. +Beyond 10◦, all properties become spatial independent. +The AA center distance also becomes very short (∼1.5 +nm at 10◦), which could be shorter than the size of some +phonon wave packets. In such a situation, those phonons +can not feel the scatterers and behave as the material +would be a homogeneous structure. With the increase of +θ, more and more phonons travel coherently, which leads +to the increase of TC with θ. +It is worth noting that although both electronic magic +angle and thermal magic angle happen at the same value, +1.08, the underlying mechanisms are quite different. In + +5 +the electronic case, it is the strong electron correlation +which happens only at extremely low temperature, at +which lattice vibrations is extremely small and will not +break down the electron correlations. In contrast, in the +thermal case, it is due to atomic vibrations and the ther- +mal magic can even exist at high temperatures as illus- +trated in our current work in Fig. 2. +In summary, we find a thermal magic angle at 1.08◦ +at which a local dip of the thermal conductivity ap- +pears. The decomposition analysis of the thermal con- +ductivity demonstrates that the rapid thermal conduc- +tivity reduction below 1.08◦ arises from the reduction +of both in-plane and out-of-plane mode contributions. +While beyond 3◦, the thermal conductivity reduction +arises mainly from the out-of-plane mode contribution. +The twist of bilayer graphene leads to non-uniform stack- +ings, which results in spatial-dependent properties and +thus scatters phonons. At small angles, the space depen- +dence of both vibrational amplitude and stress weakens +with the increase of twist angle, leading to a reduced +scattering strength of a single scatterer. On the other +hand, the scattering sites dramatically increase in num- +ber with the twist angle. The competition between these +two effects eventually result in the formation of a thermal +magic angle. The current research could help to under- +stand the origin of twist angle-dependent properties in +TBLG, especially around the magic angle, and be bene- +ficial for discovering other novel properties in TBLG. The +physical mechanisms discovered will also provide clues for +thermal management and control by using graphene and +related materials. +ACKNOWLEDGEMENTS +Y. Cheng and Z. Fan contribute equally to this work. +This work was supported by the National Natural Sci- +ence Foundation of China under Grant No. 12174276, +the Major Research Plan of the National Natural Sci- +ence Foundation of China (Grant No. 91833303), and the +Major International (Regional) Joint Research Project of +the National Natural Science Foundation of China (Grant +No. 51920105005). +† E-mail: zhugm@mail.sustech.edu.cn +∗ E-mail: libw@sustech.edu.cn +‡ E-mail: xiongshiyun216@163.com +[1] R. Bistritzer and A. H. MacDonald, Proceedings of the +National Academy of Sciences 108, 12233 (2011). +[2] Y. Cao, V. 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B 95, 214310 +(2017). + diff --git a/DdAyT4oBgHgl3EQfefik/content/tmp_files/load_file.txt b/DdAyT4oBgHgl3EQfefik/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a52f119aef87f7cee47857009792098545b6c96b --- /dev/null +++ b/DdAyT4oBgHgl3EQfefik/content/tmp_files/load_file.txt @@ -0,0 +1,568 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf,len=567 +page_content='Magic angle in thermal conductivity of twisted bilayer graphene Yajuan Cheng,1 Zheyong Fan,2 Tao Zhang,1 Masahiro Nomura,3 Sebastian Volz,3 Guimei Zhu,4, † Baowen Li,5, ∗ and Shiyun Xiong6, ‡ 1School of Physics and Materials Science, Guangzhou University, Guangzhou 510006, China 2College of Physical Science and Technology, Bohai University, Jinzhou 121013, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' China 3Laboratory for Integrated Micro Mechatronic Systems (LIMMS/CNRS-IIS), The University of Tokyo, Tokyo 153-8505, Japan 4School of Microelectronics, Southern University of Science and Technology, Shenzhen 518055, China 5Department of Materials Science and Engineering, Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China Paul M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Rady Department of Mechanical Engineering and Department of Physics, University of Colorado, Boulder, Colorado 80305-0427, USA 6Guangzhou Key Laboratory of Low-Dimensional Materials and Energy Storage Devices, School of Materials and Energy, Guangdong University of Technology, Guangzhou 510006, China In this Letter, we report a magic angle of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='08◦ in the thermal conductivity of twisted bilayer graphene (TBLG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Within the supercell of a moir´e lattice, there exist sites with different staking modes between the two graphene layers, which serve as scatterers for the phonons that reduce the thermal conductivity of TBLG compared to that of untwisted bilayer graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Our detailed study reveals that the thermal magic angle arises from the competition between the weakened spatial dependence of vibrational amplitude and stress on one hand, and the increased number of scattering sites on the other hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' The rapid decrease of spatial mismatch for atomic vibrational amplitudes and local stresses at small angles significantly weakens the scattering strength of a single scatterer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' In contrast, the reduced crystal period with twist angle dramatically increases scatterer density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' The combination of the two effects eventually leads to the apparition of this irregularity in heat conduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Our work not only enriches the research of twisted graphene, but also reveals the underlying physics of the thermal magic angle, which should impact heat conduction in two- dimensional materials in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Twisted bilayer graphene (TBLG) exhibits a moir´e pattern with a larger second lattice periodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' When the twist angle between the two graphene layers reaches 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='08◦, which called magic angle [1–3], band hybridization and avoided crossings emerge and result in the formation of flat bands near the Dirac point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' This unusual behav- ior leads to many novel phenomena that are not prevalent either in a single-layer or in a bilayer graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Among many others are electronic correlation, superconductiv- ity, spontaneous ferromagnetism, quantized anomalous Hall states, and topologically protected states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' This magic angle has attracted substantial research interests in recent years since its theoretical prediction and exper- imental observation [4–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' In this work, we report an abnormal behavior of ther- mal property in the vicinity of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='08◦, where the thermal conductivity shows a local dip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Our systematic investi- gations with homogeneous non-equilibrium molecular dy- namic (HNEMD) simulations reveal that the magic arises from the competition between the reduced spatial depen- dence of atomic vibrational amplitude and stress on one side, and the increased density of scattering sites on the other side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' The former weakens the scattering effect of a single site while the latter strengthens the scattering rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' The Moir´e lattice formed in TBLG has a lattice pa- rameter that decreases with increasing twist angle θ as amoire = a 2 sin(θ/2), a = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='452 ˚A being the lattice param- eter of monolayer graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' The unit cell of the Moir´e lattice has an hexagonal shape as shown by the black lines in the top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Inside the unit cell of the Moir´e lattice, the stacking between the two layers is not uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Typical stacking modes include AA stack, AB stack, and SP stack, which are schematically illustrated in the bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' To construct an orthogonal simulation box, we adopted a larger unit cell with a rectangle shape as indicated by the blue dashed lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' The number of atoms in such a rectangular cell is twice that in the hexagonal unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' To simulate the thermal transport properties of TBLG, we performed HNEMD simulations [14, 15] implemented in the graphics processing units molecular dynamics (GPUMD) package [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' The intralayer C-C interaction is described by the optimized Tersoff poten- tial [17] while the interlayer C-C interaction is governed by the interlayer potential (ILP) [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' The ILP poten- tial has been proven to provide a better description for the interlayer weak Van der Waals interactions than the commonly used Lennard-Jones (LJ) potential as demon- strated by Lebedeva et al [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' The cutoff of the ILP potential is chosen as 15 ˚A, beyond which atomic inter- action energies are negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' During the simulations, a time step of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='0 fs is adopted for the integration of New- ton’s equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' The details of HNEMD formulation as well as the decomposition of the TC into the in-plane and out-of-plane modes can be found in the supplemen- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='00323v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='mes-hall] 1 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Top panel: the Moir´e lattice formed in TBLG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' The relative position of AA, AB, and SP stacks are illustrated by the circles and the unit cell of the Moir´e lattice is indicated by the black parallelogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Bottom panel: the atomic ar- rangements of AA, AB, and SP stacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Red and blue atoms correspond to the atoms in the bottom and top layers, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' tary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' 2(a) illustrates the twisted angle dependent ther- mal conductivity (TC) at 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' In general, the TC of TBLG is reduced compared to the untwisted one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Be- low 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='08◦, the TC decreases rapidly with the increase of twist angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' While beyond 10◦, the TC slightly increases, which is in agreement with other simulations [19–21] and experiments [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Surprisingly, we observe an abnor- mal TC valley in the vicinity of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='08◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Beyond that the TC increases rapidly and reach a local maximum when the angle reaches around 3◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' After that the TC decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' It is well known that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='08◦ corresponds to the magic angle for electron transport, at which superconductivity emerges at low temperatures [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Thus we call this angle - thermal magic angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' What is the underlying mechanism of this coincidence?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Our study shows that the thermal magic angle also emerges at other temperatures as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' 2(c) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Moreover, to verify whether the thermal magic an- gle is related to the adopted interlayer potential, we also calculated the TC of TBLG with the LJ potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' The results emphasize that an abnormal TC increase start- ing from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='56◦ is also preserved (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Therefore, we believe that the observed thermal magic angle is an FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' (a) Total, in-plane, and out-of-plane thermal conduc- tivity of TBLG as a function of twist angle from 0◦ to 30◦ at 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' (b) Total thermal conductivity of TBLG with twist angle below 5◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' (c) Total thermal conductivity of TBLG ver- sus with twist angle at different temperature: 300 K, 400 K, and 500 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' (d) Normalized thermal conductivity with respect to the value of untwisted structure as a function of twist angle at 300 K, 400 K, and 500 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' intrinsic property of TBLG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' For two-dimensional materials, the in-plane and out- of-plane modes contribute much differently to the total TC [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' To examine such a difference and the effect of the twist angle on the in-plane and out-of-plane modes, we decompose the total TC into the contributions of in- plane and out-of-plane modes at 300 K following the method proposed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' The corresponding re- sults are reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Similarly to other two- dimensional materials, the out-of-plane modes contribute more to the total TC than the in-plane ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Interest- ingly, the in-plane TC is more sensitive to the twist angle below 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='08◦ while remaining almost constant beyond 3◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' As a result, the in-plane TC around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='08◦ corresponds to the global minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' In contrast, the out-of-plane TC rapidly decreases from 3◦ to 10◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Moreover, the total TC increase beyond 15◦ is fully contributed by the out- of-plane modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' In the following, we shall illustrate that the magic ther- Stack x AA stack AB/BA stack SP stack2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='4 (a) (b) total 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='2 out-of-plane K) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='0 in-plane W/m 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='6 (103 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='8 0 5 10 15 20 25 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='0 (degree) 0 (degree)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='5F (c) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='05 300 (p)) 300 K K 400 K 400 K 500 K 500 K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='00: K 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='80 10 15 20 25 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='0 0 5 0 (degree) θ (degree)3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Spatial distribution of in-plan atomic vibrational amplitude at (a) θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='08◦, (b) θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='56◦, and (c) θ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='0◦ (unit: ˚A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' The bright yellow regions correspond to AA stacks in (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Spatial distribution of atomic (normal) stress along the x direction (σxx) of the bottom graphene layer for (d) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='08◦, (e) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='56◦, and (f) 5◦ at 300 K (unit: eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' The red disks denote the AA stack centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' The averaged stress for all systems is shifted to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' (g) the atomic in-plane vibrational amplitude distribution at different angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' (h) the distribution of atomic normal stress for all atoms at indicated angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' (i) Twist angle dependent S-factor in TBLG calculated by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' The constant C is chosen as 6×10−6 ˚AeV/nm2 to make the value of S-factor in the same range of std(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' mal property is due to the competition between the re- duced mismatch of local properties (atomic vibrational amplitude and stress) and increased scattering center density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' In TBLG, the Moir´e unit cell is relatively large and the stacking is not uniform inside the unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' It changes from the stable AB stack to the unstable AA and SP stacks continuously (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Such a non-uniform stack can scatter phonons strongly, thus reducing the TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' On the other hand, it can also lead to spatial-dependent vi- brational and mechanical properties, such as the atomic vibrational amplitude and the atomic stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' We will show that the thermal magic angle arises from the inter- play between the increased number of scattering centers on one side, and the weakened spatial-dependent atomic vibrational amplitude and stress on the other side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' At zero temperature, the AA stack has a larger equi- librium interlayer distance (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='65 ˚A) compared to the one of the AB stack (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='38 ˚A) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' For small twist an- gles, the AA and AB stack centers are far away from each other, and the interlayer distances around AA and AB centers highlight distinct differences (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' With the increase of θ, the reduced AA and AB distances make their interlayer interactions around AA and AB regions more balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Therefore the interlayer distance becomes less spatial dependent (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' The spatial dependent interlayer distance (especially at small angles) affect the atomic vibrations in different regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' To ex- amine those, we analyze the atomic trajectories for each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Due to the fluctuations of TBLG in the out-of- plane direction (z), we only count the atomic displace- ment along the in-plane directions (x and y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' 3(a)-(c) report the time-averaged atomic in-plane vibration amplitude Rxy at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='08◦, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='56◦ and 5◦ (300 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='085 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='12 (b) 140 (a) (c) 200 140 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='051 120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='11 120 160 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='05 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='10 80 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='049 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='07 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='09 60 80 40 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='065 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='048 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='08 20 20 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='06 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='047 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='07 0 40 ¥120160 200 240 40 80 120 160 2040 08 60 80 100 120 140 x (A) x (A) x (A) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='15 140 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='15 (e) (f) (d) 140 200 120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='1 120 160 100 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='05 80 120 80 0 0 0 60 60 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='05 40 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='1 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='1 20 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='15 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='15 0 0 40 80120 160 200 240 40 80 120 160 20 6080 100 120 140 x (A) x (A) x (A) (eV-1) 40 (h) I S-factor (z-wu) I g) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='10 10 103 0° 0° 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='5° 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='5° 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='08° 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='08° S-factor AA stack density 102 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='56° 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='56° std(R) ()ps 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='06 3° 3° std() 5° 5° 101 Distribution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='04 10-2 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='02 100 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='00 10-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='3-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='2-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='0 ¥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='3 0 2 4 6 8 10 Atomic displacement Rxx (A) Atomic in-plane stress (eV) Twist angle (degree)4 At small angles, the spatial distribution of Rxy clearly represents the Moir´e patterns, where Rxy is maximized at the AA stack regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' The mismatch magnitude of the vibrational amplitudes at different regions reduces rapidly as the twist angle increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' 3(g), the distribution range of Rxy is large at small angles but re- duces rapidly with the increase of the twist angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Start- ing from 3◦, Rxy becomes a normal distribution with a small standard deviation, which is similar to the one of the untwisted structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' The averaged Rxy at θ ≥ 3◦ is reaching 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='05˚A, which is slightly larger than the corre- sponding value of the untwisted structure (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' S7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Similarly to Rxy, we find that the in-plane atomic stress is also spatial dependent because the rotation changes the periodicity of graphene along a specific spa- tial direction, leading to different period lengths along the same direction for the top and bottom layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' The lattice mismatch can cause the atoms from the top and bottom layers to attract or repel each other along a specific in- plane direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Since the attraction and repulsion occur between the two layers, the stresses of the top and bot- tom layers are directed in opposite directions with similar magnitudes at the same place (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' S8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' 3(d)-(f) illustrates the time-averaged normal stress along the x direction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=', σxx) of the bottom layer graphene for θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='08◦, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='56◦ and 5◦, respec- tively, for temperature at 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Different from The maximum/minimum of vibrational amplitudes that are located at the AA or AB stack, the maximum and min- imum of the stresses are located between the two neigh- boring AA regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Due to the symmetry of the TBLG, the maxi- mum/minimum of the normal stresses σxx and σyy are located at two edges of the triangle formed by AA cen- ters (solid lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' 3(d), while another edge of the triangle (dashed line) possesses the maximum/minimum shear stress τxy as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' At small twist angles, the atoms at different regions carry fairly differ- ent stresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Such a local stress difference reduces rapidly with the increase of the twist angle and becomes negligi- ble eventually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' To check the distribution of atomic stresses, we counted the distributions of σxx and σyy as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' 3(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' It clearly shows that the stress is distributed in a broad range in the small twist angle region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' For ex- ample, σxx and σyy are ranging from -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='2 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='2 eV when θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='5◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' With the increase of θ, the stress distribu- tion range reduces and it becomes a normal distribution with reduced stand deviations starting from 5◦, which is similar to the case of the untwisted structure (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' S10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' The structure-induced spatial inhomogeneity in TBLG can strongly scatter phonons, which induces the reduc- tion of TC in bilayer graphene with a twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Larger vibra- tional amplitudes result in stronger anharmonicity while local stresses can lead to the mismatch of phonon fre- quencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' The total phonon scattering strength depends both on the scattering strength of each individual scat- terer and the density of scatters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' In TBLG, the scatter- ing strength of a single scatterer can be characterized by the standard deviation of the corresponding distribution function, namely std(Rxy) and std(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Since the number of stress and vibrational amplitude maxima/minima is directly related to the AA stack numbers, we use the AA stack density ρAA to represent the density of scatterers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' The variation of ρAA, std(Rxy) and std(σ) as a func- tion of θ are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' S12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' At small angles, ρAA increases dramatically with θ while both std(Rxy) and std(σ) decreases rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' The increased ρAA pro- duces more scattering sites while the reduced std(Rxy) and std(σ) weakens the scattering strength of a single scattering site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' As a result, the two effects compete with each other and eventually lead to the abnormal TC in- crease after 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='08◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Here we can define a S-factor: S = C std(Rxy)std(σ)ρAA (1) to quantify the effect of the scattering, where C is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' The denominator std(Rxy)std(σ)ρAA charac- terizes the total scattering strength, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=', the product of scattering strength of a single scatterer and site density of scatterers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' 3(i) illustrates the variation of S-factor with twist angle at 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Interestingly, the twist angle dependent S-factor features the same trend as TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Starting from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='5◦, the S-factor decreases first and reaches to a local minimum around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='08◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' With further increase of the twist angle, the S-factor increases and reaches to a lo- cal maximum around 3◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' After that, it reduces with the increase of twist angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' The similar trend of S-factor and TC with the variation of twist angle quantitatively con- firms the thermal magic arises from the interplay between the reduced scattering strength of a single scatterer and the increased scattering site density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' With the further increase of the twist angle after 3◦, the weakening effect of the scattering strength of a single scattering site is negligible and the TC is governed by the increased scattering site density again, which leads to the TC decrease after 3◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' We note that beyond 10◦, the TC is slightly enhanced with the increase of the twist angle, which is related to a coherent phonon transport effect similar to case in superlattices [25–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Beyond 10◦, all properties become spatial independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' The AA center distance also becomes very short (∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='5 nm at 10◦), which could be shorter than the size of some phonon wave packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' In such a situation, those phonons can not feel the scatterers and behave as the material would be a homogeneous structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' With the increase of θ, more and more phonons travel coherently, which leads to the increase of TC with θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' It is worth noting that although both electronic magic angle and thermal magic angle happen at the same value, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='08, the underlying mechanisms are quite different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' In 5 the electronic case, it is the strong electron correlation which happens only at extremely low temperature, at which lattice vibrations is extremely small and will not break down the electron correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' In contrast, in the thermal case, it is due to atomic vibrations and the ther- mal magic can even exist at high temperatures as illus- trated in our current work in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' In summary, we find a thermal magic angle at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='08◦ at which a local dip of the thermal conductivity ap- pears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' The decomposition analysis of the thermal con- ductivity demonstrates that the rapid thermal conduc- tivity reduction below 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='08◦ arises from the reduction of both in-plane and out-of-plane mode contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' While beyond 3◦, the thermal conductivity reduction arises mainly from the out-of-plane mode contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' The twist of bilayer graphene leads to non-uniform stack- ings, which results in spatial-dependent properties and thus scatters phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' At small angles, the space depen- dence of both vibrational amplitude and stress weakens with the increase of twist angle, leading to a reduced scattering strength of a single scatterer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' On the other hand, the scattering sites dramatically increase in num- ber with the twist angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' The competition between these two effects eventually result in the formation of a thermal magic angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' The current research could help to under- stand the origin of twist angle-dependent properties in TBLG, especially around the magic angle, and be bene- ficial for discovering other novel properties in TBLG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' The physical mechanisms discovered will also provide clues for thermal management and control by using graphene and related materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' ACKNOWLEDGEMENTS Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Cheng and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' Fan contribute equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' This work was supported by the National Natural Sci- ence Foundation of China under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' 12174276, the Major Research Plan of the National Natural Sci- ence Foundation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' 91833303), and the Major International (Regional) Joint Research Project of the National Natural Science Foundation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' 51920105005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content=' † E-mail: zhugm@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='sustech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='cn ∗ E-mail: libw@sustech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAyT4oBgHgl3EQfefik/content/2301.00323v1.pdf'} +page_content='cn ‡ E-mail: xiongshiyun216@163.' metadata={'source': 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0000000000000000000000000000000000000000..3081ae85d9f722cf7400ce214440e3ee42d5ccc9 --- /dev/null +++ b/EtFRT4oBgHgl3EQfBjcA/content/tmp_files/2301.13465v1.pdf.txt @@ -0,0 +1,1745 @@ +GDOD: Effective Gradient Descent using Orthogonal +Decomposition for Multi-Task Learning +Xin Dong∗ +Ant Group +Shanghai, China +zhaoxin.dx@antgroup.com +Ruize Wu∗ +Ant Group +Hangzhou, China +kezhui.wrz@antgroup.com +Chao Xiong +Ant Group +Shanghai, China +xc272640@antgroup.com +Hai Li +Ant Group +Shanghai, China +tianshu.lh@antgroup.com +Lei Cheng +Ant Group +Hangzhou, China +lei.chenglei@antgroup.com +Yong He +Ant Group +Hangzhou, China +heyong.h@antgroup.com +Shiyou Qian +Shanghai Jiao Tong University +Shanghai, China +qshiyou@sjtu.edu.cn +Jian Cao +Shanghai Jiao Tong University +Shanghai, China +cao-jian@cs.sjtu.edu.cn +Linjian Mo† +Ant Group +Shanghai, China +linyi01@antgroup.com +ABSTRACT +Multi-task learning (MTL) aims at solving multiple related tasks +simultaneously and has experienced rapid growth in recent years. +However, MTL models often suffer from performance degeneration +with negative transfer due to learning several tasks simultaneously. +Some related work attributed the source of the problem is the con- +flicting gradients. In this case, it is needed to select useful gradient +updates for all tasks carefully. To this end, we propose a novel op- +timization approach for MTL, named GDOD, which manipulates +gradients of each task using an orthogonal basis decomposed from +the span of all task gradients. GDOD decomposes gradients into +task-shared and task-conflict components explicitly and adopts a +general update rule for avoiding interference across all task gradi- +ents. This allows guiding the update directions depending on the +task-shared components. Moreover, we prove the convergence of +GDOD theoretically under both convex and non-convex assump- +tions. Experiment results on several multi-task datasets not only +demonstrate the significant improvement of GDOD performed to +existing MTL models but also prove that our algorithm outperforms +state-of-the-art optimization methods in terms of AUC and Logloss +metrics. +CCS CONCEPTS +• Computing methodologies → Multi-task learning; Trans- +fer learning. +∗Both authors contributed equally to this research. +†Corresponding author. +Permission to make digital or hard copies of all or part of this work for personal or +classroom use is granted without fee provided that copies are not made or distributed +for profit or commercial advantage and that copies bear this notice and the full citation +on the first page. Copyrights for components of this work owned by others than the +author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or +republish, to post on servers or to redistribute to lists, requires prior specific permission +and/or a fee. Request permissions from permissions@acm.org. +CIKM ’22, October 17–21, 2022, Atlanta, GA, USA +© 2022 Copyright held by the owner/author(s). Publication rights licensed to ACM. +ACM ISBN 978-1-4503-9236-5/22/10...$15.00 +https://doi.org/10.1145/3511808.3557333 +KEYWORDS +multi-task learning, orthogonal decomposition, gradient conflict +ACM Reference Format: +Xin Dong, Ruize Wu, Chao Xiong, Hai Li, Lei Cheng, Yong He, Shiyou +Qian, Jian Cao, and Linjian Mo. 2022. GDOD: Effective Gradient Descent +using Orthogonal Decomposition for Multi-Task Learning. In Proceedings +of the 31st ACM International Conference on Information and Knowledge +Management (CIKM ’22), October 17–21, 2022, Atlanta, GA, USA. ACM, New +York, NY, USA, 10 pages. https://doi.org/10.1145/3511808.3557333 +1 +INTRODUCTION +Multi-task learning (MTL) aims to build a shared model that learns +multiple related tasks simultaneously. Compared to single-task +learning, it can significantly improve learning efficiency and pre- +diction accuracy through knowledge sharing between tasks [1]. +This allows MTL models to deploy to a wide range of real-world +applications, such as computer vision [15], natural language pro- +cessing [4], online recommendation and advertising systems [31]. +Recently, MTL has acted as a regularizer during network learn- +ing, leading to more meaningful neural representations and better +generalization [29]. +In practice, the training process of the MTL network is not al- +ways ideal. Since the competition of shared parameter updates may +harm individual tasks. The MTL approach often leads to networks +that accurately improve the performance of a subset of the tasks, +while the rest suffer, a phenomenon referred to as negative transfer +or destructive interference [24]. Minimizing the negative transfer +is a key goal for MTL models. To mitigate this problem, several +works [28, 34] opted to cluster tasks into groups based on prior +beliefs about their similarity or relatedness. +Alternatively, some related work attributed the source of the +problem to the gradient conflict [33]. Several approaches have been +proposed to minimize conflict between the updates across multiple +tasks. In this context, we split these approaches into two categories. +On one hand, gradients with different magnitudes lead to parameter +updating dominated by a subset of tasks. Consequently, a number +arXiv:2301.13465v1 [cs.LG] 31 Jan 2023 + +CIKM ’22, October 17–21, 2022, Atlanta, GA, USA +Xin Dong et al. +of approaches have been developed to tune a set of task-weighting +parameters to balance relative gradient magnitudes for different +tasks [2, 11]. However, these approaches do not solve the problem of +task gradients canceling out due to them pointing towards different +directions. On the other hand, some approaches [14, 26, 33] find +a common gradient descent direction for all tasks so that they +do not cancel each other. However, such solutions either cannot +distinguish the conflicting gradients explicitly or cannot mitigate +conflicting gradients completely. +Here, we argue the gradient conflict problem and reveal an ad- +vanced problem: how to distinguish the conflicting gradients and +mitigate their impact on each task. For this purpose, we instead +present an approach that straightly manipulates gradients and mit- +igates the interference across tasks. Specifically, we decompose the +gradient by the orthogonal basis in the subspace spanned by all +task per-example gradients. We analyze the updates of each task +according to its impact on other tasks. So that each task gradients +can be decomposed into two components: 1) task-shared compo- +nent which is helpful for all tasks; and 2) task-conflict component +which interferes with other tasks. Only the task-shared component +is used to update the network. To achieve a tractable approach, we +introduce a novel and robust algorithm, named GDOD, to estimate +the subspace spanned by all task gradients and decompose each task +update appropriately. As a result, we can integrate our approach +with existing MTL models. To evaluate the performance of GDOD, +we conduct extensive experiments on three available public multi- +task datasets and a large-scale industrial dataset. Consequently, +GDOD guarantees convergence in theory and outperforms other +state-of-the-art optimization methods across all datasets in experi- +ments. +In light of the above background, the main contributions of this +paper are the following: +• We propose an optimization approach, named GDOD, to ma- +nipulate each task gradient using an orthogonal decomposi- +tion built from the span of all task gradients. GDOD decom- +poses gradients into task-shared and task-conflict compo- +nents explicitly and adopts a general update rule for avoiding +interference across all task gradients. +• We prove the convergence of GDOD theoretically under +both convex and non-convex assumptions. +• We conduct extensive experiments on several multi-task +datasets to evaluate the effectiveness of GDOD. Experiment +results not only demonstrate the significant improvement +of GDOD performed to existing MTL models but also out- +perform state-of-the-art optimization methods across all +datasets in terms of AUC metric. +2 +RELATED WORK +Efficient multi-task learning models and optimization approaches +of MTL models are two research areas related to our work. +2.1 +Multi-Task Learning Models +The learning conception of MTL that modeling the shared repre- +sentation for related tasks brings many benefits. However, MTL +may suffer from negative transfer due to task conflicts as parame- +ters are straightforwardly shared between tasks. To deal with task +conflicts, many works design different network architectures that +allow optimal knowledge sharing between tasks. +Cross-stitch network [21] and sluice network [25] propose to +learn weights of linear combinations to fuse representations from +different tasks selectively. However, the shared representations are +combined with the same static weights in these models and the neg- +ative transfer is not addressed. More studies apply the gate structure +and attention network for representation fusion. MOE [10] splits +the shared bottom layer into experts and combines experts through +a gating network. MMoE [18] and PLE [31] extend MOE to utilize +different gate nets to aggregate experts for each task. Similarly, +MARN [35] employs multi-head self-attention to learn different +representations with different feature sets for each task. However, +none of the above works has explicitly addressed the issues of joint +optimization of shared representation learning. +There are also some works utilizing neural architecture search +(NAS) approaches to find a good MTL network architecture. SNR +framework [17] controls connections between sub-networks by +binary random variables and applies NAS [37] to search for the +optimal structure. Similarly, Gumbel-matrix routing framework [19] +learns to route MTL models formulated as a binary matrix with the +Gumbel-Softmax trick. Moreover, [23] models the routing process +as MDP and employs MARL [27] to train the routing network. In +contrast to these methods, we propose an approach to address the +negative transfer problem in multi-task learning that allows us +to learn the tasks simultaneously without the need for specific +network design. +2.2 +Optimization Methods in MTL +Similar to our work, several prior researchers utilize some optimiza- +tion methods to address the negative transfer problem in multi-task +learning. A very common solution is to balance the impact of indi- +vidual tasks on the training of the network by adaptively weighting +the task-specific losses or gradients. There have been some stud- +ies developing a set of task-weighting parameters to balance the +training procedure. Uncertainty Weights [11] devises a weighting +method dependent on the homoscedastic uncertainty inherently +linked to each task. These weights for each loss function are trained +together with the MTL model parameters. GradNorm [2] reduces +the task imbalances by weighting task losses so that their gradients +are similar in magnitude. There are several methods dynamically +weighting the loss functions of tasks by the learning speed. [15] +and [16] explicitly set a weight to a task loss using a ratio of the +current loss to the previous loss. However, these loss weighting +methods do not work well all the time in practice. Moreover, the for- +mulation design of the weighing calculation is generally empirical +and lacks theoretical derivation. +There have also been some optimization methods to improve +MTL performance by mitigating conflicting gradients. The problem +of conflicting gradients has been previously explored in multi-task +learning as well as continual learning. [7] and [5] choose to ignore +the gradients of auxiliary tasks if the direction is not similar to the +main task. [22] overcomes catastrophic forgetting by maximizing +the dot product between task gradients. MGDA [6] employs the +condition of the Pareto stationary point for multi-objective opti- +mization. It finds a linear combination of gradients that reduces + +GDOD: Effective Gradient Descent using Orthogonal Decomposition for Multi-Task Learning +CIKM ’22, October 17–21, 2022, Atlanta, GA, USA +(a) Gradient Descent +(b) GDOD +Figure 1: The overview of two optimization methods. (a) Nor- +mal gradient descent. (b) Gradient manipulation example in +the 2-D plane with GDOD. +every loss function simultaneously. PCGrad [33] projects conflict- +ing gradients to each other, which achieves a similar simultaneous +descent effect as MGDA. CAGrad [14] looks for an update vector +that maximizes the worst local improvement of any objective in a +neighborhood of the average gradient. And the performance can +shift from GD-like to MGDA-like by a hyper-parameter. These +methods deal with the gradient decent independent of the model +structure and can be combined with normal optimizers such as SGD +and Adam. However, the above methods either cannot distinguish +the conflicting gradients explicitly or cannot mitigate conflicting +gradients completely. +3 +MULTI-TASK LEARNING USING GDOD +In this section, to realize effective gradient descent, we present a +novel optimization approach that mitigates conflicting gradients +across all tasks. +3.1 +Preliminaries: Notation and Problem +Consider an MTL model with 𝐾 different tasks that we want to learn +simultaneously. For simplicity, we assume that they share an input +space X and a collection of task spaces {Y𝑖}𝑖 ∈[𝐾 ]. Each of the inputs +in X is associated with a set of labels for all 𝐾 tasks, forming a large +dataset of i.i.d. data points {𝑥𝑗,𝑦1 +𝑗, ...,𝑦𝐾 +𝑗 }𝑗 ∈[𝑁 ] of 𝑁 observations. +The MTL model can be divided into two parts: a backbone and +multiple heads. The backbone contains the shared parameters 𝜃 +and transforms the input X into a shared representation. Next, this +representation is fed to each task-specific head to produce output. +We consider the empirical loss for each task𝑖 as L𝑖 (𝜃,𝜃𝑖) = � +𝑗 ∈𝑁 𝑙 𝑗 +𝑖 , +where 𝑙 𝑗 +𝑖 is the loss for the 𝑗-th observation and 𝜃𝑖 are task-specific +head parameters. We aim to find both the backbone parameters 𝜃 +and the task-specific parameters 𝜃𝑖 to minimize the loss L𝑖. More +formally, for a set of 𝐾 tasks, the final goal is to minimize the +multi-task loss as: +𝑚𝑖𝑛 +𝜃,𝜃1,...,𝜃𝐾 +𝐾 +∑︁ +𝑖=1 +L𝑖 (𝜃,𝜃𝑖). +(1) +During training, each task competes for updating backbone pa- +rameters 𝜃 to minimize the loss for each individual task, this leads +to the occurrence of negative transfer [24]. Therefore, we focus +on the learning of the backbone parameters 𝜃 in this work. The +task-specific parameters 𝜃𝑖 are updated by normal gradient descent. +For simplicity, we omit the task-specific parameters 𝜃𝑖 in loss L𝑖. +Generally, the gradient of the loss with respect to a particular shared +parameter 𝜃 from task 𝑖 is 𝑔𝑖 = ∇𝜃L𝑖 (𝜃). Using gradient descent +to minimize the multi-task loss, we obtain the update rule for a +task-shared parameter 𝜃 as: +𝜃 = 𝜃 − 𝛾 +𝐾 +∑︁ +𝑖=1 +∇𝜃L𝑖 (𝜃). +(2) +The above expression shows that the overall success of an MTL +model is dependent on the individual task gradients and their rela- +tionship to each other. However, task gradients might cancel each +other. Or a subset of tasks might dominate the gradients and point +towards a direction that does not improve any of the individual +tasks. +In this paper, we provide an approach that straightly manipulates +gradients to mitigate the conflicting gradient problem in MTL mod- +els. Our approach decomposes each gradient by an orthogonal basis +which is a subspace spanned by all task per-example gradients. It +decomposes the gradients of all the tasks into two components, and +only the helpful component is used to update the model parameters. +3.2 +Gradient Descent using Orthogonal +Decompotision +This section introduces a new optimization method to improve +generalization for all tasks. Figure 1 shows an illustration of the core +idea of GDOD which manipulates each task gradients to maximize +their usefulness to all tasks. To ensure the convergence and stability +of the optimization process, we modify the gradients for each task so +as to minimize negative conflict with other task gradients. Exactly, +we decompose each task gradients into two components: 1) task- +shared component which is helpful for all tasks; and 2) task-conflict +component which interferes with other tasks. Then, GDOD only +utilizes the task-shared components from all tasks to update model +parameters. +At each training step, suppose there are 𝑚 samples in a mini- +batch of data. GDOD first collects the gradients of the losses with +respect to 𝜃 for individual examples from all tasks as +M𝑇 = {{∇𝜃L𝑗 +1}𝑇, ..., {∇𝜃L𝑗 +𝐾 }𝑇 }∀𝑗 ∈[𝑚], +(3) +where ∇𝜃L𝑗 +𝑖 ∈ R𝑚×𝐷 and 𝐷 is the dimension of the model pa- +rameters. Then, we define a subspace S by the span of the gradi- +ent vectors in M𝑇 . Any linear combination of each task gradients +lies in this subspace, e.g., 𝑔𝑖 = E(∇𝜃L𝑗 +𝑖 ) ∈ S. Note that, the di- +mension of the subspace S is 𝑟, which is the rank of the matrix +M ∈ R𝑟×𝐷 (𝑟 ≤ 𝑚𝐾 << 𝐷). Third, we project each task gradi- +ents onto S. This allows distinguishing between the directions of +each task update which helps other task losses and those which +have a negative impact. Consequently, we can decompose each task +gradients as +𝑔𝑖 = 𝑔𝑠ℎ +𝑖 ++ 𝑔𝑐𝑜𝑛 +𝑖 +, +(4) +where 𝑔𝑠ℎ +𝑖 +∈ S is the portion of the gradient that improves all task +results and 𝑔𝑐𝑜𝑛 +𝑖 +∈ S is the portion of the gradient that damages +some task results. + +91.4 +92 +92 +91 +91 ++92 +92 +911+92 +q +92 +Task1 Target +92 +Task2 Target +gi +91 +ContourLine of Loss1 +Contour Line of Loss2 +b2 +Task1 Gradient +gi +Task2 Gradient +UpdatedGradient +Orthogonal BasisCIKM ’22, October 17–21, 2022, Atlanta, GA, USA +Xin Dong et al. +In order to decompose each task gradient, we define an orthog- +onal basis for subspace S as {𝑏𝑢}𝑢∈[𝑟 ]. On the orthogonal basis, +we can measure whether the components of each gradient is agree +or disagree with each other. It is said that the two gradients agree +along 𝑏𝑢 if and only if 𝑠𝑖𝑔𝑛(𝑔𝑇 +𝑖 · 𝑏𝑢) = 𝑠𝑖𝑔𝑛(𝑔𝑇 +𝑗 · 𝑏𝑢) for task 𝑖 and +𝑗. This mean that the gradient components {𝑔𝑠ℎ +𝑖 }𝑖 ∈[𝐾 ] refer to the +projections of {𝑔𝑖}𝑖 ∈[𝐾 ] onto the basis vectors where all task gra- +dients agree. On the contrary, {𝑔𝑐𝑜𝑛 +𝑖 +}𝑖 ∈[𝐾 ] refer to the gradient +components where the directions are disagree. By this decomposi- +tion, 𝑔𝑠ℎ +𝑖 +helps for all task, i.e., (𝑔𝑠ℎ +𝑖 )𝑇 · 𝑔𝑗 > 0 for any other task 𝑗, +while 𝑔𝑐𝑜𝑛 +𝑖 +interferes with some tasks, i.e., (𝑔𝑐𝑜𝑛 +𝑖 +)𝑇 · 𝑔𝑘 < 0 for task +𝑘. +The remaining problem is how to select an orthogonal basis for +the span of all task gradients. There are multiple methods, such as +Schmidt Decomposition, singular vector decomposition (SVD) and +randomized approximate matrix decomposition [9], to obtain the +basis {𝑏𝑢}𝑢∈[𝑟 ] at each training time step. The method is critical +since the two components of 𝑔𝑠ℎ +𝑖 +and 𝑔𝑐𝑜𝑛 +𝑖 +are helpful or harmful de- +pending on how they agree with the projection of all task gradients +onto this basis. +As some work [20] proved that along the directions associated +with singular vectors of a neural network Jacobian can generalize +well. In this paper, the subspace S is constructed from a mini-batch +of all task gradients. In brief, we select the singular vectors of the +matrix of M as the basis. We also compare the impact of different +decomposition methods mentioned above in Section 4.4. +Algorithm 1 GDOD Update At Each Training Step +Require: 𝜃, 𝛾: model parameters shared with all tasks, learning +rate +Require: M𝑇 = {{∇𝜃L𝑗 +1}𝑇, ..., {∇𝜃L𝑗 +𝐾 }𝑇 }∀𝑗 ∈[𝑚]: gradients with +respect to 𝜃 for each task +1: 𝑔𝑖 = 1 +𝑚 +�𝑚 +𝑗=1 ∇𝜃L𝑗 +𝑖 , ∀𝑖 ∈ [𝐾] +2: 𝐵 ← 𝑆𝑉𝐷_𝑃𝑟𝑜(M) +3: 𝑝𝑖 = 𝐵(𝑔𝑖)𝑇, ∀𝑖 ∈ [𝐾] +4: 𝑝𝑠ℎ +𝑖 , 𝑝𝑐𝑜𝑛 +𝑖 += +� +1[𝑝1 ⊙...⊙𝑝𝐾 ≥0] +� +⊙ 𝑝𝑖, +� +1[𝑝1 ⊙...⊙𝑝𝐾 <0] +� +⊙ 𝑝𝑖, ∀𝑖 ∈ +[𝐾] // ⊙ is the hadamard product operator +5: 𝑔𝑠ℎ +𝑖 ,𝑔𝑐𝑜𝑛 +𝑖 += (𝑝𝑠ℎ +𝑖 )𝑇 𝐵, (𝑝𝑐𝑜𝑛 +𝑖 +)𝑇 𝐵, ∀𝑖 ∈ [𝐾] +6: return update 𝜃 = 𝜃 − 𝛾 �𝐾 +𝑖=1 𝑔𝑠ℎ +𝑖 +We summarize the full update procedure in Algorithm 1. Sup- +pose the gradient for task 𝑖 is 𝑔𝑖. GDOD proceeds as follows: At +each training step, it first obtains the orthogonal basis 𝐵 from the +span of all task gradients M by SVD. The 𝑆𝑉𝐷_𝑃𝑟𝑜 is the procedure +of SVD and 𝐵 ∈ R𝑟×𝐷 is the non-zero and right-singular vectors +of M. Secondly, GDOD decomposes each task gradient on the or- +thogonal basis 𝐵. Thirdly, it obtains the helpful component 𝑔𝑠ℎ +𝑖 +and +harmful component𝑔𝑐𝑜𝑛 +𝑖 +for each task gradient respectively. Finally, +it utilizes the helpful components from all tasks to update model +parameters. +Algorithm 1 shows that this procedure is simple to implement +and ensures that the modified gradients we update for each task +have no conflict with the other tasks in each training step. Hence, +GDOD mitigates the conflicting gradient problem in MTL models. +Moreover, to reduce the computational complexity, the rank of the +matrix M can be reduced by grouping gradients. For example, the +gradients are divided into different groups and an average pooling +is performed on the gradients in the same group. We also examine +the impact of different dimensions of subspace S in Section 4.5. +Furthermore, by replacing the original gradients �𝐾 +𝑖=1 𝑔𝑖 with the +task-shared gradients �𝐾 +𝑖=1 𝑔𝑠ℎ +𝑖 , GDOD can be combined with any +other gradient-based optimizer, such as SGD with momentum and +Adam. +3.3 +Discussion +Several gradient-based approaches have been proposed to manip- +ulate each task gradients to obtain a new update vector and have +shown improved performance on existing MTL models. MGDA [6] +finds a linear combination of gradients that reduces every loss +function simultaneously. It proposes to minimize the following +combination of task gradients: +min1 +2 ∥ +𝐾 +∑︁ +𝑖=1 +𝑤𝑖𝑔𝑖 ∥2, +𝑠.𝑡. +𝐾 +∑︁ +𝑖=1 +𝑤𝑖 = 1 +and +∀𝑖, 𝑤𝑖 ≥ 0. +(5) +From equation 5, MGDA seeks the linear combination of gradients +that results in the smallest norm. Tasks that have larger gradients +will become attenuated by MGDA. For example, if an MTL model +has two tasks where task 1 is under-optimized and task 2 is near a +local optimum. The model has a large gradient for task 1 and a rela- +tive small gradient for task 2. In this situation, even if it is possible +to improve task 1 a lot while not affecting the performance of task +2, MGDA may not take that move because the best improvement on +task 1 is bounded by its improvement on task 2. This often causes +slow improvement of MGDA in practice. +Moreover, PCGrad [33] projects conflicting gradients to the or- +thogonal direction of each other. It sets a universal gradient simi- +larity objective of zero for any two tasks explicitly. Consequently, +if 𝑔𝑖 · 𝑔𝑗 < 0, PCGrad projects these conflicting gradients to each +other. It replace 𝑔𝑖 (𝑔𝑗) by its projection onto the normal plane of +𝑔𝑗 (𝑔𝑖): +𝑔𝑖 = 𝑔𝑖 − 𝑔𝑖 · 𝑔𝑗 +||𝑔𝑗 ||2𝑔𝑗 . +(6) +It is not effective for the case of positive gradient similarities with +𝑔𝑖 · 𝑔𝑗 ≥ 0. In fact, the two tasks share positive cosine similarities +such that the precondition for PCGrad would never be satisfied. +However, GDOD alters gradients more preemptively under both +positive and negative cases, taking more proactive measurements +in updating the gradient. +CAGrad [14] finds a linear combination of a new updated vector𝑑 +that is a linear combination of the original individual task gradients. +It obtains the updated vector𝑑 by solving the following optimization +problem: +max +𝑑 ∈𝑅 min +𝑖 ∈[𝐾 ] < 𝑔𝑖,𝑑 >, 𝑠.𝑡. ∥𝑑 − 𝑔0∥ ≤ 𝑐∥𝑔0∥. +(7) +The difference between MGDA and CAGrad is that the new updated +vector 𝑑 is searched around the 0 vector for MGDA and 𝑔0 (average +gradient vector) for CAGrad. CAGrad chooses the average loss +over all tasks as the main objective. Nevertheless, we find that +CAGrad is not robust with different task weights in Section 4.6. +For our method, we find an updated vector guided by the singular + +GDOD: Effective Gradient Descent using Orthogonal Decomposition for Multi-Task Learning +CIKM ’22, October 17–21, 2022, Atlanta, GA, USA +vectors of the Jacobian matrix. As some works [20] point out that +using the principal vectors as directions of descent instead of the +mean induces a more robust algorithm since the mini-batch average +gradient is susceptible to outliers and skews from replicated data +points. +3.4 +Theoretical Analysis of GDOD +In this section, we analyze the convergence of GDOD with the +following theorem. +Theorem 1. Let L(𝜃𝑡) represents the full batch losses of all 𝐾 +tasks at training step 𝑡. Suppose the gradients {𝑔𝑖}𝑖 ∈[𝐾 ] of all 𝐾 tasks +are Lipschitz continuous with 𝐿 > 0. Then, the GDOD update rule +𝜃𝑡+1 = 𝜃𝑡 − 𝛾 �𝐾 +𝑖=1 𝑔𝑠ℎ +𝑖 +with learning rate 𝛾 ≤ 1 +𝐿 will converge to +either (1) the optimal value if L(𝜃) is convex or (2) a stationary point +if L(𝜃) is non-convex. +Proof. According to the Lipschitz smoothness assumption, we +obtain the following inequality: +L(𝜃𝑡+1) ≤ L(𝜃𝑡) + ∇𝜃L(𝜃𝑡)𝑇 (Δ𝜃) + 1 +2𝐿∥Δ𝜃 ∥2 +2 +Now, we can plug in the GDOD update by replacing Δ𝜃 = 𝜃𝑡+1−𝜃𝑡 = +−𝛾 �𝐾 +𝑖=1 𝑔𝑠ℎ +𝑖 . We then obtain: +L(𝜃𝑡+1) ≤ L(𝜃𝑡) + 𝛾( +𝐾 +∑︁ +𝑖=1 +𝑔𝑖)𝑇 (− +𝐾 +∑︁ +𝑖=1 +𝑔𝑠ℎ +𝑖 ) + 1 +2𝐿∥𝛾 +𝐾 +∑︁ +𝑖=1 +𝑔𝑠ℎ +𝑖 ∥2 +2 += L(𝜃𝑡) + 𝛾( +𝐾 +∑︁ +𝑖=1 +𝑔𝑠ℎ +𝑖 ++ +𝐾 +∑︁ +𝑖=1 +𝑔𝑐𝑜𝑛 +𝑖 +)𝑇 (− +𝐾 +∑︁ +𝑖=1 +𝑔𝑠ℎ +𝑖 ) ++ 1 +2𝐿𝛾2∥ +𝐾 +∑︁ +𝑖=1 +𝑔𝑠ℎ +𝑖 ∥2 +2 += L(𝜃𝑡) − 𝛾( +𝐾 +∑︁ +𝑖=1 +𝑔𝑠ℎ +𝑖 )2 − 𝛾( +𝐾 +∑︁ +𝑖=1 +𝑔𝑐𝑜𝑛 +𝑖 +)𝑇 ( +𝐾 +∑︁ +𝑖=1 +𝑔𝑠ℎ +𝑖 ) ++ 1 +2𝐿𝛾2∥ +𝐾 +∑︁ +𝑖=1 +𝑔𝑠ℎ +𝑖 ∥2 +2 +(8) += L(𝜃𝑡) − 𝛾( +𝐾 +∑︁ +𝑖=1 +𝑔𝑠ℎ +𝑖 )2 + 1 +2𝐿𝛾2∥ +𝐾 +∑︁ +𝑖=1 +𝑔𝑠ℎ +𝑖 ∥2 +2 +(9) += L(𝜃𝑡) − (1 − 1 +2𝐿𝛾)𝛾∥ +𝐾 +∑︁ +𝑖=1 +𝑔𝑠ℎ +𝑖 ∥2 +2 +Note that in going equation 8 to 9 in the above proof, we use the fact +that (𝑔𝑐𝑜𝑛 +𝑖 +)𝑇𝑔𝑠ℎ +𝑗 += 0 for any two tasks 𝑖 and 𝑗 due to orthogonality. +We define the updated gradient at training step 𝑡 is 𝑔𝑠ℎ +𝑡 += �𝐾 +𝑖=1 𝑔𝑠ℎ +𝑖 . +Using 𝛾 ≤ 1 +𝐿 , we know that +− (1 − 1 +2𝐿𝛾) = 1 +2𝐿𝛾 − 1 ≤ 1 +2𝐿( 1 +𝐿 ) − 1 = −1 +2. +Plugging this into the last expression above, we can conclude the +following: +L(𝜃𝑡+1) ≤ L(𝜃𝑡) − 1 +2𝛾∥𝑔𝑠ℎ +𝑡 ∥2 +2 +(10) +≤ L(𝜃𝑡) +Thus, the above theorem ensures that GDOD is minimizing L(𝜃𝑡). +If L(𝜃) is convex and differentiable, hence repeatedly applying +GDOD process can reach the optimal value. +Assume L(𝜃) is non-convex, using telescope sum to equation 10, +we have +L(𝜃𝑇 ) − L(𝜃0) ≤ −1 +2𝛾 +𝑇−1 +∑︁ +𝑡=0 +∥𝑔𝑠ℎ +𝑡 ∥2 +2 +(11) +Thus, we have: +min +0≤𝑡 ≤𝑇 ∥𝑔𝑠ℎ +𝑡 ∥2 +2 ≤ 1 +𝑇 +𝑇−1 +∑︁ +𝑡=0 +∥𝑔𝑠ℎ +𝑡 ∥2 +2 +≤ 2(L(𝜃0) − L(𝜃𝑇 )) +𝑇𝛾 +≤ 2(L(𝜃0) − L∗) +𝑇𝛾 +(12) +where L∗ is the minimal function value. Therefore, GDOD updating +with gradients 𝑔𝑠ℎ +𝑡 +can converge to a stationary point in O( 1 +𝑇 ) +steps. +□ +Therefore, we prove GDOD converges to either the optimal value +if L(𝜃) is convex or a stationary point if L(𝜃) is non-convex. +4 +EXPERIMENTS +In this section, we evaluate the performance and effectiveness of +GDOD with four multi-task datasets from different domains. We +first evaluate the performance of GDOD as well as several state- +of-the-art optimization methods. Then, we verify that GDOD is +model-agnostic and can improve performance for any MTL models +with shared parameters. Finally, we present ablation experiments +to explain the impact of hyper-parameter selection. +Table 1: The statistics of the four datasets. +Dataset +Phase +Users +Items +Samples +BookCrossing +Train +92,792 +239,029 +919,824 +Test +42,194 +99,404 +229,956 +IJCAI-15 +Train +237,295 +274,709 +2,142,528 +Test +106,023 +127,772 +544,025 +Alipay Advertising +Train +7,579,571 +1,098 +14,298,291 +Test +5,822,077 +835 +10,740,289 +Census-Income +Train +- +- +199,523 +Test +- +- +99,762 +4.1 +Datasets and Settings +4.1.1 +Datasets. We use three public multi-task datasets from dif- +ferent domains and a large-scale real-world advertising dataset to +verify the effectiveness of GDOD. The statistics of the datasets are +listed in Table 1. These datasets are described as follows: +• BookCrossing Dataset [36] collects user ratings in the +Book-Crossing community. It includes 278,858 users who +provide 1,157,112 ratings about 271,379 books. As suggested +in the original paper [36], we define the following two related + +CIKM ’22, October 17–21, 2022, Atlanta, GA, USA +Xin Dong et al. +prediction tasks based on this dataset: 1) predict whether a +user has rated a book; 2) predict whether a rating score from +a user on a book is higher than or equal to 9. +• IJCAI-15 Dataset [32] is collected from the E-commerce +website Tmall.com. It is a public dataset used in the IJCAI2015 +repeat buyers prediction competition hosted by Alibaba +Group. It contains 241,093 users with 2,295,706 instances on +237,564 items. We model two related prediction tasks involv- +ing CVR (Conversion Rate): 1) predict whether a user adds +an item to his favourites after clicking; 2) predict whether a +user buys an item after adding it to his favourites. +• Alipay Advertising Dataset is collected over three months +from user traffic logs of a commercial advertising system +in the Alipay App. It contains 7,630,003 users who produce +25,038,580 samples about 1,120 advertisements. One CTR +task to predict whether a user clicks an item and two CVR +tasks similar with IJCAI-15 dataset are modeled. +• Census-Income(KDD) Dataset [13] is a dataset extracted +from the 1994 census database. It contains 199,523 instances +with 42 demographic and employment related features. Given +a person, we model six related prediction tasks based on this +dataset contains: 1) predict whether the person’s income +exceeds $50K; 2) predict whether the person’s marital status +has never married; 3) predict whether the person’s education +level is at least college; 4) predict whether the person’s em- +ployment status is full time; 5) predict whether the person’s +gender is male; and 6) predict whether the person’s race is +white. +4.1.2 +Comparative Optimization Methods. We compare GDOD +with seven SOTA optimization methods. +• Adam is used as the baseline to compute the performance +gains of other methods. +• Uncertainty Weights (Uncert) [11] uses a joint likelihood +formulation to derive task weights based on the intrinsic +uncertainty in each task. +• GradNorm [2] reduces the task imbalances by weighting +task losses so that their gradients are similar in magnitude. +• MGDA [6] applies a multiple-gradient descent algorithm for +MTL. It finds a linear combination of gradients that reduces +every task loss simultaneously. +• Gradient Regularization (GradReg) [30] proposes a gra- +dient regularization term that minimizes task interference +by enforcing near orthogonal gradients. +• PCGrad [33] projects conflicting gradients to the orthogonal +direction of each other, so that achieving a similar simulta- +neous descent effect. +• CAGrad [14] looks for an update gradient vector in the +neighborhood of the average gradient that minimizes the +average loss and leverages the worst local improvement of +individual tasks. +4.1.3 +Baseline MTL models. We evaluate the effect of our GDOD +with the following representative MTL models. +• Shared-Bottom [1]. Shared-Bottom shares the embedding +layers and a low-level feature extraction layer (MLP) for all +tasks and each task has its own task-specific high-level layers +built on top of the shared layers. +• Cross-Stitch [21]. It fuses the tower layers of tasks by linear +transformation based on the Shared-Bottom model. +• MMOE [18]. MMOE transforms the shared low-level layers +into sub-networks and uses different gating networks for +tasks to utilize different sub-networks. +• SNR [17]. SNR modularizes the shared low-level layers into +parallel sub-networks and uses a transformation matrix mul- +tiplied by a scalar coding variable to learn their connections. +• PLE [31]. PLE separates shared components and task-specific +components and adopts a progressive routing mechanism to +achieve more effective information sharing. +4.1.4 +Evaluation Metrics . For fair comparisons, we employ AUC +and Logloss as our evaluation metrics. +• AUC is the Area Under the ROC Curve over the test set. +It measures the goodness of order by ranking all the items +with predicted CTR in the test set. It is noticeable that a +slightly higher AUC at 0.001-level is regarded as significant +for CTR/CVR prediction tasks, which has been pointed out +in existing works [3, 8, 21]. Note that, the larger AUC shows +better performance. +• Logloss is the loss value on the test set. The smaller Logloss +means better performance. +4.1.5 +Implementation Details. For all the baseline MTL models, +there are trained by the Adam optimizer [12] with an initial learning +rate of 1e-3. The mini-batch size is fixed to 256. The embedding +size of each sparse feature is set to 8. The hidden sizes of the two +shared hidden layers in shared-bottom model are [256, 32]. The +number of sub-networks/experts in SNR, PLE and MMoE is set to +8 and the hidden size of each sub-network/expert is 32. There are +two specific tower hidden layers with the size of [16, 1] for each +task. +Moreover, the weights in GradReg is tuned in [1e-1, 1e-2, 1e-3, +1e-4]. For GradNorm, the hyper-parameter 𝛼 is tuned in [0.5, 1.5]. +The hyper-parameter 𝑐 in CAGrad is tuned in [0.1, 0.3, 0.5, 0.7, +0.9]. All hyper-parameters are settled with the best performance on +each dataset. For GDOD, the training examples are divided into 16 +groups in a mini-batch at each training step. For PCGrad, CAGrad +and GDOD, they are combined with Adam by passing the computed +update to replace the original gradient. We repeat all experiments +5 times and report the averaged results. +4.2 +Optimization Method Comparison +Table 2 shows the AUC of the comparative results on the BookCross- +ing dataset, IJCAI-15 dataset and Advertising dataset. Focusing on +the detail, the shared-bottom model combined with GDOD achieves +higher AUC compared to other optimization methods. Moreover, +we have the following four observations: 1) GDOD, PCGrad and +CAGrad outperform other five optimization methods. This indi- +cates that optimization methods manipulated per-task gradients +are more practical. 2) GDOD achieves better performances than +PCGrad and CAGrad, e.g., GDOD achieves 0.0064 and 0.0084 AUC +gains compared to PCGrad and CAGrad in task2 with BookCrossing +dataset respectively. The magnitude of this improvement is fairly + +GDOD: Effective Gradient Descent using Orthogonal Decomposition for Multi-Task Learning +CIKM ’22, October 17–21, 2022, Atlanta, GA, USA +Table 2: Performance comparisons of different optimization methods. The baseline MTL model is Shared-Bottom. Gain mea- +sures the AUC improvement between Adam with other optimization methods. +Optimization Method +BookCrossing +IJCAI-15 +Alipay Advertising +Task1 +Task2 +Task1 +Task2 +Task1 +Task2 +Task3 +AUC +Gain +AUC +Gain +AUC +Gain +AUC +Gain +AUC +Gain +AUC +Gain +AUC +Gain +Adam +0.7838 +- +0.7633 +- +0.6968 +- +0.7451 +- +0.7505 +- +0.7387 +- +0.8237 +- +Uncert +0.7814 +-0.0024 +0.7663 +0.0030 +0.6631 +-0.0337 +0.7327 +-0.0124 +0.7599 +0.0094 +0.7475 +0.0088 +0.8264 +0.0027 +GradReg +0.7696 +-0.0142 +0.7466 +-0.0167 +0.6775 +-0.0193 +0.7321 +-0.0130 +0.7635 +0.0130 +0.7513 +0.0126 +0.8273 +0.0036 +GradNorm +0.7857 +0.0019 +0.7677 +0.0044 +0.7125 +0.0157 +0.7477 +0.0026 +0.7649 +0.0144 +0.7505 +0.0118 +0.8238 +0.0001 +MGDA +0.7811 +-0.0027 +0.7593 +-0.0040 +0.7122 +0.0154 +0.7548 +0.0097 +0.7600 +0.0095 +0.7440 +0.0053 +0.8300 +0.0063 +PCGrad +0.7912 +0.0074 +0.7753 +0.0120 +0.7188 +0.0220 +0.7524 +0.0073 +0.7630 +0.0125 +0.7431 +0.0044 +0.8335 +0.0098 +CAGrad +0.7900 +0.0062 +0.7733 +0.0100 +0.7204 +0.0236 +0.7542 +0.0091 +0.7699 +0.0194 +0.7456 +0.0069 +0.8384 +0.0147 +GDOD +0.7922 +0.0084 +0.7817 +0.0184 +0.7268 +0.0300 +0.7555 +0.0104 +0.7723 +0.0218 +0.7571 +0.0184 +0.8390 +0.0153 +(a) BookCrossing task1 loss. +(b) BookCrossing task2 loss. +(c) IJCAI-15 task1 loss. +(d) IJCAI-15 task2 loss. +Figure 2: Test loss comparisons about several optimization methods on BookCrossing and IJCAI-15 datasets. In all cases GDOD +outperforms all other optimization methods. +(a) Task1 on BookCrossing +(b) Task2 on BookCrossing +(c) Task1 on IJCAI-15 +(d) Task2 on IJCAI-15 +Figure 3: Test AUC comparisons about several optimization methods on BookCrossing and IJCAI-15 datasets. In all cases +GDOD outperforms all other optimization methods. +significant. Because GDOD implements a decomposition method +that can distinguish the conflicting gradients effectively. 3) Several +situations with Uncert and GradReg are proven to be worse than +Adam, showing the applicability of re-weighting methods is poor. 4) +MGDA seems to perform worse than some re-weighting methods in +some tasks. This is because MGDA will attenuate the performance +of tasks that have higher gradients. Overall, these results verify +that GDOD is a highly effective optimization method to avoid task +competition. +Moreover, Figure 2 illustrates the test loss curves during the train- +ing procedure on BookCrossing and IJCAI-15 datasets. From these +curves, GDOD can be shown to achieve the lowest LogLoss than +any other optimization method with a fixed step size. Therefore, +these results demonstrate that GDOD can accelerate convergence +and achieve good performance at the same step. We also show the +AUC curves during the training procedure on the BookCrossing +dataset and IJCAI-15 dataset in Figure 3. It is observed that GDOD +achieves the highest AUC compared to all the other optimization +methods. Moreover, with a fixed training step, GDOD performs +the best performance in most experiments. These results demon- +strate that GDOD outperforms other compared SOTA optimization +methods. +4.3 +GDOD with Multi-task Models +Table 3 shows the AUC and Logloss of the comparison results on +BookCrossing, IJCAI-15 and Alipay Advertising datasets. Focusing +on the detail of Table 3, all MTL models combined with GDOD +achieve higher AUC and lower Logloss compared to the original +MTL models. These results confirm that GDOD improves the perfor- +mance for multi-task learning benchmarks by avoiding interference +across all task gradients. For example, the Cross-Stitch model with +GDOD optimization achieves 0.0287 AUC gain compared to the + +CIKM ’22, October 17–21, 2022, Atlanta, GA, USA +Xin Dong et al. +Table 3: Performance of GDOD with MTL models on three datasets. The metrics are the average AUC and the average Logloss +on the test dataset. +Method +BookCrossing +IJCAI-15 +Alipay Advertising +Task1 +Task2 +Task1 +Task2 +Task1 +Task2 +Task3 +Basemodel +Optimizer +AUC +LogLoss +AUC +LogLoss +AUC +LogLoss +AUC +LogLoss +AUC +LogLoss +AUC +LogLoss +AUC +LogLoss +Shared-Bottom +Adam +0.7838 +0.7068 +0.7633 +0.7251 +0.6968 +0.7257 +0.7451 +0.7187 +0.7505 +0.7096 +0.7387 +0.6994 +0.8237 +0.6992 +GDOD +0.7922 +0.6969 +0.7817 +0.7211 +0.7268 +0.7231 +0.7555 +0.7159 +0.7723 +0.6991 +0.7571 +0.6951 +0.8390 +0.6956 +Cross-Stitch +Adam +0.7771 +0.7093 +0.7543 +0.7333 +0.6932 +0.7286 +0.7421 +0.7217 +0.7518 +0.7023 +0.7436 +0.6975 +0.8256 +0.6971 +GDOD +0.7954 +0.6860 +0.7830 +0.7203 +0.7114 +0.7248 +0.7567 +0.7142 +0.7708 +0.6992 +0.7577 +0.6953 +0.8384 +0.6975 +MMoE +Adam +0.7772 +0.7097 +0.7519 +0.7386 +0.6933 +0.7302 +0.7463 +0.7182 +0.7646 +0.7030 +0.7481 +0.6954 +0.8286 +0.6964 +GDOD +0.7912 +0.7019 +0.7789 +0.7216 +0.7169 +0.7243 +0.7532 +0.7157 +0.7716 +0.6997 +0.7596 +0.6944 +0.8353 +0.6973 +PLE +Adam +0.7810 +0.7078 +0.7595 +0.7300 +0.6945 +0.7315 +0.7481 +0.7173 +0.7686 +0.7060 +0.7494 +0.6950 +0.8308 +0.6973 +GDOD +0.7905 +0.7032 +0.7751 +0.7243 +0.7196 +0.7239 +0.7578 +0.7132 +0.7739 +0.6989 +0.7535 +0.6968 +0.8417 +0.6955 +SNR +Adam +0.7807 +0.7067 +0.7630 +0.7278 +0.7017 +0.7278 +0.7479 +0.7180 +0.7692 +0.7014 +0.7424 +0.6985 +0.8305 +0.6967 +GDOD +0.7895 +0.7033 +0.7736 +0.7246 +0.7103 +0.7251 +0.7541 +0.7161 +0.7745 +0.7003 +0.7566 +0.6955 +0.8388 +0.6959 +Table 4: AUC comparisons of different decomposition methods. The baseline MTL model is Shared-Bottom. Diff measures the +AUC gap between the decomposition method used in GDOD and other decomposition methods. +Decomposition Method +BookCrossing +IJCAI-15 +Alipay Advertising +Task1 +Task2 +Task1 +Task2 +Task1 +Task2 +Task3 +AUC +Diff +AUC +Diff +AUC +Diff +AUC +Diff +AUC +Diff +AUC +Diff +AUC +Diff +Random +0.5897 +-0.2025 +0.5848 +-0.1969 +0.5623 +-0.1645 +0.5477 +-0.2078 +0.5584 +-0.2139 +0.5434 +-0.2137 +0.6303 +-0.2087 +QR +0.7927 +0.0005 +0.7720 +-0.0097 +0.7091 +-0.0177 +0.7449 +-0.0106 +0.7630 +-0.0093 +0.7422 +-0.0149 +0.8287 +-0.0103 +RandDec +0.7926 +0.0004 +0.7776 +-0.0041 +0.7210 +-0.0058 +0.7538 +-0.0017 +0.7711 +-0.0012 +0.7495 +-0.0076 +0.8322 +-0.0068 +SVD +0.7922 +- +0.7817 +- +0.7268 +- +0.7555 +- +0.7723 +- +0.7571 +- +0.8390 +- +original model in task2 with the BookCrossing dataset. The magni- +tude of this improvement is fairly significant. Moreover, some MTL +networks also have addressed the negative transfer phenomenon. +For example, PLE separates shared components and task-specific +components and adopts a progressive routing mechanism to reduce +negative transfer. We can see that PLE outperforms other networks, +such as MMOE and Cross-Stitch. PLE with GDOD also achieves +0.01 AUC gain compared to the original model in most tasks. It +validates the effectiveness of GDOD and proves that mitigating +conflicting gradients can boost the performance of MTL models. +4.4 +Ablation Study: Effect of Different +Decomposition Methods +In this section, we examine the effect of different decomposition +methods in GDOD. Our approach relies on the singular vectors +from SVD to define the basis to identify the positive and negative +components of each task gradients. We compare SVD with several +decomposition methods: +• Random obtains the basis spanned by 𝑟 randomly chosen +orthogonal vectors in R. +• QR Decomposition is directly to decompose a matrix and +seek the matrix column space as the orthogonal basis. Gram- +Schmidt is a commonly used method to achieve this decom- +position. +• Randomized Approximate Matrix Decomposition (Rand- +Dec) [9] follows the framework that usually projects the orig- +inal matrix to a low-rank sample space and then computes +the approximate decomposition of the original matrix. +(a) AUC on BookCrossing +(b) AUC on IJCAI-15 +Figure 4: Performance with different dimensions of sub- +space S on BookCrossing and IJCAI-15 datasets. +Table 4 shows the AUC comparisons of different decomposi- +tion methods on BookCrossing, IJCAI-15 and Alipay Advertising +datasets. From Table 4, we can see that SVD performs the best in +most situations and Random achieves the worst performance. We +also observed a phenomenon that the magnitude of AUC diff about +task2 is greater than task1 in the BookCrossing dataset. It demon- +strates that a good choice of decomposition methods can mitigate +the negative transfer across all tasks. +4.5 +Ablation Study: Effect of Different +Dimensions of Subspace S +In this section, we examine the effect of different dimensions of +subspace S in GDOD. Figure 4 depicts the task AUC varies with dif- +ferent dimension of the subspace S on BookCrossing and IJCAI-15 +datasets. From Figure 4(b), we observe that it is better to decompose +all the task gradients in a larger dimensional subspace. In general, a + +GDOD: Effective Gradient Descent using Orthogonal Decomposition for Multi-Task Learning +CIKM ’22, October 17–21, 2022, Atlanta, GA, USA +Table 5: Performance comparisons of different optimization methods on Census-Income dataset. The baseline MTL model is +Shared-Bottom. +Method +Task1 +Task2 +Task3 +Task4 +Task5 +Task6 +AUC +Gain +AUC +Gain +AUC +Gain +AUC +Gain +AUC +Gain +AUC +Gain +Adam +0.93632 +0.99350 +0.90114 +0.98416 +0.83914 +0.84135 +Uncert +0.93997 +0.00365 +0.99351 +0.00001 +0.90711 +0.00597 +0.98429 +0.00013 +0.84424 +0.0051 +0.85916 +0.01781 +GradReg +0.94023 +0.00391 +0.99365 +0.00015 +0.90688 +0.00574 +0.98433 +0.00017 +0.85154 +0.0124 +0.86486 +0.02351 +GradNorm +0.94307 +0.00675 +0.99378 +0.00028 +0.90657 +0.00543 +0.98455 +0.00039 +0.83941 +0.00027 +0.87282 +0.03147 +MGDA +0.94060 +0.00428 +0.99390 +0.00040 +0.90480 +0.00366 +0.98440 +0.00024 +0.84690 +0.00776 +0.86000 +0.01865 +PCGrad +0.93781 +0.00149 +0.99382 +0.00032 +0.90309 +0.00195 +0.98431 +0.00015 +0.84728 +0.00814 +0.8712 +0.02985 +CAGrad +0.94145 +0.00513 +0.99415 +0.00065 +0.9067 +0.00556 +0.98437 +0.00021 +0.84979 +0.01066 +0.87646 +0.03511 +GDOD +0.94328 +0.00696 +0.99408 +0.00058 +0.90794 +0.00680 +0.98429 +0.00013 +0.8504 +0.01126 +0.87659 +0.03524 +Weighted-GDOD +0.94367 +0.00735 +0.99422 +0.00072 +0.90903 +0.00789 +0.98444 +0.00028 +0.85062 +0.01148 +0.88536 +0.04401 +(a) Task1 on BookCrossing +(b) Task2 on BookCrossing +(c) Task1 on IJCAI-15 +(d) Task2 on IJCAI-15 +Figure 5: Methods comparison with different weights of +tasks. The sum of the weights of the two tasks is one. +larger dimensional subspace possibly captures a richer description +of the matrix M. However, Figure 4(a) holds the opposite phenom- +enon. This is because a larger dimensional also creates the risk of +over-fitting especially in a limited dataset, such as the Bookcrossing +dataset. +4.6 +Ablation Study: Effect of Tasks with +Varying Weights +In this section, we examine the effect with varying weights for each +tasks. Figure 5 shows AUC varies with different weights for each +task on BookCrossing and IJCAI-15 datasets with several gradient- +based methods. The weight for each task is equally in the previous +setting. From Figure 5, we can see that a task with a higher weight +indication probability usually receives a higher AUC. It is obviously +that GDOD performs the best with varying task weights in most +situations. Moreover, for CAGrad, the performance of a task with a +smaller weight (the weight for task 2 is 0.1) reduces significantly. +This is because that CAGrad searches the new updated vector is +around 𝑔0 (average gradient vector). However, the reduction for +GDOD is smaller than other methods. It verify that GDOD is a more +robust algorithm. +4.7 +GDOD with More Tasks +In Algorithm 1, GDOD uses the helpful components which refer to +the projections of original gradients onto the basis vectors where all +task gradients agree in the direction to update the model parameters. +However, as the number of tasks increases, the components of all +tasks in the same direction will decrease. To deal with more tasks, +we propose a weighted-GDOD which defines a weight for task +components from the dimension of basis. For each basis vector, the +gradient components of all tasks are divided into two sets {𝑆+} and +{𝑆−} by the sign. Suppose {𝑆+} and {𝑆−} have 𝑎 and 𝑏 gradient +components respectively. The weight for each gradient component +is calculated as following: +• If 𝑎 ≥ 𝑏, the weights for gradient components in set 𝑆+ and +𝑆− are 𝑎−𝑏 +𝐾 +and 0. +• If 𝑎 < 𝑏, the weights for gradient components in set 𝑆+ and +𝑆− are 0 and 𝑏−𝑎 +𝐾 . +We also examine the effect of weighted-GDOD with the Census- +Income dataset that has six tasks. As shown in Table 5, we observe +that weighted-GDOD and GDOD achieve the best performance +in most tasks. Especially, weighted-GDOD and GDOD realize sig- +nificant improvements for task 5 and 6. Moreover, all results with +weighted-GDOD are proven to be better than GDOD, showing that +GDOD with a weighted policy is more effective with more tasks. +5 +CONCLUSION +In this paper, we present a novel optimization approach for MTL, +GDOD, which manipulates each task gradient using a decomposi- +tion built from the span of all task gradients. GDOD decomposes +gradients into task-shared and task-specific components explic- +itly and adopts a general update rule for avoiding interference +across all task gradients. Moreover, we present the convergence of +GDOD theoretically under both convex and non-convex assump- +tions. Experiment results on several multi-task datasets not only +demonstrate the significant improvement of GDOD performed to +existing MTL models but also outperform state-of-the-art optimiza- +tion methods in terms of AUC metric. Our future study would focus +on exploring other decomposition methods to optimize training +procedure for more effective and efficient multi-task learning. + +CIKM ’22, October 17–21, 2022, Atlanta, GA, USA +Xin Dong et al. +REFERENCES +[1] Rich Caruana. 1997. Multitask learning. 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Neural architecture search with reinforcement +learning. arXiv preprint arXiv:1611.01578 (2016). + diff --git a/EtFRT4oBgHgl3EQfBjcA/content/tmp_files/load_file.txt b/EtFRT4oBgHgl3EQfBjcA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8e5732cb48a41ba61a2d5b7818293b049d714c07 --- /dev/null +++ b/EtFRT4oBgHgl3EQfBjcA/content/tmp_files/load_file.txt @@ -0,0 +1,1138 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf,len=1137 +page_content='GDOD: Effective Gradient Descent using Orthogonal Decomposition for Multi-Task Learning Xin Dong∗ Ant Group Shanghai, China zhaoxin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='dx@antgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='com Ruize Wu∗ Ant Group Hangzhou, China kezhui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='wrz@antgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='com Chao Xiong Ant Group Shanghai, China xc272640@antgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='com Hai Li Ant Group Shanghai, China tianshu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='lh@antgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='com Lei Cheng Ant Group Hangzhou, China lei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='chenglei@antgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='com Yong He Ant Group Hangzhou, China heyong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='h@antgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='com Shiyou Qian Shanghai Jiao Tong University Shanghai, China qshiyou@sjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='cn Jian Cao Shanghai Jiao Tong University Shanghai, China cao-jian@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='sjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='cn Linjian Mo† Ant Group Shanghai, China linyi01@antgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='com ABSTRACT Multi-task learning (MTL) aims at solving multiple related tasks simultaneously and has experienced rapid growth in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' However, MTL models often suffer from performance degeneration with negative transfer due to learning several tasks simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Some related work attributed the source of the problem is the con- flicting gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' In this case, it is needed to select useful gradient updates for all tasks carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' To this end, we propose a novel op- timization approach for MTL, named GDOD, which manipulates gradients of each task using an orthogonal basis decomposed from the span of all task gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' GDOD decomposes gradients into task-shared and task-conflict components explicitly and adopts a general update rule for avoiding interference across all task gradi- ents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' This allows guiding the update directions depending on the task-shared components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Moreover, we prove the convergence of GDOD theoretically under both convex and non-convex assump- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Experiment results on several multi-task datasets not only demonstrate the significant improvement of GDOD performed to existing MTL models but also prove that our algorithm outperforms state-of-the-art optimization methods in terms of AUC and Logloss metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' CCS CONCEPTS Computing methodologies → Multi-task learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Trans- fer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' ∗Both authors contributed equally to this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' †Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='00 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='1145/3511808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='3557333 KEYWORDS multi-task learning, orthogonal decomposition, gradient conflict ACM Reference Format: Xin Dong, Ruize Wu, Chao Xiong, Hai Li, Lei Cheng, Yong He, Shiyou Qian, Jian Cao, and Linjian Mo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' GDOD: Effective Gradient Descent using Orthogonal Decomposition for Multi-Task Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' In Proceedings of the 31st ACM International Conference on Information and Knowledge Management (CIKM ’22), October 17–21, 2022, Atlanta, GA, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' ACM, New York, NY, USA, 10 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='1145/3511808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='3557333 1 INTRODUCTION Multi-task learning (MTL) aims to build a shared model that learns multiple related tasks simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Compared to single-task learning, it can significantly improve learning efficiency and pre- diction accuracy through knowledge sharing between tasks [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' This allows MTL models to deploy to a wide range of real-world applications, such as computer vision [15], natural language pro- cessing [4], online recommendation and advertising systems [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Recently, MTL has acted as a regularizer during network learn- ing, leading to more meaningful neural representations and better generalization [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' In practice, the training process of the MTL network is not al- ways ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Since the competition of shared parameter updates may harm individual tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' The MTL approach often leads to networks that accurately improve the performance of a subset of the tasks, while the rest suffer, a phenomenon referred to as negative transfer or destructive interference [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Minimizing the negative transfer is a key goal for MTL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' To mitigate this problem, several works [28, 34] opted to cluster tasks into groups based on prior beliefs about their similarity or relatedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Alternatively, some related work attributed the source of the problem to the gradient conflict [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Several approaches have been proposed to minimize conflict between the updates across multiple tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' In this context, we split these approaches into two categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' On one hand, gradients with different magnitudes lead to parameter updating dominated by a subset of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Consequently, a number arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='13465v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='LG] 31 Jan 2023 CIKM ’22, October 17–21, 2022, Atlanta, GA, USA Xin Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' of approaches have been developed to tune a set of task-weighting parameters to balance relative gradient magnitudes for different tasks [2, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' However, these approaches do not solve the problem of task gradients canceling out due to them pointing towards different directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' On the other hand, some approaches [14, 26, 33] find a common gradient descent direction for all tasks so that they do not cancel each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' However, such solutions either cannot distinguish the conflicting gradients explicitly or cannot mitigate conflicting gradients completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Here, we argue the gradient conflict problem and reveal an ad- vanced problem: how to distinguish the conflicting gradients and mitigate their impact on each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' For this purpose, we instead present an approach that straightly manipulates gradients and mit- igates the interference across tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Specifically, we decompose the gradient by the orthogonal basis in the subspace spanned by all task per-example gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' We analyze the updates of each task according to its impact on other tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' So that each task gradients can be decomposed into two components: 1) task-shared compo- nent which is helpful for all tasks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' and 2) task-conflict component which interferes with other tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Only the task-shared component is used to update the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' To achieve a tractable approach, we introduce a novel and robust algorithm, named GDOD, to estimate the subspace spanned by all task gradients and decompose each task update appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' As a result, we can integrate our approach with existing MTL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' To evaluate the performance of GDOD, we conduct extensive experiments on three available public multi- task datasets and a large-scale industrial dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Consequently, GDOD guarantees convergence in theory and outperforms other state-of-the-art optimization methods across all datasets in experi- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' In light of the above background, the main contributions of this paper are the following: We propose an optimization approach, named GDOD, to ma- nipulate each task gradient using an orthogonal decomposi- tion built from the span of all task gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' GDOD decom- poses gradients into task-shared and task-conflict compo- nents explicitly and adopts a general update rule for avoiding interference across all task gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' We prove the convergence of GDOD theoretically under both convex and non-convex assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' We conduct extensive experiments on several multi-task datasets to evaluate the effectiveness of GDOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Experiment results not only demonstrate the significant improvement of GDOD performed to existing MTL models but also out- perform state-of-the-art optimization methods across all datasets in terms of AUC metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' 2 RELATED WORK Efficient multi-task learning models and optimization approaches of MTL models are two research areas related to our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='1 Multi-Task Learning Models The learning conception of MTL that modeling the shared repre- sentation for related tasks brings many benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' However, MTL may suffer from negative transfer due to task conflicts as parame- ters are straightforwardly shared between tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' To deal with task conflicts, many works design different network architectures that allow optimal knowledge sharing between tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Cross-stitch network [21] and sluice network [25] propose to learn weights of linear combinations to fuse representations from different tasks selectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' However, the shared representations are combined with the same static weights in these models and the neg- ative transfer is not addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' More studies apply the gate structure and attention network for representation fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' MOE [10] splits the shared bottom layer into experts and combines experts through a gating network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' MMoE [18] and PLE [31] extend MOE to utilize different gate nets to aggregate experts for each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Similarly, MARN [35] employs multi-head self-attention to learn different representations with different feature sets for each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' However, none of the above works has explicitly addressed the issues of joint optimization of shared representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' There are also some works utilizing neural architecture search (NAS) approaches to find a good MTL network architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' SNR framework [17] controls connections between sub-networks by binary random variables and applies NAS [37] to search for the optimal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Similarly, Gumbel-matrix routing framework [19] learns to route MTL models formulated as a binary matrix with the Gumbel-Softmax trick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Moreover, [23] models the routing process as MDP and employs MARL [27] to train the routing network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' In contrast to these methods, we propose an approach to address the negative transfer problem in multi-task learning that allows us to learn the tasks simultaneously without the need for specific network design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='2 Optimization Methods in MTL Similar to our work, several prior researchers utilize some optimiza- tion methods to address the negative transfer problem in multi-task learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' A very common solution is to balance the impact of indi- vidual tasks on the training of the network by adaptively weighting the task-specific losses or gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' There have been some stud- ies developing a set of task-weighting parameters to balance the training procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Uncertainty Weights [11] devises a weighting method dependent on the homoscedastic uncertainty inherently linked to each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' These weights for each loss function are trained together with the MTL model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' GradNorm [2] reduces the task imbalances by weighting task losses so that their gradients are similar in magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' There are several methods dynamically weighting the loss functions of tasks by the learning speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' [15] and [16] explicitly set a weight to a task loss using a ratio of the current loss to the previous loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' However, these loss weighting methods do not work well all the time in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Moreover, the for- mulation design of the weighing calculation is generally empirical and lacks theoretical derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' There have also been some optimization methods to improve MTL performance by mitigating conflicting gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' The problem of conflicting gradients has been previously explored in multi-task learning as well as continual learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' [7] and [5] choose to ignore the gradients of auxiliary tasks if the direction is not similar to the main task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' [22] overcomes catastrophic forgetting by maximizing the dot product between task gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' MGDA [6] employs the condition of the Pareto stationary point for multi-objective opti- mization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' It finds a linear combination of gradients that reduces GDOD: Effective Gradient Descent using Orthogonal Decomposition for Multi-Task Learning CIKM ’22, October 17–21, 2022, Atlanta, GA, USA (a) Gradient Descent (b) GDOD Figure 1: The overview of two optimization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' (a) Nor- mal gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' (b) Gradient manipulation example in the 2-D plane with GDOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' every loss function simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' PCGrad [33] projects conflict- ing gradients to each other, which achieves a similar simultaneous descent effect as MGDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' CAGrad [14] looks for an update vector that maximizes the worst local improvement of any objective in a neighborhood of the average gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' And the performance can shift from GD-like to MGDA-like by a hyper-parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' These methods deal with the gradient decent independent of the model structure and can be combined with normal optimizers such as SGD and Adam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' However, the above methods either cannot distinguish the conflicting gradients explicitly or cannot mitigate conflicting gradients completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' 3 MULTI-TASK LEARNING USING GDOD In this section, to realize effective gradient descent, we present a novel optimization approach that mitigates conflicting gradients across all tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='1 Preliminaries: Notation and Problem Consider an MTL model with 𝐾 different tasks that we want to learn simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' For simplicity, we assume that they share an input space X and a collection of task spaces {Y𝑖}𝑖 ∈[𝐾 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Each of the inputs in X is associated with a set of labels for all 𝐾 tasks, forming a large dataset of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' data points {𝑥𝑗,𝑦1 𝑗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=',𝑦𝐾 𝑗 }𝑗 ∈[𝑁 ] of 𝑁 observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' The MTL model can be divided into two parts: a backbone and multiple heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' The backbone contains the shared parameters 𝜃 and transforms the input X into a shared representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Next, this representation is fed to each task-specific head to produce output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' We consider the empirical loss for each task𝑖 as L𝑖 (𝜃,𝜃𝑖) = � 𝑗 ∈𝑁 𝑙 𝑗 𝑖 , where 𝑙 𝑗 𝑖 is the loss for the 𝑗-th observation and 𝜃𝑖 are task-specific head parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' We aim to find both the backbone parameters 𝜃 and the task-specific parameters 𝜃𝑖 to minimize the loss L𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' More formally, for a set of 𝐾 tasks, the final goal is to minimize the multi-task loss as: 𝑚𝑖𝑛 𝜃,𝜃1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=',𝜃𝐾 𝐾 ∑︁ 𝑖=1 L𝑖 (𝜃,𝜃𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' (1) During training, each task competes for updating backbone pa- rameters 𝜃 to minimize the loss for each individual task, this leads to the occurrence of negative transfer [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Therefore, we focus on the learning of the backbone parameters 𝜃 in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' The task-specific parameters 𝜃𝑖 are updated by normal gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' For simplicity, we omit the task-specific parameters 𝜃𝑖 in loss L𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Generally, the gradient of the loss with respect to a particular shared parameter 𝜃 from task 𝑖 is 𝑔𝑖 = ∇𝜃L𝑖 (𝜃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Using gradient descent to minimize the multi-task loss, we obtain the update rule for a task-shared parameter 𝜃 as: 𝜃 = 𝜃 − 𝛾 𝐾 ∑︁ 𝑖=1 ∇𝜃L𝑖 (𝜃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' (2) The above expression shows that the overall success of an MTL model is dependent on the individual task gradients and their rela- tionship to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' However, task gradients might cancel each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Or a subset of tasks might dominate the gradients and point towards a direction that does not improve any of the individual tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' In this paper, we provide an approach that straightly manipulates gradients to mitigate the conflicting gradient problem in MTL mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Our approach decomposes each gradient by an orthogonal basis which is a subspace spanned by all task per-example gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' It decomposes the gradients of all the tasks into two components, and only the helpful component is used to update the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='2 Gradient Descent using Orthogonal Decompotision This section introduces a new optimization method to improve generalization for all tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Figure 1 shows an illustration of the core idea of GDOD which manipulates each task gradients to maximize their usefulness to all tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' To ensure the convergence and stability of the optimization process, we modify the gradients for each task so as to minimize negative conflict with other task gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Exactly, we decompose each task gradients into two components: 1) task- shared component which is helpful for all tasks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' and 2) task-conflict component which interferes with other tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Then, GDOD only utilizes the task-shared components from all tasks to update model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' At each training step, suppose there are 𝑚 samples in a mini- batch of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' GDOD first collects the gradients of the losses with respect to 𝜃 for individual examples from all tasks as M𝑇 = {{∇𝜃L𝑗 1}𝑇, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=', {∇𝜃L𝑗 𝐾 }𝑇 }∀𝑗 ∈[𝑚], (3) where ∇𝜃L𝑗 𝑖 ∈ R𝑚×𝐷 and 𝐷 is the dimension of the model pa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Then, we define a subspace S by the span of the gradi- ent vectors in M𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Any linear combination of each task gradients lies in this subspace, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=', 𝑔𝑖 = E(∇𝜃L𝑗 𝑖 ) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Note that, the di- mension of the subspace S is 𝑟, which is the rank of the matrix M ∈ R𝑟×𝐷 (𝑟 ≤ 𝑚𝐾 << 𝐷).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Third, we project each task gradi- ents onto S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' This allows distinguishing between the directions of each task update which helps other task losses and those which have a negative impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Consequently, we can decompose each task gradients as 𝑔𝑖 = 𝑔𝑠ℎ 𝑖 + 𝑔𝑐𝑜𝑛 𝑖 , (4) where 𝑔𝑠ℎ 𝑖 ∈ S is the portion of the gradient that improves all task results and 𝑔𝑐𝑜𝑛 𝑖 ∈ S is the portion of the gradient that damages some task results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='4 92 92 91 91 +92 92 911+92 q 92 Task1 Target 92 Task2 Target gi 91 ContourLine of Loss1 Contour Line of Loss2 b2 Task1 Gradient gi Task2 Gradient UpdatedGradient Orthogonal BasisCIKM ’22, October 17–21, 2022, Atlanta, GA, USA Xin Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' In order to decompose each task gradient, we define an orthog- onal basis for subspace S as {𝑏𝑢}𝑢∈[𝑟 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' On the orthogonal basis, we can measure whether the components of each gradient is agree or disagree with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' It is said that the two gradients agree along 𝑏𝑢 if and only if 𝑠𝑖𝑔𝑛(𝑔𝑇 𝑖 · 𝑏𝑢) = 𝑠𝑖𝑔𝑛(𝑔𝑇 𝑗 · 𝑏𝑢) for task 𝑖 and 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' This mean that the gradient components {𝑔𝑠ℎ 𝑖 }𝑖 ∈[𝐾 ] refer to the projections of {𝑔𝑖}𝑖 ∈[𝐾 ] onto the basis vectors where all task gra- dients agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' On the contrary, {𝑔𝑐𝑜𝑛 𝑖 }𝑖 ∈[𝐾 ] refer to the gradient components where the directions are disagree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' By this decomposi- tion, 𝑔𝑠ℎ 𝑖 helps for all task, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=', (𝑔𝑠ℎ 𝑖 )𝑇 · 𝑔𝑗 > 0 for any other task 𝑗, while 𝑔𝑐𝑜𝑛 𝑖 interferes with some tasks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=', (𝑔𝑐𝑜𝑛 𝑖 )𝑇 · 𝑔𝑘 < 0 for task 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' The remaining problem is how to select an orthogonal basis for the span of all task gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' There are multiple methods, such as Schmidt Decomposition, singular vector decomposition (SVD) and randomized approximate matrix decomposition [9], to obtain the basis {𝑏𝑢}𝑢∈[𝑟 ] at each training time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' The method is critical since the two components of 𝑔𝑠ℎ 𝑖 and 𝑔𝑐𝑜𝑛 𝑖 are helpful or harmful de- pending on how they agree with the projection of all task gradients onto this basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' As some work [20] proved that along the directions associated with singular vectors of a neural network Jacobian can generalize well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' In this paper, the subspace S is constructed from a mini-batch of all task gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' In brief, we select the singular vectors of the matrix of M as the basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' We also compare the impact of different decomposition methods mentioned above in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Algorithm 1 GDOD Update At Each Training Step Require: 𝜃, 𝛾: model parameters shared with all tasks, learning rate Require: M𝑇 = {{∇𝜃L𝑗 1}𝑇, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=', {∇𝜃L𝑗 𝐾 }𝑇 }∀𝑗 ∈[𝑚]: gradients with respect to 𝜃 for each task 1: 𝑔𝑖 = 1 𝑚 �𝑚 𝑗=1 ∇𝜃L𝑗 𝑖 , ∀𝑖 ∈ [𝐾] 2: 𝐵 ← 𝑆𝑉𝐷_𝑃𝑟𝑜(M) 3: 𝑝𝑖 = 𝐵(𝑔𝑖)𝑇, ∀𝑖 ∈ [𝐾] 4: 𝑝𝑠ℎ 𝑖 , 𝑝𝑐𝑜𝑛 𝑖 = � 1[𝑝1 ⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='⊙𝑝𝐾 ≥0] � ⊙ 𝑝𝑖, � 1[𝑝1 ⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='⊙𝑝𝐾 <0] � ⊙ 𝑝𝑖, ∀𝑖 ∈ [𝐾] // ⊙ is the hadamard product operator 5: 𝑔𝑠ℎ 𝑖 ,𝑔𝑐𝑜𝑛 𝑖 = (𝑝𝑠ℎ 𝑖 )𝑇 𝐵, (𝑝𝑐𝑜𝑛 𝑖 )𝑇 𝐵, ∀𝑖 ∈ [𝐾] 6: return update 𝜃 = 𝜃 − 𝛾 �𝐾 𝑖=1 𝑔𝑠ℎ 𝑖 We summarize the full update procedure in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Sup- pose the gradient for task 𝑖 is 𝑔𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' GDOD proceeds as follows: At each training step, it first obtains the orthogonal basis 𝐵 from the span of all task gradients M by SVD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' The 𝑆𝑉𝐷_𝑃𝑟𝑜 is the procedure of SVD and 𝐵 ∈ R𝑟×𝐷 is the non-zero and right-singular vectors of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Secondly, GDOD decomposes each task gradient on the or- thogonal basis 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Thirdly, it obtains the helpful component 𝑔𝑠ℎ 𝑖 and harmful component𝑔𝑐𝑜𝑛 𝑖 for each task gradient respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Finally, it utilizes the helpful components from all tasks to update model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Algorithm 1 shows that this procedure is simple to implement and ensures that the modified gradients we update for each task have no conflict with the other tasks in each training step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Hence, GDOD mitigates the conflicting gradient problem in MTL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Moreover, to reduce the computational complexity, the rank of the matrix M can be reduced by grouping gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' For example, the gradients are divided into different groups and an average pooling is performed on the gradients in the same group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' We also examine the impact of different dimensions of subspace S in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Furthermore, by replacing the original gradients �𝐾 𝑖=1 𝑔𝑖 with the task-shared gradients �𝐾 𝑖=1 𝑔𝑠ℎ 𝑖 , GDOD can be combined with any other gradient-based optimizer, such as SGD with momentum and Adam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='3 Discussion Several gradient-based approaches have been proposed to manip- ulate each task gradients to obtain a new update vector and have shown improved performance on existing MTL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' MGDA [6] finds a linear combination of gradients that reduces every loss function simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' It proposes to minimize the following combination of task gradients: min1 2 ∥ 𝐾 ∑︁ 𝑖=1 𝑤𝑖𝑔𝑖 ∥2, 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' 𝐾 ∑︁ 𝑖=1 𝑤𝑖 = 1 and ∀𝑖, 𝑤𝑖 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' (5) From equation 5, MGDA seeks the linear combination of gradients that results in the smallest norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Tasks that have larger gradients will become attenuated by MGDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' For example, if an MTL model has two tasks where task 1 is under-optimized and task 2 is near a local optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' The model has a large gradient for task 1 and a rela- tive small gradient for task 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' In this situation, even if it is possible to improve task 1 a lot while not affecting the performance of task 2, MGDA may not take that move because the best improvement on task 1 is bounded by its improvement on task 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' This often causes slow improvement of MGDA in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Moreover, PCGrad [33] projects conflicting gradients to the or- thogonal direction of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' It sets a universal gradient simi- larity objective of zero for any two tasks explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Consequently, if 𝑔𝑖 · 𝑔𝑗 < 0, PCGrad projects these conflicting gradients to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' It replace 𝑔𝑖 (𝑔𝑗) by its projection onto the normal plane of 𝑔𝑗 (𝑔𝑖): 𝑔𝑖 = 𝑔𝑖 − 𝑔𝑖 · 𝑔𝑗 ||𝑔𝑗 ||2𝑔𝑗 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' (6) It is not effective for the case of positive gradient similarities with 𝑔𝑖 · 𝑔𝑗 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' In fact, the two tasks share positive cosine similarities such that the precondition for PCGrad would never be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' However, GDOD alters gradients more preemptively under both positive and negative cases, taking more proactive measurements in updating the gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' CAGrad [14] finds a linear combination of a new updated vector𝑑 that is a linear combination of the original individual task gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' It obtains the updated vector𝑑 by solving the following optimization problem: max 𝑑 ∈𝑅 min 𝑖 ∈[𝐾 ] < 𝑔𝑖,𝑑 >, 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' ∥𝑑 − 𝑔0∥ ≤ 𝑐∥𝑔0∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' (7) The difference between MGDA and CAGrad is that the new updated vector 𝑑 is searched around the 0 vector for MGDA and 𝑔0 (average gradient vector) for CAGrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' CAGrad chooses the average loss over all tasks as the main objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Nevertheless, we find that CAGrad is not robust with different task weights in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' For our method, we find an updated vector guided by the singular GDOD: Effective Gradient Descent using Orthogonal Decomposition for Multi-Task Learning CIKM ’22, October 17–21, 2022, Atlanta, GA, USA vectors of the Jacobian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' As some works [20] point out that using the principal vectors as directions of descent instead of the mean induces a more robust algorithm since the mini-batch average gradient is susceptible to outliers and skews from replicated data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='4 Theoretical Analysis of GDOD In this section, we analyze the convergence of GDOD with the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Let L(𝜃𝑡) represents the full batch losses of all 𝐾 tasks at training step 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Suppose the gradients {𝑔𝑖}𝑖 ∈[𝐾 ] of all 𝐾 tasks are Lipschitz continuous with 𝐿 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Then, the GDOD update rule 𝜃𝑡+1 = 𝜃𝑡 − 𝛾 �𝐾 𝑖=1 𝑔𝑠ℎ 𝑖 with learning rate 𝛾 ≤ 1 𝐿 will converge to either (1) the optimal value if L(𝜃) is convex or (2) a stationary point if L(𝜃) is non-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' According to the Lipschitz smoothness assumption, we obtain the following inequality: L(𝜃𝑡+1) ≤ L(𝜃𝑡) + ∇𝜃L(𝜃𝑡)𝑇 (Δ𝜃) + 1 2𝐿∥Δ𝜃 ∥2 2 Now, we can plug in the GDOD update by replacing Δ𝜃 = 𝜃𝑡+1−𝜃𝑡 = −𝛾 �𝐾 𝑖=1 𝑔𝑠ℎ 𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' We then obtain: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='L(𝜃𝑡+1) ≤ L(𝜃𝑡) + 𝛾( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝐾 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='∑︁ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑖=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑔𝑖)𝑇 (− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝐾 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='∑︁ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑖=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑔𝑠ℎ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑖 ) + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='2𝐿∥𝛾 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝐾 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='∑︁ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑖=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑔𝑠ℎ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑖 ∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='= L(𝜃𝑡) + 𝛾( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝐾 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='∑︁ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑖=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑔𝑠ℎ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝐾 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='∑︁ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑖=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑔𝑐𝑜𝑛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=')𝑇 (− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝐾 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='∑︁ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑖=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑔𝑠ℎ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑖 ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='2𝐿𝛾2∥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝐾 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='∑︁ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑖=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑔𝑠ℎ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑖 ∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='= L(𝜃𝑡) − 𝛾( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝐾 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='∑︁ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑖=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑔𝑠ℎ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑖 )2 − 𝛾( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝐾 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='∑︁ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑖=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑔𝑐𝑜𝑛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=')𝑇 ( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝐾 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='∑︁ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑖=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑔𝑠ℎ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑖 ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='2𝐿𝛾2∥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝐾 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='∑︁ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑖=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑔𝑠ℎ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑖 ∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='(8) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='= L(𝜃𝑡) − 𝛾( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝐾 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='∑︁ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑖=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑔𝑠ℎ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑖 )2 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='2𝐿𝛾2∥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝐾 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='∑︁ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑖=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑔𝑠ℎ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑖 ∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='(9) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='= L(𝜃𝑡) − (1 − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='2𝐿𝛾)𝛾∥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝐾 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='∑︁ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑖=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑔𝑠ℎ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='𝑖 ∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='Note that in going equation 8 to 9 in the above proof,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' we use the fact that (𝑔𝑐𝑜𝑛 𝑖 )𝑇𝑔𝑠ℎ 𝑗 = 0 for any two tasks 𝑖 and 𝑗 due to orthogonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' We define the updated gradient at training step 𝑡 is 𝑔𝑠ℎ 𝑡 = �𝐾 𝑖=1 𝑔𝑠ℎ 𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Using 𝛾 ≤ 1 𝐿 , we know that − (1 − 1 2𝐿𝛾) = 1 2𝐿𝛾 − 1 ≤ 1 2𝐿( 1 𝐿 ) − 1 = −1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Plugging this into the last expression above, we can conclude the following: L(𝜃𝑡+1) ≤ L(𝜃𝑡) − 1 2𝛾∥𝑔𝑠ℎ 𝑡 ∥2 2 (10) ≤ L(𝜃𝑡) Thus, the above theorem ensures that GDOD is minimizing L(𝜃𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' If L(𝜃) is convex and differentiable, hence repeatedly applying GDOD process can reach the optimal value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Assume L(𝜃) is non-convex, using telescope sum to equation 10, we have L(𝜃𝑇 ) − L(𝜃0) ≤ −1 2𝛾 𝑇−1 ∑︁ 𝑡=0 ∥𝑔𝑠ℎ 𝑡 ∥2 2 (11) Thus, we have: min 0≤𝑡 ≤𝑇 ∥𝑔𝑠ℎ 𝑡 ∥2 2 ≤ 1 𝑇 𝑇−1 ∑︁ 𝑡=0 ∥𝑔𝑠ℎ 𝑡 ∥2 2 ≤ 2(L(𝜃0) − L(𝜃𝑇 )) 𝑇𝛾 ≤ 2(L(𝜃0) − L∗) 𝑇𝛾 (12) where L∗ is the minimal function value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Therefore, GDOD updating with gradients 𝑔𝑠ℎ 𝑡 can converge to a stationary point in O( 1 𝑇 ) steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' □ Therefore, we prove GDOD converges to either the optimal value if L(𝜃) is convex or a stationary point if L(𝜃) is non-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' 4 EXPERIMENTS In this section, we evaluate the performance and effectiveness of GDOD with four multi-task datasets from different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' We first evaluate the performance of GDOD as well as several state- of-the-art optimization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Then, we verify that GDOD is model-agnostic and can improve performance for any MTL models with shared parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Finally, we present ablation experiments to explain the impact of hyper-parameter selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Table 1: The statistics of the four datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Dataset Phase Users Items Samples BookCrossing Train 92,792 239,029 919,824 Test 42,194 99,404 229,956 IJCAI-15 Train 237,295 274,709 2,142,528 Test 106,023 127,772 544,025 Alipay Advertising Train 7,579,571 1,098 14,298,291 Test 5,822,077 835 10,740,289 Census-Income Train 199,523 Test 99,762 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='1 Datasets and Settings 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='1 Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' We use three public multi-task datasets from dif- ferent domains and a large-scale real-world advertising dataset to verify the effectiveness of GDOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' The statistics of the datasets are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' These datasets are described as follows: BookCrossing Dataset [36] collects user ratings in the Book-Crossing community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' It includes 278,858 users who provide 1,157,112 ratings about 271,379 books.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' As suggested in the original paper [36], we define the following two related CIKM ’22, October 17–21, 2022, Atlanta, GA, USA Xin Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' prediction tasks based on this dataset: 1) predict whether a user has rated a book;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' 2) predict whether a rating score from a user on a book is higher than or equal to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' IJCAI-15 Dataset [32] is collected from the E-commerce website Tmall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' It is a public dataset used in the IJCAI2015 repeat buyers prediction competition hosted by Alibaba Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' It contains 241,093 users with 2,295,706 instances on 237,564 items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' We model two related prediction tasks involv- ing CVR (Conversion Rate): 1) predict whether a user adds an item to his favourites after clicking;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' 2) predict whether a user buys an item after adding it to his favourites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Alipay Advertising Dataset is collected over three months from user traffic logs of a commercial advertising system in the Alipay App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' It contains 7,630,003 users who produce 25,038,580 samples about 1,120 advertisements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' One CTR task to predict whether a user clicks an item and two CVR tasks similar with IJCAI-15 dataset are modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Census-Income(KDD) Dataset [13] is a dataset extracted from the 1994 census database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' It contains 199,523 instances with 42 demographic and employment related features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Given a person, we model six related prediction tasks based on this dataset contains: 1) predict whether the person’s income exceeds $50K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' 2) predict whether the person’s marital status has never married;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' 3) predict whether the person’s education level is at least college;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' 4) predict whether the person’s em- ployment status is full time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' 5) predict whether the person’s gender is male;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' and 6) predict whether the person’s race is white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='2 Comparative Optimization Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' We compare GDOD with seven SOTA optimization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Adam is used as the baseline to compute the performance gains of other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Uncertainty Weights (Uncert) [11] uses a joint likelihood formulation to derive task weights based on the intrinsic uncertainty in each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' GradNorm [2] reduces the task imbalances by weighting task losses so that their gradients are similar in magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' MGDA [6] applies a multiple-gradient descent algorithm for MTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' It finds a linear combination of gradients that reduces every task loss simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Gradient Regularization (GradReg) [30] proposes a gra- dient regularization term that minimizes task interference by enforcing near orthogonal gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' PCGrad [33] projects conflicting gradients to the orthogonal direction of each other, so that achieving a similar simulta- neous descent effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' CAGrad [14] looks for an update gradient vector in the neighborhood of the average gradient that minimizes the average loss and leverages the worst local improvement of individual tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='3 Baseline MTL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' We evaluate the effect of our GDOD with the following representative MTL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Shared-Bottom [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Shared-Bottom shares the embedding layers and a low-level feature extraction layer (MLP) for all tasks and each task has its own task-specific high-level layers built on top of the shared layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Cross-Stitch [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' It fuses the tower layers of tasks by linear transformation based on the Shared-Bottom model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' MMOE [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' MMOE transforms the shared low-level layers into sub-networks and uses different gating networks for tasks to utilize different sub-networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' SNR [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' SNR modularizes the shared low-level layers into parallel sub-networks and uses a transformation matrix mul- tiplied by a scalar coding variable to learn their connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' PLE [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' PLE separates shared components and task-specific components and adopts a progressive routing mechanism to achieve more effective information sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='4 Evaluation Metrics .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' For fair comparisons, we employ AUC and Logloss as our evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' AUC is the Area Under the ROC Curve over the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' It measures the goodness of order by ranking all the items with predicted CTR in the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' It is noticeable that a slightly higher AUC at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='001-level is regarded as significant for CTR/CVR prediction tasks, which has been pointed out in existing works [3, 8, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Note that, the larger AUC shows better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Logloss is the loss value on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' The smaller Logloss means better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='5 Implementation Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' For all the baseline MTL models, there are trained by the Adam optimizer [12] with an initial learning rate of 1e-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' The mini-batch size is fixed to 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' The embedding size of each sparse feature is set to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' The hidden sizes of the two shared hidden layers in shared-bottom model are [256, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' The number of sub-networks/experts in SNR, PLE and MMoE is set to 8 and the hidden size of each sub-network/expert is 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' There are two specific tower hidden layers with the size of [16, 1] for each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Moreover, the weights in GradReg is tuned in [1e-1, 1e-2, 1e-3, 1e-4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' For GradNorm, the hyper-parameter 𝛼 is tuned in [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' The hyper-parameter 𝑐 in CAGrad is tuned in [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' All hyper-parameters are settled with the best performance on each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' For GDOD, the training examples are divided into 16 groups in a mini-batch at each training step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' For PCGrad, CAGrad and GDOD, they are combined with Adam by passing the computed update to replace the original gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' We repeat all experiments 5 times and report the averaged results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='2 Optimization Method Comparison Table 2 shows the AUC of the comparative results on the BookCross- ing dataset, IJCAI-15 dataset and Advertising dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Focusing on the detail, the shared-bottom model combined with GDOD achieves higher AUC compared to other optimization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Moreover, we have the following four observations: 1) GDOD, PCGrad and CAGrad outperform other five optimization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' This indi- cates that optimization methods manipulated per-task gradients are more practical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' 2) GDOD achieves better performances than PCGrad and CAGrad, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=', GDOD achieves 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='0064 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='0084 AUC gains compared to PCGrad and CAGrad in task2 with BookCrossing dataset respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' The magnitude of this improvement is fairly GDOD: Effective Gradient Descent using Orthogonal Decomposition for Multi-Task Learning CIKM ’22, October 17–21, 2022, Atlanta, GA, USA Table 2: Performance comparisons of different optimization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' The baseline MTL model is Shared-Bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Gain mea- sures the AUC improvement between Adam with other optimization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Optimization Method BookCrossing IJCAI-15 Alipay Advertising Task1 Task2 Task1 Task2 Task1 Task2 Task3 AUC Gain AUC Gain AUC Gain AUC Gain AUC Gain AUC Gain AUC Gain Adam 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='7838 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='7633 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='6968 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='7451 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='7505 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='7387 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='8237 Uncert 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='0124 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='7599 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='0094 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='7475 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='0088 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='8264 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='0027 GradReg 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='0130 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='7635 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='0130 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='7513 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='0126 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='8273 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='0036 GradNorm 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' (b) BookCrossing task2 loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' (c) IJCAI-15 task1 loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' (d) IJCAI-15 task2 loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Figure 2: Test loss comparisons about several optimization methods on BookCrossing and IJCAI-15 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' In all cases GDOD outperforms all other optimization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' (a) Task1 on BookCrossing (b) Task2 on BookCrossing (c) Task1 on IJCAI-15 (d) Task2 on IJCAI-15 Figure 3: Test AUC comparisons about several optimization methods on BookCrossing and IJCAI-15 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' In all cases GDOD outperforms all other optimization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Because GDOD implements a decomposition method that can distinguish the conflicting gradients effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' 3) Several situations with Uncert and GradReg are proven to be worse than Adam, showing the applicability of re-weighting methods is poor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' 4) MGDA seems to perform worse than some re-weighting methods in some tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' This is because MGDA will attenuate the performance of tasks that have higher gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Overall, these results verify that GDOD is a highly effective optimization method to avoid task competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Moreover, Figure 2 illustrates the test loss curves during the train- ing procedure on BookCrossing and IJCAI-15 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' From these curves, GDOD can be shown to achieve the lowest LogLoss than any other optimization method with a fixed step size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Therefore, these results demonstrate that GDOD can accelerate convergence and achieve good performance at the same step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' We also show the AUC curves during the training procedure on the BookCrossing dataset and IJCAI-15 dataset in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' It is observed that GDOD achieves the highest AUC compared to all the other optimization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Moreover, with a fixed training step, GDOD performs the best performance in most experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' These results demon- strate that GDOD outperforms other compared SOTA optimization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='3 GDOD with Multi-task Models Table 3 shows the AUC and Logloss of the comparison results on BookCrossing, IJCAI-15 and Alipay Advertising datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Focusing on the detail of Table 3, all MTL models combined with GDOD achieve higher AUC and lower Logloss compared to the original MTL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' These results confirm that GDOD improves the perfor- mance for multi-task learning benchmarks by avoiding interference across all task gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' For example, the Cross-Stitch model with GDOD optimization achieves 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='0287 AUC gain compared to the CIKM ’22, October 17–21, 2022, Atlanta, GA, USA Xin Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Table 3: Performance of GDOD with MTL models on three datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' The metrics are the average AUC and the average Logloss on the test dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Method BookCrossing IJCAI-15 Alipay Advertising Task1 Task2 Task1 Task2 Task1 Task2 Task3 Basemodel Optimizer AUC LogLoss AUC LogLoss AUC LogLoss AUC LogLoss AUC LogLoss AUC LogLoss AUC LogLoss Shared-Bottom Adam 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} 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decomposition methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' The baseline MTL model is Shared-Bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Diff measures the AUC gap between the decomposition method used in GDOD and other decomposition methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Decomposition Method BookCrossing IJCAI-15 Alipay Advertising Task1 Task2 Task1 Task2 Task1 Task2 Task3 AUC Diff AUC Diff AUC Diff AUC Diff AUC Diff AUC Diff AUC Diff Random 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='5897 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='7711 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='0012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='7495 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='0076 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='8322 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='0068 SVD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='7922 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='7817 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='7268 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='7555 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='7723 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='7571 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='8390 original model in task2 with the BookCrossing dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' The magni- tude of this improvement is fairly significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Moreover, some MTL networks also have addressed the negative transfer phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' For example, PLE separates shared components and task-specific components and adopts a progressive routing mechanism to reduce negative transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' We can see that PLE outperforms other networks, such as MMOE and Cross-Stitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' PLE with GDOD also achieves 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='01 AUC gain compared to the original model in most tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' It validates the effectiveness of GDOD and proves that mitigating conflicting gradients can boost the performance of MTL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='4 Ablation Study: Effect of Different Decomposition Methods In this section, we examine the effect of different decomposition methods in GDOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Our approach relies on the singular vectors from SVD to define the basis to identify the positive and negative components of each task gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' We compare SVD with several decomposition methods: Random obtains the basis spanned by 𝑟 randomly chosen orthogonal vectors in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' QR Decomposition is directly to decompose a matrix and seek the matrix column space as the orthogonal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Gram- Schmidt is a commonly used method to achieve this decom- position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Randomized Approximate Matrix Decomposition (Rand- Dec) [9] follows the framework that usually projects the orig- inal matrix to a low-rank sample space and then computes the approximate decomposition of the original matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' (a) AUC on BookCrossing (b) AUC on IJCAI-15 Figure 4: Performance with different dimensions of sub- space S on BookCrossing and IJCAI-15 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Table 4 shows the AUC comparisons of different decomposi- tion methods on BookCrossing, IJCAI-15 and Alipay Advertising datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' From Table 4, we can see that SVD performs the best in most situations and Random achieves the worst performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' We also observed a phenomenon that the magnitude of AUC diff about task2 is greater than task1 in the BookCrossing dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' It demon- strates that a good choice of decomposition methods can mitigate the negative transfer across all tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='5 Ablation Study: Effect of Different Dimensions of Subspace S In this section, we examine the effect of different dimensions of subspace S in GDOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Figure 4 depicts the task AUC varies with dif- ferent dimension of the subspace S on BookCrossing and IJCAI-15 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' From Figure 4(b), we observe that it is better to decompose all the task gradients in a larger dimensional subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' In general, a GDOD: Effective Gradient Descent using Orthogonal Decomposition for Multi-Task Learning CIKM ’22, October 17–21, 2022, Atlanta, GA, USA Table 5: Performance comparisons of different optimization methods on Census-Income dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' The 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='03524 Weighted-GDOD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='94367 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='00735 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='99422 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='00072 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='90903 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='00789 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='98444 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='00028 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='85062 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='01148 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='88536 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='04401 (a) Task1 on BookCrossing (b) Task2 on BookCrossing (c) Task1 on IJCAI-15 (d) Task2 on IJCAI-15 Figure 5: Methods comparison with different weights of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' The sum of the weights of the two tasks is one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' larger dimensional subspace possibly captures a richer description of the matrix M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' However, Figure 4(a) holds the opposite phenom- enon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' This is because a larger dimensional also creates the risk of over-fitting especially in a limited dataset, such as the Bookcrossing dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='6 Ablation Study: Effect of Tasks with Varying Weights In this section, we examine the effect with varying weights for each tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Figure 5 shows AUC varies with different weights for each task on BookCrossing and IJCAI-15 datasets with several gradient- based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' The weight for each task is equally in the previous setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' From Figure 5, we can see that a task with a higher weight indication probability usually receives a higher AUC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' It is obviously that GDOD performs the best with varying task weights in most situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Moreover, for CAGrad, the performance of a task with a smaller weight (the weight for task 2 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='1) reduces significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' This is because that CAGrad searches the new updated vector is around 𝑔0 (average gradient vector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' However, the reduction for GDOD is smaller than other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' It verify that GDOD is a more robust algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content='7 GDOD with More Tasks In Algorithm 1, GDOD uses the helpful components which refer to the projections of original gradients onto the basis vectors where all task gradients agree in the direction to update the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' However, as the number of tasks increases, the components of all tasks in the same direction will decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' To deal with more tasks, we propose a weighted-GDOD which defines a weight for task components from the dimension of basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' For each basis vector, the gradient components of all tasks are divided into two sets {𝑆+} and {𝑆−} by the sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Suppose {𝑆+} and {𝑆−} have 𝑎 and 𝑏 gradient components respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' The weight for each gradient component is calculated as following: If 𝑎 ≥ 𝑏, the weights for gradient components in set 𝑆+ and 𝑆− are 𝑎−𝑏 𝐾 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' If 𝑎 < 𝑏, the weights for gradient components in set 𝑆+ and 𝑆− are 0 and 𝑏−𝑎 𝐾 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' We also examine the effect of weighted-GDOD with the Census- Income dataset that has six tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' As shown in Table 5, we observe that weighted-GDOD and GDOD achieve the best performance in most tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Especially, weighted-GDOD and GDOD realize sig- nificant improvements for task 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Moreover, all results with weighted-GDOD are proven to be better than GDOD, showing that GDOD with a weighted policy is more effective with more tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' 5 CONCLUSION In this paper, we present a novel optimization approach for MTL, GDOD, which manipulates each task gradient using a decomposi- tion built from the span of all task gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' GDOD decomposes gradients into task-shared and task-specific components explic- itly and adopts a general update rule for avoiding interference across all task gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Moreover, we present the convergence of GDOD theoretically under both convex and non-convex assump- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Experiment results on several multi-task datasets not only demonstrate the significant improvement of GDOD performed to existing MTL models but also outperform state-of-the-art optimiza- tion methods in terms of AUC metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' Our future study would focus on exploring other decomposition methods to optimize training procedure for more effective and efficient multi-task learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtFRT4oBgHgl3EQfBjcA/content/2301.13465v1.pdf'} +page_content=' 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b/FdAyT4oBgHgl3EQfe_hY/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:40e0e91b8832913cbae57af901efb0cf6e7f5f42082417ec8a03aec2017e0c8d +size 4980781 diff --git a/FdE4T4oBgHgl3EQf7A74/content/tmp_files/2301.05337v1.pdf.txt b/FdE4T4oBgHgl3EQf7A74/content/tmp_files/2301.05337v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..eeb61d701b92be8d3b7aeaa5ac1a224ec6a0dea2 --- /dev/null +++ b/FdE4T4oBgHgl3EQf7A74/content/tmp_files/2301.05337v1.pdf.txt @@ -0,0 +1,1083 @@ +Exceptional degeneracies in non-Hermitian Rashba semiconductors +Jorge Cayao∗ +Department of Physics and Astronomy, Uppsala University, Box 516, S-751 20 Uppsala, Sweden +(Dated: January 16, 2023) +Exceptional points are spectral degeneracies of non-Hermitian systems where eigenvalues and +eigenvectors coalesce and induce unique topological phases that have no counterpart in the Hermitian +realm. +Here we consider a non-Hermitian system by coupling a two-dimensional semiconductor +with Rashba spin-orbit coupling to a ferromagnet lead and show the emergence of highly tunable +exceptional points along rings in momentum space. Interestingly, these exceptional degeneracies +are the endpoints of lines formed by the eigenvalue coalescence at finite real energy, resembling the +bulk Fermi arcs commonly defined at zero real energy. We then show that an in-plane Zeeman field +provides a way to control these exceptional degeneracies although higher values of non-Hermiticity +are required in contrast to the zero Zeeman field regime. +Furthermore, we find that the spin +projections also coalescence at the exceptional degeneracies and can acquire larger values than in +the Hermitian regime. +Our results thus reveal the potential of systems with Rashba spin-orbit +coupling for realizing non-Hermitian bulk phenomena. +I. +INTRODUCTION +The effect of dissipation, often seen as detrimental, +has recently attracted a paramount attention in physics +due its potential to induce novel phenomena with tech- +nological applications [1–7]. +Dissipation naturally oc- +curs in open systems and is effectively described by non- +Hermitian (NH) Hamiltonians [8–10]. The most salient +property of these NH models is the emergence of a com- +plex spectrum with degeneracies known as exceptional +points (EPs) [11–20], where eigenstates and eigenvalues +coalesce, in stark contrast to Hermitian systems. While +EPs were initially seen as a mathematical curiosity, it has +been recently shown that they represent truly topological +objects enabling topological phases with no counterpart +in Hermitian setups [3, 4, 7]. +The concept of EPs and their topological properties +have recently been generalized to higher dimensions, giv- +ing rise to exceptional degeneracies in the form of lines, +rings, and surfaces as generic and stable bulk phenom- +ena. These exceptional degeneracies have already proven +crucial to enable unique topological effects [3, 4], such as +enhanced sensing [21, 22], unidirectional lasing [23, 24], +and bulk Fermi arcs [25–35], which do not have a Her- +mitian analog. Despite the numerous theoretical and ex- +perimental studies, however, the majority of them has +investigated exceptional degeneracies mostly in optical +and photonic systems [1, 2, 5, 6]. +Material junctions have been shown to offer another +powerful and experimentally relevant platform for the +realization of exceptional degeneracies [36–44]. +Mate- +rial junctions constitute electronic open systems with a +clear NH description that is well-established in quan- +tum transport [45]. In this regard, open semiconductor- +superconductor junctions have been shown to host sev- +eral classes of exceptional degeneracies [38, 39, 44], which +∗ jorge.cayao@physics.uu.se +Figure 1. Schematics of studied non-Hermitian Rashba sys- +tem: a 2D Rashba semiconductor (orange) is coupled to a +semi-infinite ferromagnet lead (gray). A Zeeman field along +x is applied (brown) in order to control the emergent non- +Hermitian degeneracies. +characterize distinct NH topological phases without ana- +log in the Hermitian regime. Notwithstanding the im- +portance of this study, it only focused on the impact of +non-Hermiticity on the superconducting properties, such +as on its particle-hole symmetry and energy gap, leav- +ing largely unexplored the role of non-Hermiticity on the +semiconductor. Of particular importance in such semi- +conductors is their intrinsic Rashba spin-orbit coupling +(SOC) [46–48], which arises due to the lack of structural +inversion symmetry and induces a spin-momentum lock- +ing [49, 50]. This property of the Rashba SOC has been +shown to enable a great control of the electron’s spin, a +crucial ingredient for several spintronics and topological +phenomena [51], already proven useful in recent exper- +iments [50–55]. However, despite the advances, the in- +terplay between Rashba SOC and non-Hermiticity still +remains unknown, specially the potential of this combi- +nation for inducing exceptional degeneracies. +In this work we consider a realistic NH Rashba semi- +conductor and discover the formation of stable and highly +tunable bulk exceptional degeneracies. +In particular, +we engineer a NH Rashba system by coupling a two- +dimensional (2D) semiconductor with Rashba SOC to a +semi-infinite ferromagnet lead, an easily achievable het- +erostructure using e.g., InAs or InSb semiconductors [50– +arXiv:2301.05337v1 [cond-mat.mes-hall] 13 Jan 2023 + +2DRashbasemiconductor +Z +X +y +Ferromagnetlead2 +55], see also [56]. +We discover that EPs appear along +rings in 2D momentum space and mark the ends of lines +formed by the coalescence of eigenvalues at finite real en- +ergy. The emergence of eigenvalues at the same real en- +ergy resembles the formation of bulk Fermi arcs, which, +although initially conceived at zero real energy, have re- +cently been generalized to finite real energies [57]. We +also show that the exceptional degeneracies found here +can be controlled by an in-plane Zeeman field but then +higher values of non-Hermiticity are required. Moreover, +we also find that the spin projections coalesce at the ex- +ceptional degeneracies and can even develop larger values +than in the Hermitian phase due to non-Hermiticity. +II. +NON-HERMITIAN EFFECTIVE MODEL +We consider an open system by coupling a 2D semicon- +ductor with Rashba SOC to a semi-infinite ferromagnet +lead, as schematically shown in Fig. 1. This open system +is modelled by the following effective NH Hamiltonian +Heff = HR + Σr(ω = 0) , +(1) +where HR describes the closed system, which is Hermi- +tian, and Σr(ω = 0) is the zero-frequency retarded self- +energy due to the coupling to the semi-infinite ferromag- +net lead. More specifically, the closed system corresponds +to a 2D Rashba semiconductor described by +HR = ξk + α(kyσx − kxσy) , +(2) +where ξk = ℏ2(k2 +x+k2 +y)/2m−µ is the kinetic energy, kx(y) +the momentum along x(y), µ is the chemical potential, +α is the Rashba SOC strength, and σj the j-th Pauli +matrix in spin space. The Hamiltonian HR in Eq. (2) de- +scribes well the Rashba SOC in 2D semiconductors, such +as in InAs or InSb, which are also within experimental +reach [50–56]. As an external control knob, we also con- +sider that the closed system is subjected to an applied +magnetic field along x which produces a Zeeman field B, +denoted by the brown arrow in Fig. 1. The effect of this +Zeeman field is modelled by adding Bσx to HR in Eq. (2) +which induces a renormalization to the SOC term αkyσx. +The zero-frequency self-energy Σr(ω = 0) in Eq. (1), +whose independence of frequency ω is well justified in the +wide-band limit [45], is analytically obtained and given +by [42, 43] +Σr(ω = 0) = −iΓσ0 − iγσz , +(3) +where Γ = (Γ↑ + Γ↓)/2 and γ = (Γ↑ − Γ↓)/2, with +Γσ = π|t′|2ρσ +L, being t′ the hopping amplitude into the +lead from the 2D Rashba semiconductor and ρσ +L the sur- +face density of states of the lead for spin σ =↑, ↓. +It +is thus evident that Γσ characterizes the coupling ampli- +tude between the lead and the 2D Rashba semiconductor. +The self-energy in Eq. (3) is imaginary and thus NH, +a unique effect emerging due to the coupling to the +Figure 2. Exceptional degeneracies in 2D Rashba semicon- +ductors at zero Zeeman field: (a) Real (Re) and imaginary +(Im) parts of the eigenvalues as a function of kx depicted in +solid blue and dashed red curves at ky = 0. +Green curve +represents the absolute value of the overlap between the two +wavefunctions ψ+− = ⟨ψ+|ψ−⟩. Gray curves show eigenvalues +without non-Hermiticity, Γ↑/↓ = 0. (b,d) Real and imaginary +parts of the energy differences ∆E = (E+ −E−) as a function +of kx and ky. (c) represents ψ+− as a function of kx and ky. +Parameters: α = 1, Γ↑ = 3, Γ↓ = 0, µ = 1, B = 0. +semi-infinite ferromagnet lead. Thus, the imaginary self- +energy renders the total effective Hamiltonian Heff to be +NH, introducing dramatic changes in the properties of +the closed system HR, which is the focus of this work +here. In particular, we are interested in investigating the +interplay between non-Hermiticity and Rashba SOC in +2D semiconductors and how it leads to the formation of +bulk exceptional degeneracies. +III. +EXCEPTIONAL DEGENERACIES +To identify the emergence of bulk exceptional degen- +eracies, we obtain the eigenvalues and eigenvectors of the +effective Hamiltonian Heff given by Eq. (1). At zero Zee- +man field B = 0, they are given by +E± = ξk − iΓ ± +� +α2|k|2 − γ2 , +(4) +Ψ± = +1 +√ +2 +� +1 +iγ±√ +α2|k|2−γ2 +α(ky+ikx) +� +, +(5) +where |k|2 = k2 +x + k2 +y and ± labels the two distinct bands +which have a mixture of ↑ and ↓ spins. At finite Zeeman +fields B, the eigenvalues and eigenvectors can be obtained +by replacing αky → B + αky in Eqs. (4) and (5). An im- +mediate observation in the energies and wavefunctions +is their dependence on the couplings Γ↑,↓ via γ and Γ, + +(a) +(b) +4 +Re△E +ky=0 +2 +2 +10.0 +7.5 +0 +0 +5.0 +2 +2.5 +ReE +ImE +4 +T +-2 +0 +-4 +0 +2 +4 +-2 +-4 +0 +2 +4 +(c) +4 +(d) +4 +14+-1 +Im△E +1.0 +3.0 +2 +2 +ky +2.5 +0.8 +Momentum +Momentum +2.0 +0 +0.6 +0 +1.5 +0.4 +-2 +.2 +1.0 +一 +0.2 +0.5 +4 +0 +0 +-4 +-2 +2 +4 +-4 +0 +-2 +0 +2 +4 +Momentum kx +Momentum kx3 +already revealing a clear impact of the NH self energy +given by Eq. (3). This can be visualized in Fig. 2, where +we plot the eigenvalues and eigenvectors as a function of +momenta kx and ky at zero Zeeman field. At Γ↑,↓ = 0, +the system described by Eq. (1) is Hermitian and its two +eigenvalues in Eq. (4) are real: they correspond to two +parabolas shifted by ±ksoc = ±mα/ℏ2 that intersect at +kx,y = 0, see gray curves in Fig. 2(a). Here, their respec- +tive eigenvectors are orthogonal as expected for Hermi- +tian systems, see Eq. (5). While this Rashba system is +gapless at zero momenta, finite values of ky opens a gap +at kx = 0 even at zero Zeeman field. A finite in-plane +Zeeman field opens a gap at kx,y = 0, also known as he- +lical gap, where states are counter propagating and have +distinct spins [56, 58–60]. +At any Γ↑,↓ ̸= 0, the two eigenvalues E± acquire fi- +nite imaginary parts that strongly depend on momenta, +see Eq. (4). The formation of eigenvalues with imaginary +terms signals the emergence of NH physics as a pure effect +due to the ferromagnet lead [42–44]. The inverse of these +imaginary parts define the quasiparticle lifetime in the +2D Rashba semiconductor, thus offering a clear physical +meaning of non-Hermiticity [45]. From the dependence of +the eigenvalues on γ in Eq. (4), we note that their imag- +inary parts exhibit a non trivial behaviour. In fact, at +γ = 0, which is satisfied when Γ↓ = Γ↓, the two eigenval- +ues acquire the same imaginary part equal to −iΓ. This +situation remains for γ ̸= 0 only when |γ| < α|k|. For +these conditions, therefore, quasiparticles in the Rashba +semiconductor have the same and constant lifetime. +The behaviour of the eigenvalues becomes more inter- +esting when γ ̸= 0 and |γ| > α|k|, which then allows +the two eigenvalues to acquire distinct imaginary parts. +This is visualized in Fig. 2(a) where we plot the real and +imaginary parts of the eigenvalues at zero Zeeman field +and at ky = 0, see solid blue and dashed red curves. Sur- +prisingly, we observe that both the real and imaginary +parts simultaneously merge at finite energy into a single +value at special positive and negative momenta. We also +see that the wavefunctions become parallel instead of or- +thogonal at these special momenta, see the green curve +in Fig. 2(a) depicting the overlap ψ+− = ⟨ψ+|ψ−⟩ where +ψ+/− is given by Eq. (5). +These spectral degeneracies +signal the emergence of EPs, whose formation can be un- +derstood by noting that they occur when the square root +in Eq. (4) vanishes. At zero Zeeman field, this condition +for EPs is given by +α2(k2 +x + k2 +y) − γ2 = 0 , +(6) +while at finite Zeeman field we have to change αky → +B+αky. At this EP condition, the eigenvalues and eigen- +vectors become, +EEP +± += ξk − iΓ , +ΨEP +± = +1 +√ +2 +� +1 +iγ +α(ky+ikx) +� +, +(7) +where the values of momenta satisfy the condition given +Figure 3. Tunability of exceptional degeneracies in 2D semi- +conductors with Rashba SOC: (a,b) Real part of the energy +difference ∆E = (E+ − E−) as a function of α and Γ↑ at +B = 0 and B = 1. (c,d) Same quantity as in (a,b) but as a +function of B and Γ↑ at α = 1 (c), and as a function of α and +kx at B = 0 (d). Parameters: Γ↓ = 0, µ = 1. +by Eq. (6). Thus, the two eigenvalues (eigenvectors) coa- +lesce at EPs: instead of having two eigenvalues (eigenvec- +tors), only one eigenvalue (eigenvector) remains at EPs, +see Fig. 2(a). For ky = 0, the EP occur at positive and +negative momenta given by ±kEP +x += ±(γ/α)2 at B = 0, +marking the ends of the cyan region in Fig. 2(a). Between +these two EP points, the eigenvalues have the same real +part determined by the quadratic dispersion ξk and differ- +ent imaginary parts determined by −iΓ±i +� +γ2 − α2|k|2, +see cyan region in Fig. 2(a). We note that having bulk +energy lines due to eigenvalues with the same real part +resembles the formation of bulk Fermi arcs [25–35], al- +though here they occur at finite real energy in contrast +to the common expectation at zero real energy. In this re- +gard, very recently, the definition of bulk Fermi arcs has +been generalized to any two eigenvalues with the same +real energy [57], suggesting that the bulk energy lines +found here might be an example of bulk Fermi arcs. +Furthermore, another property of the EPs determined +by the condition in Eq. (6) is that they occur along a +ring defined by α2(k2 +x + k2 +y) = γ2 at B = 0 or by +α2k2 +x + (αky + B)2 = γ2 at B ̸= 0. +To support this +idea, in Fig. 2(b,d) we plot the difference between real +and imaginary parts of the eigenvalues, namely, Re∆E = +Re(E+ − E−) and Im∆E = Im(E+ − E−), as a func- +tion of kx and ky. In this case, the blue regions indicate +Re∆E = 0 and Im∆E = 0, with their borders mark- +ing the occurrence of rings. To highlight these rings, in +Fig. 2(b,d) we also plot the condition given by Eq. (6) +in dashed cyan color. +Together with the coalescence +of eigenvalues, along these rings their associated wave- +functions become parallel as revealed in Fig. 2(c), thus + +(a) +(b) +4 +4 +Re△E +Re△E +kxv=0.5 +B=0 +kx.v=0.5 +B=1 +6 +α +2 +2 +SOC strength + strength +6 +0 +0 +4 +00 +2 +-2 +2 +S +S +2 +4 +4 +0 +0 +0 +2 +4 +0 +2 +4 +Coupling 「 +Coupling 「 +(c) +(d) +4 +Re△E +Re△E +ky=0.5 +kx.v=0.5 +2 +B +2 +8 +sOC strength +field +30 +Zeeman t +0 +6 +0 +20 +4 +-2 +S +10 +2 +4 +4 +0 +0 +0 +2 +4 +-4 +-2 +0 +2 +4 +Coupling [ +Momentum kx4 +demonstrating that these rings represent exceptional NH +degeneracies. Hence, the bulk exceptionally degeneracies +in 2D Rashba semiconductors acquire the form of excep- +tional rings. +To induce the formation of the exceptional degenera- +cies discussed above it is sufficient the interplay between +non-Hermiticity and SOC, as clearly seen in Eq. (6). +While this conclusion is already evident in Fig. 2, to fur- +ther support it, in Fig. 3(a) we present Re∆E as a func- +tion of the SOC strength α and coupling Γ↑ at finite mo- +menta and zero Zeeman field. We obtain that the region +with Re∆E = 0 increases following a triangular-shaped +profile depicted in blue, which is delimited by ± +� +γ2/|k|2 +indicated by cyan dashed lines. At fixed momenta, the +SOC drives the formation of EPs, requiring lower SOC +when non-Hermiticity is small. By fixing only one mo- +mentum coordinate, it is also possible to induce EPs, +as seen in Fig. 3(d). Furthermore, another possibility to +control the appearance of EPs is by an in-plane Zeeman +field along x as considered in Fig. 1. Thus, in Fig. 3(b) we +show Re∆E as a function of α and Γ↑ at finite B, while +in Fig. 3(c) we show Re∆E as a function of B and Γ↑. In +this case, we identify two relevant features. First, at finite +SOC and finite momenta, the non-Hermiticity needed to +induce EPs needs to overcome the effect of the Zeeman +field B, thus requiring larger non-Hermiticity than in the +absence of B [Fig. 3(b)]. +Second, at all fixed parame- +ters, the Zeeman field can drive the emergence of EPs +[Fig. 3(b)]; the EPs here are marked by the cyan curves +which correspond to −αky ± +� +γ2 + α2k2x. In sum, the +bulk exceptional degeneracies found in 2D Rashba semi- +conductors exhibit a high degree of tunability by SOC, +momenta, and Zeeman field, which could be relevant for +their realization and subsequent observation. +IV. +SPIN PROJECTIONS +Having established the emergence of exceptional de- +generacies in the bulk of 2D Rashba semiconductors, now +we turn our attention to how the spins here behave un- +der non-Hermiticity. This is motivated by the fact that it +is the spin an important quantity for several phenomena +in semiconductors, useful for spintronics and topological +phenomena. In particular, in this part we focus on the +spin expectation values, which here will be referred to as +spin projections and are obtained by +Sη +j = Ψ† +ησjΨη +(8) +where Ψ† +η is given by Eq. (5) with η = ± and σj the +j-th spin Pauli matrix. +Thus, Sη +j represents the spin +projection along j axis associated to η = ±. By plugging +Eq. (5) into Eq. (8), we obtain +S± +x = +1 +2α|k|2 +� +kxγ ± ky +� +α2|k|2 − γ2 +� +, +S± +y = +1 +2α|k|2 +� +kyγ ∓ kx +� +α2|k|2 − γ2 +� +, +(9) +Figure 4. +Spin projection along y, S± +y : +(a,b) Real (Re) +and imaginary (Im) parts of the spin projection, Re[S± +y ] and +Im[S± +y ], as a function of kx for distinct values of ky at B = 0 +where solid (dashed) curves correspond to the spin projec- +tions obtained with ψ+ (ψ−). Also, light gray (brown) curves +showing a sharp (smooth) transition across kx = 0 corre- +spond to Γ↑,↓ = 0 and ky = 0 (ky = 0.5). +(c,d) Same as +(a,b) but now as a function of ky at distinct values of kx. +(e,f) Real and imaginary parts of the spin-projection differ- +ences ∆Sy = S− +y − S+ +y as a function of B and γ at finite mo- +menta kx,y = 0.5. The dashed cyan lines indicate the regimes +where exceptional points occur, which then mark the ends +of Re∆Sy = 0 (uniform blue region). Parameters: Γ↑ = 3, +α = 1, Γ↓ = 0, µ = 1. +for the spin projections along x and y, respectively, while +S± +z = 0 along z. Note that here |k|2 = k2 +x + k2 +y and γ = +(Γ↑ − Γ↓)/2 characterizes the amount of non-Hermiticity +due to the ferromagnet lead, see Eqs. (1) and (3). +As +before, the effect of the Zeeman field along x considered +in Fig. 1 can be included by replacing αky → B + αky. +The expressions given by Eqs. (9) are relatively simple +and permit us to identify the impact of non-Hermiticity +on the spin projections by naked eye. +In the Hermi- +tian regime, when γ = 0, the spin projections reduce to +S± +x = ±ky/|k| and S± +y = ∓kx/|k|, as expected [49, 50]. +Note that S+ +y(x) and S− +y(x) change their sign when kx(y) +varies from negative to positive values passing through +kx(y) = 0, see gray and pink curves in Fig. 4(a) showing +the behaviour of S+ +y . The sign of S± +y(x) remains, how- +ever, upon variations of ky(x), as depicted in gray and + +(a) +(b) +ReSt +_ImSt +--- ImSy +-- ReSy +Spin projection +Spin projection +0 +0 +y +0 +Ky +0.5 +0 +0.8 +0.8 +0.5 +1 +-2 +0 +2 +-2 +0 +2 +Momentum kx +Momentum kx +(c) +(d) +_ ReSt +_ ImSt +--- ReSy +--- ImSy +Spin projection +Spin projection +O +0 +kx +0 +0.5 +0.5 +0.8 +0.8 +1 +-2 +0 +2 +-2 +0 +2 +Momentum ky +Momentum ky +(e) +(f) +4 +Im△Sy +kx.,y=0.5 +kx,y=0.5 +1.0 +6 +B +2 +B +2 +Zeeman field +0.8 +Zeeman field +4 +0 +0.6 +0 +0.4 +-2 +-2 +2 +0.2 +.4 +-4 +0 +0 +0 +1 +2 +3 +0 +1 +2 +3 +Coupling y +Coupling y5 +pink curves in Fig. 4(c). +For finite non-Hermiticity, characterized by γ ̸= 0, the +behaviour of the spin projections S± +x(y) is highly unusual. +A finite γ generates a linear in momentum term propor- +tional to kx(y)γ for Sx(y) and renormalizes the Hermi- +tian component with +� +α2|k|2 − γ2, see first and second +terms in Eqs. (9). Both terms reveal a unique effect of +non-Hermiticity. The first part of S± +x(y), proportional to +kx(y)γ, is always real and appears along the same di- +rection of the spin projection, in contrast to the Hermi- +tian contribution where S± +x(y) is only finite along y(x). +The second part of S± +x(y) is real for |α||k| > |γ|, which +then adds up to the first part, but becomes imaginary +for |α||k| < |γ|. Thus, the appearance of an imaginary +part in the spin projections for |α||k| < |γ| can be inter- +preted as a signal of their lifetime, which becomes highly +anisotropic in momentum space. At α2|k|2 − γ2 = 0, the +second term in Eqs. (9) vanish and the spin projections +S± +j coalesce, namely, they merge into a single value that +is given by +S±,EP +x(y) += kx(y) +2|k|2 . +(10) +Interestingly, the condition α2|k|2 − γ2 = 0, which leads +to this spin projection coalescence, is the same condition +that determines the formation of exceptional degenera- +cies discussed in previous section, see Eq. (6). Thus, the +coalescence effect of S± +j can be seen as unique NH effect +without analog in Hermitian systems. We also note that +along the lines connecting these exceptional degenera- +cies, which correspond to energy lines that resemble bulk +Fermi arcs, the spin projections acquire a finite imaginary +part with a natural physical interpretation as discussed +in previous paragraph. +In order to gain visual understanding of the spin pro- +jection coalescence, in Fig. 4 we plot the real and imagi- +nary parts of S± +y as a function of momenta (a-d) and in +the B − γ plane (e,f). At ky = 0, the real part of the +spin projections S± +y vanishes along a line of kx and the +ends of such line mark the EP momenta obtained from +Eq. (6) and given by |kEP +x | = |γ|/|α|, see solid and dashed +blue curves in Fig. 4(a); see also Eqs. (9) and (10). The +imaginary part of S± +y undergoes a coalescence effect as +well at the EP momenta ±kEP +x +but acquires large val- +ues between them and vanish at kx = 0 [Fig. 4(b)]. For +ky > 0, the coalescence effect persists, with smaller imag- +inary parts, but the real part does not vanish anymore +and, instead, develops a maximum at kx = 0 favouring +a large positive spin projection along y, see Eqs. (9) and +(10). +For ky < 0, the spin projection S± +y has instead +a minimum at kx = 0, favouring a large negative spin +projection along y. The coalescence of spin projections +is also observed in Fig. 4(c,d), where we plot the real and +imaginary parts of S± +y as a function of ky at fixed values +of kx. At kx = 0, no EP transition is observed in S± +y +because the square root term that gives rise to EPs is +multiplied by zero and hence vanishes, see blue curves in +Fig. 4(c,d) and also Eqs. (9). However, for finite kx, the +spin projections develop a clear EP transition, revealing +that their coalescence is a highly tunable NH bulk effect. +The +spin +projection +coalescence +discussed +above +requires +finite +momenta, +Rashba +SOC, +and +non- +Hermiticity, a combination of ingredients inherent to NH +Rashba semiconductors. +Furthermore, it is also possi- +ble to tune and control the spin projections by an in- +plane Zeeman field B, e.g., along x as sketched in Fig. 1, +see also discussions below Eqs. (2) and (9). +To sup- +port this idea, in Fig. 4 we present the Re and the Im +parts of the difference between spin projection along y, +Re∆Sy = Re(S− +y − S+ +y) and Im∆Sy = Im(S− +y − S+ +y), at +fixed kx,y as a function of B and γ. Here, the blue re- +gions indicate Re∆Sy = 0 and Im∆Sy = 0 and their +borders show the exceptional degeneracies, indicated in +cyan dashed curves. At fixed non-Hermiticity (γ), the +Zeeman field B induces the coalescence of spin projec- +tions S± +y at EPs, which requires small (large) B for weak +(strong) non-Hermiticity. Therefore, Zeeman fields offer +another possibility for tuning and controlling the spin- +projection coalescence at exceptional degeneracies in 2D +Rashba semiconductors. +V. +CONCLUSIONS +We have demonstrated that the interplay between non- +Hermiticity and Rashba spin-orbit coupling in semicon- +ductors gives rise to the emergence of stable and highly +tunable bulk exceptional degeneracies. We have found +that these degeneracies form rings in two-dimensional +momentum space and signal the ends of lines forming +due to the coalescence of eigenvalues at finite real energy. +Interestingly, the lines at finite real energies have the ap- +pearance of bulk Fermi arcs but now at finite energies, +suggesting new possibilities for non-Hermitian bulk phe- +nomena [57]. We have also shown that the exceptional +degeneracies and bulk Fermi arcs can be controlled by +an in-plane Zeeman field, albeit larger non-Hermiticity +values are then needed. +Furthermore, we have discov- +ered that the spin projections coalesce at the exceptional +degeneracies and can easily achieve higher values than +in the Hermitian regime. +Taken together, the results +presented here put semiconductors with Rashba SOC as +an interesting arena for the realization of highly tunable +non-Hermitian bulk phenomena. +ACKNOWLEDGMENTS +We thank A. M. Black-Schaffer and P. Oppeneer for +useful discussions. +We acknowledge financial support +from the Swedish Research Council (Vetenskapsrådet +Grant No. 2021-04121), the Göran Gustafsson Founda- +tion (Grant No. 2216), the Scandinavia-Japan Sasakawa +Foundation (Grant No. GA22-SWE-0028), the Royal +Swedish Academy of Sciences (Grant No. PH2022-0003). + +6 +[1] R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. +Musslimani, S. Rotter, and D. N. Christodoulides, Non- +hermitian physics and pt symmetry, Nat. Phys. 14, 11 +(2018). +[2] Ş. K. Özdemir, S. Rotter, F. Nori, and L. Yang, Parity– +time symmetry and exceptional points in photonics, Nat. +Mater. 18, 783 (2019). +[3] E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Ex- +ceptional topology of non-hermitian systems, Rev. Mod. +Phys. 93, 015005 (2021). +[4] Y. Ashida, +Z. Gong, and M. Ueda, Non-hermitian +physics, Adv. Phys. 69, 249 (2020). +[5] M. Parto, Y. G. Liu, B. Bahari, M. Khajavikhan, and +D. N. 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Syst. +Nanostruct. 114, 113615 (2019). + diff --git a/FdE4T4oBgHgl3EQf7A74/content/tmp_files/load_file.txt b/FdE4T4oBgHgl3EQf7A74/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8b7c43458dbe995edc890f62bfc6155e8bfc54ef --- /dev/null +++ b/FdE4T4oBgHgl3EQf7A74/content/tmp_files/load_file.txt @@ -0,0 +1,772 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf,len=771 +page_content='Exceptional degeneracies in non-Hermitian Rashba semiconductors Jorge Cayao∗ Department of Physics and Astronomy, Uppsala University, Box 516, S-751 20 Uppsala, Sweden (Dated: January 16, 2023) Exceptional points are spectral degeneracies of non-Hermitian systems where eigenvalues and eigenvectors coalesce and induce unique topological phases that have no counterpart in the Hermitian realm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Here we consider a non-Hermitian system by coupling a two-dimensional semiconductor with Rashba spin-orbit coupling to a ferromagnet lead and show the emergence of highly tunable exceptional points along rings in momentum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Interestingly, these exceptional degeneracies are the endpoints of lines formed by the eigenvalue coalescence at finite real energy, resembling the bulk Fermi arcs commonly defined at zero real energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' We then show that an in-plane Zeeman field provides a way to control these exceptional degeneracies although higher values of non-Hermiticity are required in contrast to the zero Zeeman field regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Furthermore, we find that the spin projections also coalescence at the exceptional degeneracies and can acquire larger values than in the Hermitian regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Our results thus reveal the potential of systems with Rashba spin-orbit coupling for realizing non-Hermitian bulk phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' INTRODUCTION The effect of dissipation, often seen as detrimental, has recently attracted a paramount attention in physics due its potential to induce novel phenomena with tech- nological applications [1–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Dissipation naturally oc- curs in open systems and is effectively described by non- Hermitian (NH) Hamiltonians [8–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' The most salient property of these NH models is the emergence of a com- plex spectrum with degeneracies known as exceptional points (EPs) [11–20], where eigenstates and eigenvalues coalesce, in stark contrast to Hermitian systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' While EPs were initially seen as a mathematical curiosity, it has been recently shown that they represent truly topological objects enabling topological phases with no counterpart in Hermitian setups [3, 4, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' The concept of EPs and their topological properties have recently been generalized to higher dimensions, giv- ing rise to exceptional degeneracies in the form of lines, rings, and surfaces as generic and stable bulk phenom- ena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' These exceptional degeneracies have already proven crucial to enable unique topological effects [3, 4], such as enhanced sensing [21, 22], unidirectional lasing [23, 24], and bulk Fermi arcs [25–35], which do not have a Her- mitian analog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Despite the numerous theoretical and ex- perimental studies, however, the majority of them has investigated exceptional degeneracies mostly in optical and photonic systems [1, 2, 5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Material junctions have been shown to offer another powerful and experimentally relevant platform for the realization of exceptional degeneracies [36–44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Mate- rial junctions constitute electronic open systems with a clear NH description that is well-established in quan- tum transport [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' In this regard, open semiconductor- superconductor junctions have been shown to host sev- eral classes of exceptional degeneracies [38, 39, 44], which ∗ jorge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='cayao@physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='uu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='se Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Schematics of studied non-Hermitian Rashba sys- tem: a 2D Rashba semiconductor (orange) is coupled to a semi-infinite ferromagnet lead (gray).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' A Zeeman field along x is applied (brown) in order to control the emergent non- Hermitian degeneracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' characterize distinct NH topological phases without ana- log in the Hermitian regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Notwithstanding the im- portance of this study, it only focused on the impact of non-Hermiticity on the superconducting properties, such as on its particle-hole symmetry and energy gap, leav- ing largely unexplored the role of non-Hermiticity on the semiconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Of particular importance in such semi- conductors is their intrinsic Rashba spin-orbit coupling (SOC) [46–48], which arises due to the lack of structural inversion symmetry and induces a spin-momentum lock- ing [49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' This property of the Rashba SOC has been shown to enable a great control of the electron’s spin, a crucial ingredient for several spintronics and topological phenomena [51], already proven useful in recent exper- iments [50–55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' However, despite the advances, the in- terplay between Rashba SOC and non-Hermiticity still remains unknown, specially the potential of this combi- nation for inducing exceptional degeneracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' In this work we consider a realistic NH Rashba semi- conductor and discover the formation of stable and highly tunable bulk exceptional degeneracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' In particular, we engineer a NH Rashba system by coupling a two- dimensional (2D) semiconductor with Rashba SOC to a semi-infinite ferromagnet lead, an easily achievable het- erostructure using e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=', InAs or InSb semiconductors [50– arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='05337v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='mes-hall] 13 Jan 2023 2DRashbasemiconductor Z X y Ferromagnetlead2 55], see also [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' We discover that EPs appear along rings in 2D momentum space and mark the ends of lines formed by the coalescence of eigenvalues at finite real en- ergy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' The emergence of eigenvalues at the same real en- ergy resembles the formation of bulk Fermi arcs, which, although initially conceived at zero real energy, have re- cently been generalized to finite real energies [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' We also show that the exceptional degeneracies found here can be controlled by an in-plane Zeeman field but then higher values of non-Hermiticity are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Moreover, we also find that the spin projections coalesce at the ex- ceptional degeneracies and can even develop larger values than in the Hermitian phase due to non-Hermiticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' NON-HERMITIAN EFFECTIVE MODEL We consider an open system by coupling a 2D semicon- ductor with Rashba SOC to a semi-infinite ferromagnet lead, as schematically shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' This open system is modelled by the following effective NH Hamiltonian Heff = HR + Σr(ω = 0) , (1) where HR describes the closed system, which is Hermi- tian, and Σr(ω = 0) is the zero-frequency retarded self- energy due to the coupling to the semi-infinite ferromag- net lead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' More specifically, the closed system corresponds to a 2D Rashba semiconductor described by HR = ξk + α(kyσx − kxσy) , (2) where ξk = ℏ2(k2 x+k2 y)/2m−µ is the kinetic energy, kx(y) the momentum along x(y), µ is the chemical potential, α is the Rashba SOC strength, and σj the j-th Pauli matrix in spin space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' The Hamiltonian HR in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' (2) de- scribes well the Rashba SOC in 2D semiconductors, such as in InAs or InSb, which are also within experimental reach [50–56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' As an external control knob, we also con- sider that the closed system is subjected to an applied magnetic field along x which produces a Zeeman field B, denoted by the brown arrow in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' The effect of this Zeeman field is modelled by adding Bσx to HR in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' (2) which induces a renormalization to the SOC term αkyσx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' The zero-frequency self-energy Σr(ω = 0) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' (1), whose independence of frequency ω is well justified in the wide-band limit [45], is analytically obtained and given by [42, 43] Σr(ω = 0) = −iΓσ0 − iγσz , (3) where Γ = (Γ↑ + Γ↓)/2 and γ = (Γ↑ − Γ↓)/2, with Γσ = π|t′|2ρσ L, being t′ the hopping amplitude into the lead from the 2D Rashba semiconductor and ρσ L the sur- face density of states of the lead for spin σ =↑, ↓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' It is thus evident that Γσ characterizes the coupling ampli- tude between the lead and the 2D Rashba semiconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' The self-energy in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' (3) is imaginary and thus NH, a unique effect emerging due to the coupling to the Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Exceptional degeneracies in 2D Rashba semicon- ductors at zero Zeeman field: (a) Real (Re) and imaginary (Im) parts of the eigenvalues as a function of kx depicted in solid blue and dashed red curves at ky = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Green curve represents the absolute value of the overlap between the two wavefunctions ψ+− = ⟨ψ+|ψ−⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Gray curves show eigenvalues without non-Hermiticity, Γ↑/↓ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' (b,d) Real and imaginary parts of the energy differences ∆E = (E+ −E−) as a function of kx and ky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' (c) represents ψ+− as a function of kx and ky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Parameters: α = 1, Γ↑ = 3, Γ↓ = 0, µ = 1, B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' semi-infinite ferromagnet lead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Thus, the imaginary self- energy renders the total effective Hamiltonian Heff to be NH, introducing dramatic changes in the properties of the closed system HR, which is the focus of this work here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' In particular, we are interested in investigating the interplay between non-Hermiticity and Rashba SOC in 2D semiconductors and how it leads to the formation of bulk exceptional degeneracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' EXCEPTIONAL DEGENERACIES To identify the emergence of bulk exceptional degen- eracies, we obtain the eigenvalues and eigenvectors of the effective Hamiltonian Heff given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' At zero Zee- man field B = 0, they are given by E± = ξk − iΓ ± � α2|k|2 − γ2 , (4) Ψ± = 1 √ 2 � 1 iγ±√ α2|k|2−γ2 α(ky+ikx) � , (5) where |k|2 = k2 x + k2 y and ± labels the two distinct bands which have a mixture of ↑ and ↓ spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' At finite Zeeman fields B, the eigenvalues and eigenvectors can be obtained by replacing αky → B + αky in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' (4) and (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' An im- mediate observation in the energies and wavefunctions is their dependence on the couplings Γ↑,↓ via γ and Γ, (a) (b) 4 Re△E ky=0 2 2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='5 0 0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='0 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='5 ReE ImE 4 T 2 0 4 0 2 4 2 4 0 2 4 (c) 4 (d) 4 14+-1 Im△E 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='0 2 2 ky 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='8 Momentum Momentum 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='6 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='4 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='0 一 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='5 4 0 0 4 2 2 4 4 0 2 0 2 4 Momentum kx Momentum kx3 already revealing a clear impact of the NH self energy given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' This can be visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' 2, where we plot the eigenvalues and eigenvectors as a function of momenta kx and ky at zero Zeeman field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' At Γ↑,↓ = 0, the system described by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' (1) is Hermitian and its two eigenvalues in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' (4) are real: they correspond to two parabolas shifted by ±ksoc = ±mα/ℏ2 that intersect at kx,y = 0, see gray curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Here, their respec- tive eigenvectors are orthogonal as expected for Hermi- tian systems, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' While this Rashba system is gapless at zero momenta, finite values of ky opens a gap at kx = 0 even at zero Zeeman field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' A finite in-plane Zeeman field opens a gap at kx,y = 0, also known as he- lical gap, where states are counter propagating and have distinct spins [56, 58–60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' At any Γ↑,↓ ̸= 0, the two eigenvalues E± acquire fi- nite imaginary parts that strongly depend on momenta, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' The formation of eigenvalues with imaginary terms signals the emergence of NH physics as a pure effect due to the ferromagnet lead [42–44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' The inverse of these imaginary parts define the quasiparticle lifetime in the 2D Rashba semiconductor, thus offering a clear physical meaning of non-Hermiticity [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' From the dependence of the eigenvalues on γ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' (4), we note that their imag- inary parts exhibit a non trivial behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' In fact, at γ = 0, which is satisfied when Γ↓ = Γ↓, the two eigenval- ues acquire the same imaginary part equal to −iΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' This situation remains for γ ̸= 0 only when |γ| < α|k|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' For these conditions, therefore, quasiparticles in the Rashba semiconductor have the same and constant lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' The behaviour of the eigenvalues becomes more inter- esting when γ ̸= 0 and |γ| > α|k|, which then allows the two eigenvalues to acquire distinct imaginary parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' This is visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' 2(a) where we plot the real and imaginary parts of the eigenvalues at zero Zeeman field and at ky = 0, see solid blue and dashed red curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Sur- prisingly, we observe that both the real and imaginary parts simultaneously merge at finite energy into a single value at special positive and negative momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' We also see that the wavefunctions become parallel instead of or- thogonal at these special momenta, see the green curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' 2(a) depicting the overlap ψ+− = ⟨ψ+|ψ−⟩ where ψ+/− is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' These spectral degeneracies signal the emergence of EPs, whose formation can be un- derstood by noting that they occur when the square root in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' (4) vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' At zero Zeeman field, this condition for EPs is given by α2(k2 x + k2 y) − γ2 = 0 , (6) while at finite Zeeman field we have to change αky → B+αky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' At this EP condition, the eigenvalues and eigen- vectors become, EEP ± = ξk − iΓ , ΨEP ± = 1 √ 2 � 1 iγ α(ky+ikx) � , (7) where the values of momenta satisfy the condition given Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Tunability of exceptional degeneracies in 2D semi- conductors with Rashba SOC: (a,b) Real part of the energy difference ∆E = (E+ − E−) as a function of α and Γ↑ at B = 0 and B = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' (c,d) Same quantity as in (a,b) but as a function of B and Γ↑ at α = 1 (c), and as a function of α and kx at B = 0 (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Parameters: Γ↓ = 0, µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Thus, the two eigenvalues (eigenvectors) coa- lesce at EPs: instead of having two eigenvalues (eigenvec- tors), only one eigenvalue (eigenvector) remains at EPs, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' For ky = 0, the EP occur at positive and negative momenta given by ±kEP x = ±(γ/α)2 at B = 0, marking the ends of the cyan region in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Between these two EP points, the eigenvalues have the same real part determined by the quadratic dispersion ξk and differ- ent imaginary parts determined by −iΓ±i � γ2 − α2|k|2, see cyan region in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' We note that having bulk energy lines due to eigenvalues with the same real part resembles the formation of bulk Fermi arcs [25–35], al- though here they occur at finite real energy in contrast to the common expectation at zero real energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' In this re- gard, very recently, the definition of bulk Fermi arcs has been generalized to any two eigenvalues with the same real energy [57], suggesting that the bulk energy lines found here might be an example of bulk Fermi arcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Furthermore, another property of the EPs determined by the condition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' (6) is that they occur along a ring defined by α2(k2 x + k2 y) = γ2 at B = 0 or by α2k2 x + (αky + B)2 = γ2 at B ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' To support this idea, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' 2(b,d) we plot the difference between real and imaginary parts of the eigenvalues, namely, Re∆E = Re(E+ − E−) and Im∆E = Im(E+ − E−), as a func- tion of kx and ky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' In this case, the blue regions indicate Re∆E = 0 and Im∆E = 0, with their borders mark- ing the occurrence of rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' To highlight these rings, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' 2(b,d) we also plot the condition given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' (6) in dashed cyan color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Together with the coalescence of eigenvalues, along these rings their associated wave- functions become parallel as revealed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' 2(c), thus (a) (b) 4 4 Re△E Re△E kxv=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='5 B=0 kx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='v=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='5 B=1 6 α 2 2 SOC strength strength 6 0 0 4 00 2 2 2 S S 2 4 4 0 0 0 2 4 0 2 4 Coupling 「 Coupling 「 (c) (d) 4 Re△E Re△E ky=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='5 kx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='v=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='5 2 B 2 8 sOC strength field 30 Zeeman t 0 6 0 20 4 2 S 10 2 4 4 0 0 0 2 4 4 2 0 2 4 Coupling [ Momentum kx4 demonstrating that these rings represent exceptional NH degeneracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Hence, the bulk exceptionally degeneracies in 2D Rashba semiconductors acquire the form of excep- tional rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' To induce the formation of the exceptional degenera- cies discussed above it is sufficient the interplay between non-Hermiticity and SOC, as clearly seen in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' While this conclusion is already evident in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' 2, to fur- ther support it, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' 3(a) we present Re∆E as a func- tion of the SOC strength α and coupling Γ↑ at finite mo- menta and zero Zeeman field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' We obtain that the region with Re∆E = 0 increases following a triangular-shaped profile depicted in blue, which is delimited by ± � γ2/|k|2 indicated by cyan dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' At fixed momenta, the SOC drives the formation of EPs, requiring lower SOC when non-Hermiticity is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' By fixing only one mo- mentum coordinate, it is also possible to induce EPs, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' 3(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Furthermore, another possibility to control the appearance of EPs is by an in-plane Zeeman field along x as considered in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Thus, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' 3(b) we show Re∆E as a function of α and Γ↑ at finite B, while in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' 3(c) we show Re∆E as a function of B and Γ↑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' In this case, we identify two relevant features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' First, at finite SOC and finite momenta, the non-Hermiticity needed to induce EPs needs to overcome the effect of the Zeeman field B, thus requiring larger non-Hermiticity than in the absence of B [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' 3(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Second, at all fixed parame- ters, the Zeeman field can drive the emergence of EPs [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' 3(b)];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' the EPs here are marked by the cyan curves which correspond to −αky ± � γ2 + α2k2x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' In sum, the bulk exceptional degeneracies found in 2D Rashba semi- conductors exhibit a high degree of tunability by SOC, momenta, and Zeeman field, which could be relevant for their realization and subsequent observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' SPIN PROJECTIONS Having established the emergence of exceptional de- generacies in the bulk of 2D Rashba semiconductors, now we turn our attention to how the spins here behave un- der non-Hermiticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' This is motivated by the fact that it is the spin an important quantity for several phenomena in semiconductors, useful for spintronics and topological phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' In particular, in this part we focus on the spin expectation values, which here will be referred to as spin projections and are obtained by Sη j = Ψ† ησjΨη (8) where Ψ† η is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' (5) with η = ± and σj the j-th spin Pauli matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Thus, Sη j represents the spin projection along j axis associated to η = ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' By plugging Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' (5) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' (8), we obtain S± x = 1 2α|k|2 � kxγ ± ky � α2|k|2 − γ2 � , S± y = 1 2α|k|2 � kyγ ∓ kx � α2|k|2 − γ2 � , (9) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Spin projection along y, S± y : (a,b) Real (Re) and imaginary (Im) parts of the spin projection, Re[S± y ] and Im[S± y ], as a function of kx for distinct values of ky at B = 0 where solid (dashed) curves correspond to the spin projec- tions obtained with ψ+ (ψ−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Also, light gray (brown) curves showing a sharp (smooth) transition across kx = 0 corre- spond to Γ↑,↓ = 0 and ky = 0 (ky = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' (c,d) Same as (a,b) but now as a function of ky at distinct values of kx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' (e,f) Real and imaginary parts of the spin-projection differ- ences ∆Sy = S− y − S+ y as a function of B and γ at finite mo- menta kx,y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' The dashed cyan lines indicate the regimes where exceptional points occur, which then mark the ends of Re∆Sy = 0 (uniform blue region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Parameters: Γ↑ = 3, α = 1, Γ↓ = 0, µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' for the spin projections along x and y, respectively, while S± z = 0 along z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Note that here |k|2 = k2 x + k2 y and γ = (Γ↑ − Γ↓)/2 characterizes the amount of non-Hermiticity due to the ferromagnet lead, see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' (1) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' As before, the effect of the Zeeman field along x considered in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' 1 can be included by replacing αky → B + αky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' The expressions given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' (9) are relatively simple and permit us to identify the impact of non-Hermiticity on the spin projections by naked eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' In the Hermi- tian regime, when γ = 0, the spin projections reduce to S± x = ±ky/|k| and S± y = ∓kx/|k|, as expected [49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Note that S+ y(x) and S− y(x) change their sign when kx(y) varies from negative to positive values passing through kx(y) = 0, see gray and pink curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' 4(a) showing the behaviour of S+ y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' The sign of S± y(x) remains, how- ever, upon variations of ky(x), as depicted in gray and (a) (b) ReSt _ImSt --- ImSy -- ReSy Spin projection Spin projection 0 0 y 0 Ky 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='5 1 2 0 2 2 0 2 Momentum kx Momentum kx (c) (d) _ ReSt _ ImSt --- ReSy --- ImSy Spin projection Spin projection O 0 kx 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='8 1 2 0 2 2 0 2 Momentum ky Momentum ky (e) (f) 4 Im△Sy kx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=',y=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='5 kx,y=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='0 6 B 2 B 2 Zeeman field 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='8 Zeeman field 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='4 2 2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='4 4 0 0 0 1 2 3 0 1 2 3 Coupling y Coupling y5 pink curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' 4(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' For finite non-Hermiticity, characterized by γ ̸= 0, the behaviour of the spin projections S± x(y) is highly unusual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' A finite γ generates a linear in momentum term propor- tional to kx(y)γ for Sx(y) and renormalizes the Hermi- tian component with � α2|k|2 − γ2, see first and second terms in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Both terms reveal a unique effect of non-Hermiticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' The first part of S± x(y), proportional to kx(y)γ, is always real and appears along the same di- rection of the spin projection, in contrast to the Hermi- tian contribution where S± x(y) is only finite along y(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' The second part of S± x(y) is real for |α||k| > |γ|, which then adds up to the first part, but becomes imaginary for |α||k| < |γ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Thus, the appearance of an imaginary part in the spin projections for |α||k| < |γ| can be inter- preted as a signal of their lifetime, which becomes highly anisotropic in momentum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' At α2|k|2 − γ2 = 0, the second term in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' (9) vanish and the spin projections S± j coalesce, namely, they merge into a single value that is given by S±,EP x(y) = kx(y) 2|k|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' (10) Interestingly, the condition α2|k|2 − γ2 = 0, which leads to this spin projection coalescence, is the same condition that determines the formation of exceptional degenera- cies discussed in previous section, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Thus, the coalescence effect of S± j can be seen as unique NH effect without analog in Hermitian systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' We also note that along the lines connecting these exceptional degenera- cies, which correspond to energy lines that resemble bulk Fermi arcs, the spin projections acquire a finite imaginary part with a natural physical interpretation as discussed in previous paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' In order to gain visual understanding of the spin pro- jection coalescence, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' 4 we plot the real and imagi- nary parts of S± y as a function of momenta (a-d) and in the B − γ plane (e,f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' At ky = 0, the real part of the spin projections S± y vanishes along a line of kx and the ends of such line mark the EP momenta obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' (6) and given by |kEP x | = |γ|/|α|, see solid and dashed blue curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' 4(a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' see also Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' (9) and (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' The imaginary part of S± y undergoes a coalescence effect as well at the EP momenta ±kEP x but acquires large val- ues between them and vanish at kx = 0 [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' 4(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' For ky > 0, the coalescence effect persists, with smaller imag- inary parts, but the real part does not vanish anymore and, instead, develops a maximum at kx = 0 favouring a large positive spin projection along y, see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' (9) and (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' For ky < 0, the spin projection S± y has instead a minimum at kx = 0, favouring a large negative spin projection along y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' The coalescence of spin projections is also observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' 4(c,d), where we plot the real and imaginary parts of S± y as a function of ky at fixed values of kx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' At kx = 0, no EP transition is observed in S± y because the square root term that gives rise to EPs is multiplied by zero and hence vanishes, see blue curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' 4(c,d) and also Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' However, for finite kx, the spin projections develop a clear EP transition, revealing that their coalescence is a highly tunable NH bulk effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' The spin projection coalescence discussed above requires finite momenta, Rashba SOC, and non- Hermiticity, a combination of ingredients inherent to NH Rashba semiconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Furthermore, it is also possi- ble to tune and control the spin projections by an in- plane Zeeman field B, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=', along x as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' 1, see also discussions below Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' (2) and (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' To sup- port this idea, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' 4 we present the Re and the Im parts of the difference between spin projection along y, Re∆Sy = Re(S− y − S+ y) and Im∆Sy = Im(S− y − S+ y), at fixed kx,y as a function of B and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Here, the blue re- gions indicate Re∆Sy = 0 and Im∆Sy = 0 and their borders show the exceptional degeneracies, indicated in cyan dashed curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' At fixed non-Hermiticity (γ), the Zeeman field B induces the coalescence of spin projec- tions S± y at EPs, which requires small (large) B for weak (strong) non-Hermiticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Therefore, Zeeman fields offer another possibility for tuning and controlling the spin- projection coalescence at exceptional degeneracies in 2D Rashba semiconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' CONCLUSIONS We have demonstrated that the interplay between non- Hermiticity and Rashba spin-orbit coupling in semicon- ductors gives rise to the emergence of stable and highly tunable bulk exceptional degeneracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' We have found that these degeneracies form rings in two-dimensional momentum space and signal the ends of lines forming due to the coalescence of eigenvalues at finite real energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Interestingly, the lines at finite real energies have the ap- pearance of bulk Fermi arcs but now at finite energies, suggesting new possibilities for non-Hermitian bulk phe- nomena [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' We have also shown that the exceptional degeneracies and bulk Fermi arcs can be controlled by an in-plane Zeeman field, albeit larger non-Hermiticity values are then needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Furthermore, we have discov- ered that the spin projections coalesce at the exceptional degeneracies and can easily achieve higher values than in the Hermitian regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Taken together, the results presented here put semiconductors with Rashba SOC as an interesting arena for the realization of highly tunable non-Hermitian bulk phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' ACKNOWLEDGMENTS We thank A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Black-Schaffer and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Oppeneer for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' We acknowledge financial support from the Swedish Research Council (Vetenskapsrådet Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' 2021-04121), the Göran Gustafsson Founda- tion (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' 2216), the Scandinavia-Japan Sasakawa Foundation (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' GA22-SWE-0028), the Royal Swedish Academy of Sciences (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' PH2022-0003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' 6 [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' El-Ganainy, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE4T4oBgHgl3EQf7A74/content/2301.05337v1.pdf'} +page_content=' Makris, M.' metadata={'source': 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Collective opinions affect civic participation, governance, and societal norms. Due to the influence of +opinion dynamics, many models of their formation and evolution have been developed. A commonly +used approach for the study of opinion dynamics is bounded-confidence models. In these models, +individuals are influenced by the opinions of others in their network. They generally assume that +individuals will formulate their opinions to resemble those of their peers. In this paper, inspired by +the dynamics of partisan politics, we introduce a bounded-confidence model in which individuals may +be repelled by the opinions of their peers rather than only attracted to them. We prove convergence +properties of our model and perform simulations to study the behavior of our model on various types +of random networks. In particular, we observe that including opinion repulsion leads to a higher +degree of opinion fragmentation than in standard bounded-confidence models. +Key words. opinion dynamics, bounded confidence, mathematical political science, congressional voting +AMS subject classifications. 91D30, 91F10 +1. Introduction. Opinions dictate how individuals interact with society. They influence +who we are friends with, how we vote, and what we consume. At the individual and collective +level, opinions shape our lives and our social interactions. Understanding how opinions are +formed and their dynamics provides a framework for studying changes in our society. The +role of opinions in politics and governance is a prominent part of public discourse in the U.S. +Inspired by discussions of political polarization and partisan politics, this paper presents a +mathematical approach to modelling polarized opinion dynamics where individuals feel both +attractive and repulsive forces. +The influence of public opinion on politics have been studied by philosophers, sociologists, +and social theorists [6,15,30]. Contemporary approaches to studying opinions frequently seek +to quantify them. In this paper, we focus on the dynamics of opinions. We are interested +in studying how opinions in a society shift as a result of relationships between individuals. +Various models for studying individual opinions exist [9,14,17,21]. We will focus on bounded- +confidence models. Bounded-confidence models are a class of models that suppose individuals +change their opinions based on their relationships, when their opinions are already close to +those of their peers. That is, if someone’s opinion is very far away from my own, even if +I have a relationship with them, I will not base my opinions on theirs. +Many bounded- +confidence models have been developed and studied. They include examinations of consensus +formation [11,13], polarization [16,29], and a large variety of model extensions for application +to real-world opinions [1,8,18,20]. +We consider polarization, and the notion that individuals may form their opinions by +∗Submitted to the editors December 8, 2022. +Funding: This work was partially funded by the James S. McDonnell Foundation Postdoctoral Fellowship +†Department of Humanities and Social Sciences, California Institute of Technology (ckann@caltech.edu). +‡Computing + Mathematical Sciences Department, California Institute of Technology (mfeng@caltech.edu). +1 +This manuscript is for review purposes only. +arXiv:2301.02210v1 [math.DS] 5 Jan 2023 + +2 +C. KANN AND M. FENG +being contrarian. If I have an adversarial relationship with someone, I may specifically choose +to hold an opinion that is different from their’s. Similar to other bounded-confidence models, +we maintain the idea that individuals are mostly influenced by others whose opinions are +already somewhat close to our own. We are most interested in understanding how collective +opinions in this model behave. What types of relationships and community structures lead to +strong polarization within a society? How might we extend those observations to real-world +applications and data? +The paper is organized as follows. We introduce the motivation for our model in section 2 +and define our model in section 3. We present analytical results in section 4, and perform +numerical simulations on synthetic networks (section 5). Conclusions follow in section 6. +2. Background and motivation. In this section, we introduce the motivation for our +proposed model of opinion dynamics. In subsection 2.1, we discuss political science research +which motivates our modelling choices, and in subsection 2.2 we introduce the Hegselmann– +Krause model for opinion dynamics, which we use as a starting point in the formulation of +our model. +2.1. Political Science motivation. In political science it is common to think of ideologies +as points in space, as being on the left or the right, liberal or conservative. This spatial view of +politicians and individuals drives much of the work that is done on voting behavior, both at the +individual and legislative levels, as well as the models of strategic behavior within Congress. +The original conception of this model is often attributed to Downs and his median voter +theory [12]. This work was followed by further theoretical work on legislative organization +[4,19,25,26], electoral competition [2], and the courts [22] to name a few. +The most common method of obtaining ideological spacial estimates for members of +congress is NOMINATE [23]. It uses the observed voting choices and an item response model +(IRT) to recover spatial distances. This work has been expanded to include bridges over time +to estimate changes in the distribution og congressional representatives across congresses [24]. +More recently, such bridging techniques and new data sources have been used in order to get +consistent measurements for politicians in different chambers as well as candidates who do +not win their election [3,7,10,27]. +In this article we present a bounded confidence model in which there are both attractive +and repulsive links between members. This is motivated by the idea of varying salience of +issues among members of congress. While representatives may have ideological positions that +can be uncovered through voting behavior, there is reason to believe that politicians are drawn +to fellow representatives with similar priorities. Therefore, working with other members of +congress causes their ideologies to converge. In contrast, they make a point of distancing +themselves from representatives who’s salient issues run in opposition to them, regardless of +other similarities. This would cause them to attempt to distinguish themselves. From an +electoral perspective, this distinguishing is important and has not yet, to the our knowledge, +been accounted for in spatial models. +2.2. Bounded-Confidence models. The model we propose is a variant of the Hegselmann– +Krause (HK) model [17]. The HK model considers the opinions of a group of interacting agents +who influence each other. In the HK model, agents are modelled in a network, with connec- +This manuscript is for review purposes only. + +3 +tions between them. Agents who are connected to each other will affect each others’ opinions, +but only if their opinions are sufficiently close. That is, even if two agents are connected, if +their opinions are far apart, they will not take each other into consideration as they form new +opinions. +The precise mathematical statement of HK is as follows. Suppose G = (V, E) is a network, +with associated adjacency matrix A. Then at each time step t, we denote the opinions of nodes +i ∈ V with the opinion vector x(t). We associate to the model a confidence bound c. Opinions +are updated according to the following rule: +(2.1) +xi(t + 1) = +� +j∈V Aijxj(t)1|xj(t)−xi(t)| 0 +sign(j − i)c +Aij = −1, xj(t) = xi(t) +Intuitively, Mij represents a signed distance which node i will potentially travel because of +node j. The effect of M is that repulsive forces grow weaker as nodes move farther away from +This manuscript is for review purposes only. + +4 +C. KANN AND M. FENG +Fragmentation +Polarization +Consensus +0 +2500 5000 7500100000 +2500 5000 7500100000 +2500 5000 750010000 +0.00 +0.25 +0.50 +0.75 +1.00 +Time +Position +Figure 1: In this figure a Erd˝os–Renyi Random Graph is created with connection probability of +25%, the evolution three confidence intervals (0.05, 0.2, 0.6) are shown in order to demonstrate +the three steady states of the model +each other. Note that the third row of Mij covers the case where two nodes have the same +opinion and repulse each other. In this case, the node with the higher index is pushed towards +a higher opinion, while the node with the lower index is pushed towards a lower opinion. In +simulation, this situation is unlikely, as it is rare that two nodes which are repulsed share the +precise same value. We updated opinions using the following rule: +(3.2) +xi(t + 1) = xi(t) + +� +j∈V AijMij(t)1|xj(t)−xi(t)|= c, so that after this time, these two nodes will no longer affect each other, +and cannot push each other further, so the model has converged, and maxi,j |x0(T) − x1(T)| ≤ +c. +If A01 = 1, the two nodes attract each other, and the model is equivalent to standard +Hegselmann–Krause, so that we have convergence to a single point and +|x0(T) − x1(T)| = 0 +This covers all possible cases, and the proposition is proven. +The main point to note from this two-node proof is that the repulsive forces between any +two nodes will contribute to attempting to push them apart to a distance of precisely c. Also +This manuscript is for review purposes only. + +7 +note that any node cannot move more than c in any direction over the course of one timestep, +because |Mij(t)| ≤ c. +We now prove several lemmas that we will use to prove Theorem 4.10. +Lemma 4.2. Suppose i ∈ V a node. Define the following sets: +V + +i (t) = {j ∈ V : Aij = 1 and |xj(t) − xi(t)| < c} +Ui(t) = {j ∈ V : Aij = −1 and [(0 < xj(t) − xi(t) < c) or (xj(t) = xi(t) and j > i)]} +Li(t) = {j ∈ V : Aij = −1 and [(0 < xi(t) − xj(t) < c) or (xj(t) = xi(t) and i > j)]} +Then +(4.1) +xi(t + 1) = +� +j∈V + +i (t) xj(t) + � +j∈Ui(t)(xj(t) − c) + � +j∈Li(xj(t) + c) +|V + +i (t)| + |Ui(t) + |Li(t)| +Proof. From (3.2), we rearrange +xi(t + 1) = xi(t) + +� +j∈V AijMij(t)1|xj(t)−xi(t)| xj(t) for all other +nodes j, so that i is the node with the highest opinion value at time t. Then xi(t+1) > xj(t+1) +for all j. +Proof. Note that V + +k (t) = {xk} for all k, t, since every edge in G is repulsive. For conve- +nience we define the following sets: +Uij(t) = Uj(t) +� +Ui(t) +U ′ +ij(t) = Ui(t) \ Uij(t) +Lij(t) = Li(t) +� +Lj(t) +L′ +ji(t) = Lj(t) \ Lij(t) +Wij(t) = Li(t) +� +Uj(t) +Unpacking this notation, Uij(t) consists of all nodes that repel both i and j downward, while +Lij(t) consists of all nodes that repel both i and j upward. U ′ +ij(t) consists of nodes which +repel i downward, but not j (note that if xi(t) < xj(t), this is automatically empty), while +L′ +ji(t) consists of nodes which repel j upward, but not i (again, if xi(t) < xj(t), this is empty). +Finally, Wij(t) consists of nodes which repel i upward and j downward (empty if xj(t) > xi(t)). +Now, suppose xi(t) > xj(t) for all j ∈ V . Then we can write +V + +i (t) = {i} +Ui(t) = ∅ +Li(t) = Lij(t) +� +Wij(t) +� +{j} +This manuscript is for review purposes only. + +9 +and +V + +j (t) = {j} +Ui(t) = {i} +� +Wij(t) +Li(t) = Lij(t) +� +L′ +ji(t) +Then we observe the following from the knowledge that nodes only effect each other if +they are within confidence of each other. +xj(t) + c > xi(t) +Lij(t) + c > xi(t) +Wij(t) + c > xi(t) +xi(t) > L′ +ji + c +Then applying Lemma 4.2 and Lemma 4.3: +xi(t + 1) = xi(t) + (xj(t) + c) + |Lij(t)|(Lij(t) + c) + |Wij(t)|(Wij(t) + c) +2 + |Lij(t)| + |Wij(t)| +> +xi(t) + (xj(t) + c) + |Lij(t)|(Lij(t) + c) + |Wij(t)|(Wij(t) + c) + |L′ +ji(t)|(L′ +ji(t) + c) +2 + |Lij(t)| + |Wij(t)| + |L′ +ji(t)| +(4.2) +> +(xi(t) − c) + xj(t) + |Lij(t)|(Lij(t) + c) + |Wij(t)|(Wij(t) − c) + |L′ +ji(t)|(L′ +ji(t) + c) +2 + |Lij(t)| + |Wij(t)| + |L′ +ji(t)| += xj(t + 1) +where the inequality in (4.2) follows because xi(t+1) is a weighted average, and L′ +ji(t) is less +than all of the other values being averaged in the previous line. The next inequality follows +straightforwardly by replacing each value in the average with a smaller or equal value. +So if xi(t) has the highest value opinion at time t, it will always have the highest value +opinion. +Corollary 4.5. Let G = (V, E) be a network with n nodes and m edges with confidence bound +c. Suppose that every edge in G is repulsive. At time t, let M = {i : xi(t) ≥ xj(t)∀j ∈ V }. +Then xmaxM i(t + 1) > xj(t + 1)∀j ∈ V . +Proof. From the definitions of Ui(t), Li(t), we can observe that the member of M with +highest index will have the largest corresponding set Li(t) and the smallest corresponding +Ui(t), so that at time t + 1, that member of M will have the highest-valued opinion of all +members of M. By the same logic as in the proof of Lemma 4.4, that opinion will also be the +highest-valued opinion overall. +Corollary 4.6. Let G = (V, E) be the complete network with n nodes with confidence bound +c. Suppose that every edge in G is repulsive. At time t, let M = {i : xi(t) ≤ xj(t)∀j ∈ V }. +Then xminM i(t + 1) < xj(t + 1)∀j ∈ V . +This manuscript is for review purposes only. + +10 +C. KANN AND M. FENG +Proof. Proves that xi(t) < xj(t) for all j ∈ V , then xi(t + 1) < xj(t + 1) for all j ∈ V , by +segmenting Ui(t), Li(t), Uj(t), Lj(t) into the appropriate subsets and reversing inequalities as +needed as in Lemma 4.4. Then the same logic as in Corollary 4.5 proves the statement. +Lemma 4.7. Let G = (V, E) be a network with n nodes and m edges with confidence bound +c. Suppose that every edge in G is repulsive. At time t, suppose xi(t) > xj(t) for all other +nodes j ∈ V , so that i is the node with the highest-valued opinion at time t. Suppose that +there is some node j such that xi(t) − xj(t) < c, and that j has the highest-valued opinion of +all such nodes. Then +2c +2 + |Lij(t)| + |L′ +ji(t)| ≤ xi(t + 1) − xj(t + 1) ≤ +(|L′ +ji(t)| + 2)c +2 + |Lij(t)| + |L′ +ji(t)| +Proof. By assumption, since j has the highest-valued opinion of all nodes within confidence +of i, Wij(t) is empty. To prove the lower bound, +xi(t + 1) = xi(t) + (xj(t) + c) + |Lij(t)|(Lij(t) + c) +2 + |Lij(t)| +≥ +xi(t) + (xj(t) + c) + |Lij(t)|(Lij(t) + c) + |L′ +ji(t)|(L′ +ji(t) + c) +2 + |Lij(t)| + |L′ +ji(t)| +xj(t + 1) = +(xi(t) − c) + xj(t) + |Lij(t)|(Lij(t) + c) + |L′ +ji(t)|(L′ +ji(t) + c) +2 + |Lij(t)| + |L′ +ji(t)| +xi(t + 1) − xj(t + 1) ≥ +2c +2 + |Lij(t)| + |L′ +ji(t)| +To prove the upper bound, +xi(t + 1) = xi(t) + (xj(t) + c) + |Lij(t)|(Lij(t) + c)+ +2 + |Lij(t)| +≤ +xi(t) + (xj(t) + c) + |Lij(t)|(Lij(t) + c) + |L′ +ji(t)|(xj(t) + c) +2 + |Lij(t)| + |L′ +ji(t)| +xj(t + 1) = +(xi(t) − c) + xj(t) + |Lij(t)|(Lij(t) + c) + |L′ +ji(t)|(L′ +ji(t) + c) +2 + |Lij(t)| + |L′ +ji(t)| +xi(t + 1) − xj(t + 1) ≤ +c + c + |L′ +ji(t)|(xj(t) − L′ +ji(t)) +2 + |Lij(t)| + |L′ +ji(t)| +≤ +(|L′ +ji(t)| + 2)c +2 + |Lij(t)| + |L′ +ji(t)| +Notice that if L′ +ji(t) is empty, both inequalities become equalities, so that +xi(t + 1) − xj(t + 1) = +2c +2 + |Lij(t)| +Notice also that if both Lij(t) and L′ +ji(t) are empty, that the distance between xi(t + 1) − +xj(t + 1) is precisely c. +This manuscript is for review purposes only. + +11 +Corollary 4.8. Let G = (V, E) be a network with n nodes and m edges with confidence +bound c. Suppose that every edge in G is repulsive. At time t, suppose xi(t) < xj(t) for all +other nodes j ∈ V , so that i is the node with the lowest-valued opinion at time t. Suppose that +there is some node j such that xj(t) − xi(t) < c, and that j has the lowest-valued opinion of +all such nodes. Then +2c +2 + |Uij(t)| + |U ′ +ij(t)| ≤ xi(t + 1) − xj(t + 1) ≤ +(|U ′ +ij(t)| + 2)c +2 + |Uij(t)| + |U ′ +ij(t)| +Lemma 4.7 and Corollary 4.8 are interesting because they give us precise conditions under +which the nodes with the most extreme opinions will no longer be within confidence of any +other nodes. Specifically, in order for the node with the highest-value opinion to lose connec- +tion with all other nodes, it must be true that the only node it is still influenced by is the node +with the second-highest-value opinion, and that neither of the two nodes is influenced by any +other nodes. Otherwise, they will remain within confidence of each other, even as the node +with highest-value opinion remains the most extreme node and continues to have its opinion +pushed upward. +We conclude with one more lemma about the bound on the width of the gap between +consecutive nodes. +Lemma 4.9. Let G = (V, E) be the complete network with n nodes and confidence bound +c. Suppose that every edge in G is repulsive. At time t, suppose that i and j are nodes such +that (i, j) ∈ E, xi(t) > xj(t), and xi(t) − xj(t) < c, and there exist no nodes k connected to i +or j such that xi(t) > xk(t) > xj(t). Then +|xi(t + 1) − xj(t + 1)| ≤ c +Proof. By the assumption that no nodes have values between xi(t) and xj(t), we have +that Wij(t) = Wji(t) = ∅. Then to prove one direction of the bound, +xi(t + 1) = |U ′ +ij(t)|(U ′ +ij(t) − c) + |Uij(t)|(Uij(t) − c) + xi(t) + (xj(t) + c) + |Lij(t)|(Lij(t) + c) +2 + |U ′ +ij(t)| + |Uij(t)| + |Lij(t)| +≤ |U ′ +ij(t)|(U ′ +ij(t) − c) + |Uij(t)|(Uij(t) − c) + xi(t) + (xj(t) + c) + |Lij(t)|(Lij(t) + c) + |L′ +ji(t)|(xj(t) + c) +2 + |U ′ +ij(t)| + |Uij(t)| + |Lij(t)| + |L′ +ji(t)| +xj(t + 1) = |Uij(t)|(Uij(t) − c) + (xi(t) − c) + xj(t) + |Lij(t)|(Lij(t) + c) + |L′ +ji(t)|(L′ +ji(t) + c) +2 + |Uij(t)| + |Lij(t)| + |L′ +ji(t)| +≥ |U ′ +ij(t)|(xi(t) − c) + |Uij(t)|(Uij(t) − c) + (xi(t) − c) + xj(t) + |Lij(t)|(Lij(t) + c) + |L′ +ji(t)|(L′ +ji(t) + c) +2 + |U ′ +ij(t)| + |Uij(t)| + |Lij(t)| + |L′ +ji(t)| +This manuscript is for review purposes only. + +12 +C. KANN AND M. FENG +Combining both equations, +xi(t + 1) − xj(t + 1) ≤ +|U ′ +ij(t)| +� +U ′ +ij(t) − xi(t) +� ++ c + c + |L′ +ji(t)| +� +xj(t) − L′ +ji(t) +� +2 + |U ′ +ij(t)| + |Uij(t)| + |Lij(t)| + |L′ +ji(t)| +≤ +� +2 + |U ′ +ij(t)| + |L′ +ji(t)| +� +c +2 + |U ′ +ij(t)| + |Uij(t)| + |Lij(t)| + |L′ +ji(t)| +≤ c +To prove the other direction, +xj(t + 1) = |Uij(t)|(Uij(t) − c) + (xi(t) − c) + xj(t) + |Lij(t)|(Lij(t) + c) + |L′ +ji(t)|(L′ +ji(t) + c) +2 + |Uij(t)| + |Lij(t)| + |L′ +ji(t)| +≤ |U ′ +ij(t)|(Lij(t) + c) + |Uij(t)|(Uij(t) − c) + (xi(t) − c) + xj(t) + |Lij(t)|(Lij(t) + c) + |L′ +ji(t)|(L′ +ji(t) + c) +2 + |U ′ +ij(t)| + |Uij(t)| + |Lij(t)| + |L′ +ji(t)| +xi(t + 1) = |U ′ +ij(t)|(U ′ +ij(t) − c) + |Uij(t)|(Uij(t) − c) + xi(t) + (xj(t) + c) + |Lij(t)|(Lij(t) + c) +2 + |U ′ +ij(t)| + |Uij(t)| + |Lij(t)| +≥ |U ′ +ij(t)|(U ′ +ij(t) − c) + |Uij(t)|(Uij(t) − c) + xi(t) + (xj(t) + c) + |Lij(t)|(Lij(t) + c) + |L′ +ji(t)|(Uij(t) − c) +2 + |U ′ +ij(t)| + |Uij(t)| + |Lij(t)| + |L′ +ji(t)| +Combining both inequalities yields +xj(t + 1) − xi(t + 1) ≤ +|U ′ +ij(t)| +� +Lij(t) − U ′ +ij(t) + 2c +� ++ c + c + |L′ +ji(t)| +� +L′ +ji(t) − Uij(t) + 2c +� +2 + |U ′ +ij(t)| + |Uij(t)| + |Lij(t)| + |L′ +ji(t)| +Note, however, that +2c = (xi(t) + c) − (xi(t) − c) +> U ′ +ij(t) − Lij(t) +> (xj(t) + c) − xj(t) = c +and similarly c < Uij(t) − L′ +ji(t) < 2c so that we have +xj(t + 1) − xi(t + 1) ≤ +|U ′ +ij(t)| +� +Lij(t) − U ′ +ij(t) + 2c +� ++ c + c + |L′ +ji(t)| +� +L′ +ji(t) − Uij(t) + 2c +� +2 + |U ′ +ij(t)| + |Uij(t)| + |Lij(t)| + |L′ +ji(t)| +≤ +� +2 + |U ′ +ij(t)| + |L′ +ji(t)| +� +c +2 + |U ′ +ij(t)| + |Uij(t)| + |Lij(t)| + |L′ +ji(t)| +≤ c +and the proof is finished. +Theorem 4.10. Let G = (V, E) be a network with n nodes and confidence bound c. Suppose +that G is the complete graph, and that every edge (i, j) ∈ E is repulsive (that is Aij = −1). +Suppose also that we have initial opinions xi(0) such that |xi(0) − xj(0)| < c. Then the model +converges, and maxi,j |xi(T) − xj(T)| = (n − 1)c. +This manuscript is for review purposes only. + +13 +Proof. The intuition for this theorem is as follows: for any repulsive edge (i, j), nodes i +and j will repel each other until +|xi(t) − xj(t)| >= c +at some future timestep t. If every edge is repulsive, we must have distance at least c between +every pair of nodes connected by an edge in order for the model to converge. Intuitively, the +nodes will always continue to push each other outward until they reach a distance of c, and +no further, so that the final convergent state of the model will occur when there are gaps of +at least c between all of the m edges in the original graph. However, from Lemma 4.7, the +gaps will have precisely width c, so that the bound holds. +From Corollary 4.5 and Corollary 4.6, at time 1, there must be a highest and lowest- +value opinion node. By Lemma 4.4, for t > 1, these nodes will always be the highest and +lowest-value opinion nodes. Call these nodes imax, imin. +Because G is the complete graph, and all edges are repulsive, we can observe that imax and +imin will have their opinions pushed outward, since initially every node is within confidence +of every node. Additionally, from Lemma 4.3, we can observe that imax will be pushed in the +direction of {j ̸= i}(0)+c, so that the nodes with opinions much lower valued than the average +will start to drop out of confidence of imax. Further, from Lemma 4.7, imax will remain within +confidence of at least one node as long as it is within confidence of at least 2 nodes in the +previous timestep. Combining these lemmas, we can see that eventually at time t′, imax will +be within confidence of exactly one other node. +Let i′ +max be the singular node for which ximax(t′) − xi′max(t′) < c. Then we can follow the +same proof procedure as in Lemma 4.4 to prove that xi′max(t′ + 1) > xj(t′ + 1) for all j ∈ V +other than j = imax, and that none of the remaining nodes can be pushed into confidence +of imax. We do not include the procedure here because of its similarity to Lemma 4.4, but +the key observation that drives the proof is that there is only a single node imax exerting +downward pressure on i′ +max (if a very high number of nodes were exerting downward pressure +on i′ +max, it would be possible for i′ +max to lose its position as the node with second-highest-value +opinion). This allows us to rewrite the xi′max as an average of values which preserve the order +of imax, i′ +max, and the remaining nodes. Similarly, we can show that there is some time after +which the node with the second-lowest-value opinion will always remain the node with the +second-lowest-value opinion. +We continue in this manner, proceeding from the nodes with the highest and lowest-value +opinions inwards until we show that after some time, the nodes’ opinions must remain in a +fixed order. +From this point on, we observe that from Lemma 4.9, the gap between any two consecutive +nodes is bounded by c. Because of our initial conditions on xi(t), it is impossible for any gap +between consecutive nodes to be larger at any point. If any two nodes have a gap smaller +than c, we will not have converged, as the repulsion between the two nodes will push them +apart in the next time step. All nodes will push each other apart until the gap between any +two consecutive nodes is precisely c, at which point the model has converged. Because there +This manuscript is for review purposes only. + +14 +C. KANN AND M. FENG +are n nodes, this tells us +max{|xi(T) − xj(T)|} = (n − 1)c +The proof for Theorem 4.10 relies on all edges being repulsive, thereby preserving the +ordering of the nodes. This property does not necessarily hold when there are both attractive +and repulsive edges. However, we suspect based on numerics that the following theorem is +also true: +Theorem 4.11. Suppose G = (V, E) is a network with n nodes, m edges, and confidence +bound c. Let mr be the number of repulsive edges in the network. Then the model converges +and +max +i,j∈V {xi(T) − xj(T)} ≤ max{max +i,j∈V {xi(0) − xj(0)}, mc} +Intuition. The worst case for this model assumes that all repulsive nodes end up at least +c apart from each other, so if all nodes start out within confidence of each other, the worst +case is one in which all nodes with repulsive edges are chained together in consecutive order +along a line of m edges, in which case the width of their opinions cannot exceed mc, since the +bounds in Lemma 4.9 should apply and prevent any individual gap from growing wider. The +only way a gap could grow wider is if there are attractive nodes pulling the repulsed nodes +further apart, in which case those attractive nodes either have repulsive forces between them, +and have already been considered in the line, or must have started farther apart to begin with, +in which case we look at maxi,j∈V {xi(0) − xj(0)}. +Because we cannot rely on nodes remaining in fixed order in this case, we cannot use the +same technique as in Theorem 4.10 to prove convergence and a bound. However, in practice, +we observe that the range of final opinions increases with the number of repulsive edges, and +that in practice the bound of mc is not very tight (this is to be expected, as, for example, in +the case of the complete graph in Theorem 4.10, the bound is considerably smaller). To see +numerics showing that the range of final opinions scales with number of repulsive edges and +c, see Figure 5 and associated discussion. +5. Numerical results on synthetic networks. In this section we present analysis of nu- +merical simulations on a variety of random networks, chosen for their usage in modelling social +structures [28]. +5.1. Erd˝os–Renyi. We begin with an adaptation of Erd˝os–Renyi (ER) networks as a +simple random network model. To achieve a random network with both positive and negative +edges, we generate two ER networks, G1 = G(n, p1) and G2 = G(n, p2), with associated +adjacency matrices A1 and A2. The total network G, is then the network derived from the +adjacency matrix A1 −A2. A visual of this method can be seen in Figure 4. In the subsequent +network the probability of each type of edge between any set of nodes can be written as: +(5.1) +P((i, j) ∈ E) = +� +� +� +0 +(1 − p1)(1 − p2) + p1p2 +1 +p1(1 − p2) +−1 +(1 − p1)p2 +This manuscript is for review purposes only. + +15 +(a) G1: positive node network +with p1 = 0.6 +(b) G2: negative node network +with p2 = 0.2 +(c) G3: +final network with +p1 = 0.6 and p2 = 0.2. +Figure 4: An example of the generation of the ER network with attractive and repulsive edges +To create the simulation results, 100 trials were run with all combinations of the following +parameters: +p1 ∈ (0.2, 0.4, 0.6, 0.8, 1) +p2 ∈ (0.0, 0.2, 0.4, 0.6, 0.8) +c ∈ (0.05, 0.4, 0.8, 1.2, 1.6) +For each trial, a random set of initial opinions is generated and the model is applied for +10000 iterations. In Figure 5, one trial is shown for each set of parameters when p1 is set to +0.4. This trial was chosen randomly and all other trials qualitatively look the same. +The final range of opinions gets wider with both p2 and c once repulsive edges are included. +These results are in line with expectations. As p2 increases, so does the number of negative +connections, resulting in more repulsive forces between nodes, pushing opinions apart. As +c increases, nodes have more neighbors. For p2 << p1, the attractive forces overpower the +repulsive ones, so that higher c leads to more consensus, as in standard HK models. For +p2 >> p1, the opposite is true – repulsive forces overpower attractive ones, and nodes push +each other further apart for higher c, resulting in a wider spread of opinions. +In particular, we observe that the proportion of p2 +p1 seems to be the driving factor in the +range of final opinions. To draw clearer conclusions, we look at opinion spread, which we +define as the following quantity: +(5.2) +maxi,j |xi(T) − xj(T)| +maxi,j |xi(0) − xj(0)| +In Figure 6, we plot average opinion spread across trials as a function of the proportion +p2 +p1 and confidence bound c in a heat map. We observe similar trends as in Figure 5, with +higher proportions p2 +p1 leading to higher values of opinion spread, and the influence of c on +opinion spread depending on p2 +p1 . In the following examples, we will similarly see that opinion +spread is largely controlled by the negative edges in the network, but that the addition of +more structure to the network will influence opinion formation in interesting ways. +This manuscript is for review purposes only. + +16 +C. KANN AND M. FENG +0 +0.2 +0.4 +0.6 +0.8 +0.05 +0.4 +0.8 +1.2 +1.6 +−2.5 +0.0 +2.5 +−2.5 +0.0 +2.5 +−2.5 +0.0 +2.5 +−2.5 +0.0 +2.5 +−2.5 +0.0 +2.5 +Time +Position +Figure 5: For all plots, p1 = .4. The horizontal axis represents p2, while the vertical axis +represents c. Note that as p2 increases, the range of final opinions gets wider. For low values +of p2, as c increases, the range of final opinions becomes narrower (closer to consensus). By +contrast, for high values of p2, as c increases, the range of final opinions becomes wider. +0.05 +0.4 +0.8 +1.2 +1.6 +0 +0.2 +0.4 +0.6 +0.8 +0 +0.2 +0.4 +0.6 +0.8 +0 +0.2 +0.4 +0.6 +0.8 +0 +0.2 +0.4 +0.6 +0.8 +0 +0.2 +0.4 +0.6 +0.8 +0.2 +0.4 +0.6 +0.8 +1 +Negative Connection Probability +Positive Connection Probability +3 +6 +9 +Spread +Figure 6: Heat map of opinion spread as a function of the probability of a connection in G1 +and G2 iterated over confidence bound c. Data drawn from the mean of 100 trials for each +set of parameters with parameters p1 ∈ {0.2, 0.4, 0.6, 0.8, 1}, p2 ∈ {0, 0.2, 0.4, 0.6, 0.8, 1}, and +c ∈ {0.05, 0.4, 0.8, 1.2, 1.6}. +This manuscript is for review purposes only. + +17 +5.2. Stochastic Block Models. Next, we adapt a Stochastic Block Model (SBM) in order +to incorporate both positive and negative edges. In these networks, each node is assigned to +a group k ∈ K. The probabilities of connections when ik = jk is different than when ik ̸= jk. +This enforces structure within the network. Similarly to in subsection 5.1 we generate this +network through two sub networks. In this case, the process begins with two SBM networks, +G1 = G(n, p1, ρ) and G2 = G(n, p2, ρ), with associated adjacency matrices A1 and A2. The +variable p is the probability of having a connection with another node in the same cluster +while pρ is the probability of having a connection with a node in a different cluster. If G is +network represented by the adjacency matrix given by A1 −A2 we have the generated network +edge probabilities: +P((i, j) ∈ E)ik=ij = +� +� +� +0 +(1 − p1)(1 − p2) + p1p2 +1 +p1(1 − p2) +−1 +(1 − p1)p2 +(5.3) +P((i, j) ∈ E)ik̸=ij = +� +� +� +0 +(1 − p1ρ)(1 − p2ρ) + ρ2p1p2 +1 +p1ρ(1 − p2ρ) +−1 +(1 − p1ρ)p2ρ +(5.4) +A sample of the generative process can be seen in Figure 7, where the blue edges represent +positive edges and the black negative. Each color of nodes represents a group k ∈ K. +(a) G1: positive node network +with p1 = 0.85 and ρ = 0.05 +(b) G2: negative node network +with p2 = 0.3 and ρ = 0.05 +(c) G3: +final network p1 = +0.85, p2 = 0.3 and ρ = 0.05 +Figure 7: An example of the generation of the SBM network with attractive and repulsive +edges +Again, in order to create simulation results, 100 trials are run for all combinations of the +parameters +ρ ∈ (0, 0.2, 0.4, 0.6, 0.8, 1) +p1 ∈ (0.2, 0.4, 0.6, 0.8, 1) +p2 ∈ (0.0, 0.2, 0.4, 0.6, 0.8) +c ∈ (0.05, 0.4, 0.8, 1.2, 1.6) +This manuscript is for review purposes only. + +18 +C. KANN AND M. FENG +For each trial, a random set of initial opinions is generated and the model is applied. The +full histories of an example run when p1 = 0.8 and p2 = 0.2 can be seen in Figure 8. It can be +seen that each group fully converges on itself, while the confidence interval determines how +dispersed the groups are from each other. As ρ increases, the full set begins to converge. As +is the case with the ER models, for certain combinations the spread at steady-state is larger +than that at the start. From these values we can see that when p1 and p2 are locked, higher +confidence bounds result in a larger terminal spread, as does lower percentages of cross-group +edges (ρ). +0 +0.2 +0.4 +0.6 +0.8 +1 +0.05 +0.4 +0.8 +1.2 +1.6 +−1 +0 +1 +2 +−1 +0 +1 +2 +−1 +0 +1 +2 +−1 +0 +1 +2 +−1 +0 +1 +2 +Time +Position +Figure 8: Example of the paths taken when the graph is distributed according to the Stochastic +Block Model scheme laid out in 5.4. In this case p1 = 0.8 and p2 = 0.2 are both locked the +confidence bound varies between the rows and the percent of cross group links varies over +columns. Each color represents a different group. There are five groups in these examples. +As with the ER model, with the SBM we first look at steady-state opinion spread. The +results can be seen in Figure 9. Note, that when ρ = 1 the SBM model is equivalent to the ER +This manuscript is for review purposes only. + +19 +model for the same parameters. We therefore have that the final row is identical to Figure 6. +We note that as the value of ρ shrinks, the final spread increases. Otherwise, the trends found +for the ER model are consistent. +0.05 +0.4 +0.8 +1.2 +1.6 +0 +0.2 +0.4 +0.8 +1 +0 0.2 0.4 0.6 0.8 +0 0.2 0.4 0.6 0.8 +0 0.2 0.4 0.6 0.8 +0 0.2 0.4 0.6 0.8 +0 0.2 0.4 0.6 0.8 +0.2 +0.4 +0.6 +0.8 +1 +0.2 +0.4 +0.6 +0.8 +1 +0.2 +0.4 +0.6 +0.8 +1 +0.2 +0.4 +0.6 +0.8 +1 +0.2 +0.4 +0.6 +0.8 +1 +Negative Connection Probability +Positive Connection Probability +3 +6 +9 +Spread +Figure 9: Heat map of the opinion spread of Stochastic Block Model trials. The rows repre- +sent ρ, the percent of positive(negative) connection values that are out (in) group, while the +columns are c the confidence bound. +In the case of SBMs, since each vertex i ∈ V is assigned to a group in k, we are also +interested in clustering in addition to opinion spread. We introduce a measures of how close +This manuscript is for review purposes only. + +20 +C. KANN AND M. FENG +vertices are to in-group vertices versus out-group vertices. In order to calculate this, which +we call proportional spread, first we find the average in and out-group distances as: +(5.5) +IT = 1 +|k| +� +ki∈k +1 +|ki|2 +� +j∈ki⊂V +� +ℓ∈ki∈⊂V +|xj(T) − xℓ(T)| +(5.6) +OT = 1 +|k| +� +ki∈k +1 +|ki| +� +j∈ki⊂V +1 +|k| − |ki| +� +ℓ∈⊂V/ki +|xj(T) − xℓ(T)| +Since the end-spread of the samples differ, in order to appropriately compare them we +look at the ratio, that is +(5.7) +PST = IT +OT +Smaller values of proportional spread imply same-group nodes are significantly closer to +each other than different-group nodes. +Thus, small values imply increased separation by +group. In Figure 10 the spread as well as O0 and I0 can be seen. It is clear that the values +range most of the space, in addition O0 ≈ I0. Thus, uniformly, at the beginning of the trials +we have PS0 = 1. The plots of proportional spread at steady-state over trials can be seen in +Figure 11. We observe that clustering is most clear with higher values of p1 and lower values +of ρ. To a lesser extent, clustering also increases with lower values of p2. This implies that +the higher rates of in-group connection and fewer cross group connections lead to increased +polarization. +0.999000.999250.999500.999751.00000 +Range of Original Values +0.330 +0.333 +0.336 +Mean Out Group Distance +0.330 +0.333 +0.336 +Mean In Group Distance +Figure 10: Histograms of metrics at the the start of each trial, useful for comparing the final +results +This manuscript is for review purposes only. + +21 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.2 +0.4 +0.6 +0.8 +0.00 0.25 0.50 0.75 1.000.00 0.25 0.50 0.75 1.000.00 0.25 0.50 0.75 1.000.00 0.25 0.50 0.75 1.000.00 0.25 0.50 0.75 1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +Percent Cross Group Edges +Proportional Spread +CI +0.05 +0.4 +0.8 +1.2 +1.6 +Figure 11: Proportional spread for Stochastic Block Model trials. The rows represent p2, +the base probability of an out-group negative connection, while the columns are p1 the base +in-group positive probability. The x-axis is ρ the percent of cross group edges and the color +represents the confidence bound. A value of 1 means that nodes are equally as close to in and +out-group nodes. Smaller values mean groups are increasingly clustered. +This manuscript is for review purposes only. + +22 +C. KANN AND M. FENG +6. Future Work and Conclusions. Previous models of opinion dynamics have focused +on the attractive nature of network connections. These result in a complete convergence in +the steady-state for each receptivity subgraph. When the term polarization is used, it still +implies a shrinking of the overall opinion space. +The model introduced in this paper, in +contrast, acknowledges that there are circumstances in which individuals seek to differentiate +themselves from those in their network. This behavior leads to the possibility of an overall +opinion space expansion. +In this paper, the basic bounds of this expansion were proven analytically in certain cases, +with intuition provided for the general case. In addition, the steady-state behavior for random +networks were analyzed numerically. These simulation results offered insight into the effects of +various structural parameters on the model. It was clear that when there was structure in the +network links (for instance in the Stochastic Block Model) group-based clustering emerged. +This clustering was despite the random initial conditions provided. In addition, the size of the +confidence bounds and the density of repulsive edges are both pivotal in the opinion spread. +The model introduced in this paper lends itself to complex opinion dynamics, where politi- +cians or individuals want to differentiate themselves due to factors orthogonal to their ex- +pressed opinions. In future work we hope to explore how this model can help us understand +political behavior. Initial ideas include using informative initial conditions for congressional +networks. Alternatively, this model can be used to look at ideological opinions of individu- +als who are influenced by pop culture associating ideological beliefs with other factors. This +would introduce a variable connection to an ideal point, which then attracts or repulses the +individual. +The addition of repulsive forces make bounded-confidence models increasingly +relevant for empirical and theoretical studies of human behavior. +Acknowledgments. One of the authors (M. Feng) on this project is funded by the James +S. McDonnell Foundation Postdoctoral Fellowship. In addition, we would like to thank Danny +Ebanks, R. Michael Alvarez, Jonathan Katz, and Mason A. Porter for helpful comments and +insights. +REFERENCES +[1] C. Altafini and F. 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Giannotti, and J. Kert´esz, Algorithmic bias amplifies opinion fragmen- +tation and polarization: A bounded confidence model, PLOS ONE, 14 (2019), pp. 1–20. +[30] H. Speier, Historical development of public opinion, American Journal of Sociology, 55 (1950), pp. 376– +388. +This manuscript is for review purposes only. + diff --git a/H9E0T4oBgHgl3EQfRgD4/content/tmp_files/load_file.txt b/H9E0T4oBgHgl3EQfRgD4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2bf3380918cd4381fc8feddefdb6d607da12f505 --- /dev/null +++ b/H9E0T4oBgHgl3EQfRgD4/content/tmp_files/load_file.txt @@ -0,0 +1,1004 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf,len=1003 +page_content='A Repulsive Bounded-Confidence Model of Opinion Dynamics in Polarized Communities∗ Claudia Kann† and Michelle Feng‡ Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' Collective opinions affect civic participation, governance, and societal norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' Due to the influence of opinion dynamics, many models of their formation and evolution have been developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' A commonly used approach for the study of opinion dynamics is bounded-confidence models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' In these models, individuals are influenced by the opinions of others in their network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' They generally assume that individuals will formulate their opinions to resemble those of their peers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' In this paper, inspired by the dynamics of partisan politics, we introduce a bounded-confidence model in which individuals may be repelled by the opinions of their peers rather than only attracted to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' We prove convergence properties of our model and perform simulations to study the behavior of our model on various types of random networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' In particular, we observe that including opinion repulsion leads to a higher degree of opinion fragmentation than in standard bounded-confidence models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' opinion dynamics, bounded confidence, mathematical political science, congressional voting AMS subject classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' 91D30, 91F10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' Opinions dictate how individuals interact with society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' They influence who we are friends with, how we vote, and what we consume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' At the individual and collective level, opinions shape our lives and our social interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' Understanding how opinions are formed and their dynamics provides a framework for studying changes in our society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' The role of opinions in politics and governance is a prominent part of public discourse in the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' Inspired by discussions of political polarization and partisan politics, this paper presents a mathematical approach to modelling polarized opinion dynamics where individuals feel both attractive and repulsive forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' The influence of public opinion on politics have been studied by philosophers, sociologists, and social theorists [6,15,30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' Contemporary approaches to studying opinions frequently seek to quantify them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' In this paper, we focus on the dynamics of opinions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' We are interested in studying how opinions in a society shift as a result of relationships between individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' Various models for studying individual opinions exist [9,14,17,21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' We will focus on bounded- confidence models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' Bounded-confidence models are a class of models that suppose individuals change their opinions based on their relationships, when their opinions are already close to those of their peers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' That is, if someone’s opinion is very far away from my own, even if I have a relationship with them, I will not base my opinions on theirs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' Many bounded- confidence models have been developed and studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' They include examinations of consensus formation [11,13], polarization [16,29], and a large variety of model extensions for application to real-world opinions [1,8,18,20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' We consider polarization, and the notion that individuals may form their opinions by ∗Submitted to the editors December 8, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' Funding: This work was partially funded by the James S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' McDonnell Foundation Postdoctoral Fellowship †Department of Humanities and Social Sciences, California Institute of Technology (ckann@caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' ‡Computing + Mathematical Sciences Department, California Institute of Technology (mfeng@caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' 1 This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content='02210v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content='DS] 5 Jan 2023 2 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' KANN AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' FENG being contrarian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' If I have an adversarial relationship with someone, I may specifically choose to hold an opinion that is different from their’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' Similar to other bounded-confidence models, we maintain the idea that individuals are mostly influenced by others whose opinions are already somewhat close to our own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' We are most interested in understanding how collective opinions in this model behave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' What types of relationships and community structures lead to strong polarization within a society?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' How might we extend those observations to real-world applications and data?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' We introduce the motivation for our model in section 2 and define our model in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' We present analytical results in section 4, and perform numerical simulations on synthetic networks (section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' Conclusions follow in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' Background and motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' In this section, we introduce the motivation for our proposed model of opinion dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' In subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content='1, we discuss political science research which motivates our modelling choices, and in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content='2 we introduce the Hegselmann– Krause model for opinion dynamics, which we use as a starting point in the formulation of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' Political Science motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' In political science it is common to think of ideologies as points in space, as being on the left or the right, liberal or conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' This spatial view of politicians and individuals drives much of the work that is done on voting behavior, both at the individual and legislative levels, as well as the models of strategic behavior within Congress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' The original conception of this model is often attributed to Downs and his median voter theory [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' This work was followed by further theoretical work on legislative organization [4,19,25,26], electoral competition [2], and the courts [22] to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' The most common method of obtaining ideological spacial estimates for members of congress is NOMINATE [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' It uses the observed voting choices and an item response model (IRT) to recover spatial distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' This work has been expanded to include bridges over time to estimate changes in the distribution og congressional representatives across congresses [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' More recently, such bridging techniques and new data sources have been used in order to get consistent measurements for politicians in different chambers as well as candidates who do not win their election [3,7,10,27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' In this article we present a bounded confidence model in which there are both attractive and repulsive links between members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' This is motivated by the idea of varying salience of issues among members of congress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' While representatives may have ideological positions that can be uncovered through voting behavior, there is reason to believe that politicians are drawn to fellow representatives with similar priorities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' Therefore, working with other members of congress causes their ideologies to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' In contrast, they make a point of distancing themselves from representatives who’s salient issues run in opposition to them, regardless of other similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' This would cause them to attempt to distinguish themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' From an electoral perspective, this distinguishing is important and has not yet, to the our knowledge, been accounted for in spatial models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' Bounded-Confidence models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' The model we propose is a variant of the Hegselmann– Krause (HK) model [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' The HK model considers the opinions of a group of interacting agents who influence each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' In the HK model, agents are modelled in a network, with connec- This manuscript is for review purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' 3 tions between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' Agents who are connected to each other will affect each others’ opinions, but only if their opinions are sufficiently close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' That is, even if two agents are connected, if their opinions are far apart, they will not take each other into consideration as they form new opinions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' The precise mathematical statement of HK is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' Suppose G = (V, E) is a network, with associated adjacency matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' Then at each time step t, we denote the opinions of nodes i ∈ V with the opinion vector x(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' We associate to the model a confidence bound c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content=' Opinions are updated according to the following rule: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfRgD4/content/2301.02210v1.pdf'} +page_content='1) xi(t + 1) = � j∈V Aijxj(t)1|xj(t)−xi(t)| +n +0.70 +0. +Qr=0.00050±0.00034 +-0.00034 +80000 +0.0006 +0.0004 +0.0002 +Am=0.00252 ++0.00169 +0.001/3 +0.002 +0.001 +A^=-0.00251±0.00172 +0.00169 +0.001 +0 +0.68 +20 +.69 +O. +70 +0. +-0. +-0. +2m +h +Am +A^FIG. 2. The MCMC simulation results for the diffusive model’s which is given in Eq. 15 for cosmological +free parameters (Ωm0, ¯h, Ωr0, ∆m and ∆Λ), with the “true” values for Ωm0 = 0.315, ¯h = 0.674, and +Ωr0 = 2.47 × 10−5 /¯h2 provided by the Planck2018 collaboration data release [33]. We used 100 random +walkers and 10000 iterations. +an impact on the full range predicted by them. Additionally, Figs. 5 and 6 display the residuals +obtained in the above two cases. +It can clearly be seen that at no point do the models over-or +under-estimate the resulting distance modulus for each supernovae. We also note that the average +off-set the model has, compared to the data, is ¯xres = −0.0374 Mpc in both cases with a standard +deviation of σres = 0.2148 and σres = 0.2152, respectively. These results show that these are very +strong relationships between the models and the data points. +Figs. 7 and 8 shows the evolution of the Hubble parameter across the Redshift for two cosmological +7 + +Qm=0.31341±0.05310 +0.04780 +h=0.69551±0.00471 +0.00472 +0.71 +n +0.70 +0.68 +Qr=0.00050+0.00034 +-0.00034 +: +0.0008 +0.0006 +--- +0.0002 +0.08878 +- +0.24 +0.18 +0.06 +.10969 +0.06 +0.12 +-0.18 +0.24 +0.48 +0.68 +.40 +10 +O. +O. +O +O. +0. +2m +h +Am +AFIG. 3. +The diffusive model’s Eq. +15 for best- +fitting free parameters for the Supernovae Type +1A data with cosmological parameter values as +h = 0.6966+0.0047 +−0.0047, Ωm0 = 0.2678+0.0248 +−0.0237, Ωr0 = +0.0005+0.0003 +−0.0003, +∆m += +0.0025+0.0169 +−0.0173 and ∆Λ += +−0.0025+0.0172 +−0.0169 +of the MCMC simulation result +shown in Fig. 1. +FIG. 4. +The diffusive model’s Eq. +15 for best- +fitting free parameters for the Supernovae Type +1A data with cosmological parameter values as +h = 0.6955+0.0047 +−0.0047, Ωm0 = 0.3134+0.0531 +−0.0478, Ωr0 = +0.0005+0.0003 +−0.0003, +∆m += +0.1246+0.1098 +−0.0887 and ∆Λ += +−0.1244+0.0877 +−0.1096 +of the MCMC simulation result +shown in Fig. 2. +FIG. 5. This is the residuals distance in Mpc between +the predicted model values and the data points for +the diffusive model’s Eq. 15 for best-fitting free pa- +rameters shown in Fig. 3. +FIG. 6. This is the residuals distance in Mpc between +the predicted model values and the data points for +the diffusive model’s Eq. 15 for best-fitting free pa- +rameters shown in Fig. 4. +8 + +Theoreticalpredictionsforthediffusivemode +Model:Qm=0.2678,h=0.6966,Qr=0.00050, +46- +△m= 0.00252,A^=-0.00251 +(odw) w +44 +modulus: m- +42 +40 +38 +Distance +36 +34 +10~2 +10~1 +100 +Redshift: zTheoreticalpredictionsforthediffusivemode +Model:Qm=0.3134,h=0.6955,Q,=0.00050, +46- +△m= 0.12469, A^ = -0.12440 +(odw) w +44 +modulus: m- +42 +40 +38 +Distance +36 +34 +10~2 +10~1 +100 +Redshift: zResiduasforthediffusivemoderesultsonthedata +Ave = -0.0374 and o = 0.21479 +2 +10~2 +10-1 +100 +Redshift: zResiduasforthediffusivemoderesultsonthedata +Ave = -0.0374 and o = 0.21525 +0 +2 +.3 +10~2 +10~1 +100 +Redshift: zFIG. 7. The Hubble parameter vs Redshift for the +Model displayed in Fig. 1. The blue curve repre- +sent the result obtained by considering diffusive fluid +and employing MCMC simulation, with 1-σ devia- +tion result displayed in yellowish shaded region. The +red curve represent ΛCDM cosmology result using +MCMC simulation where as the green curve repre- +sent one obtained directly by using the Planck 2018 +data for the purpose of comparison. +FIG. 8. The Hubble parameter vs Redshift for the +Model displayed in Fig. 2. The blue curve repre- +sent the result obtained by considering diffusive fluid +and employing MCMC simulation, with 1-σ devia- +tion result displayed in yellowish shaded region. The +red curve represent ΛCDM cosmology result using +MCMC simulation where as the green curve repre- +sent one obtained directly by using the Planck 2018 +data for the purpose of comparison. +model cases discussed above. The blue curves represent the results obtained by considering diffusive +fluid and employing MCMC simulations, with 1-σ deviation results displayed in yellowish shaded +regions. The red curves represent ΛCDM cosmology result using the average values obtained from +the MCMC simulations where as the green curves represent those which are obtained directly by +using the Planck 2018 data. In Fig. 7 a complete overlap is observed between the two curves which +are obtained by using the MCMC simulation data in the ΛCDM model and the average value of +MCMC simulation data in diffusive model. In contrast in the results of Fig. 8 we begin to notice +a deviation between the two curves from ∼ z of 0.75 onward. Even though it is not expected to +have a complete overlap between the ΛCDM model using Planck 2018 values put directly in the +cosmological equations (green curves) and the MCMC results (blue curves and yellowish shaded +regions), the difference between them is observed to be more prominent in the case of Fig. 8 than +that of Fig. 7. +Fig. 9 and 10 shows the evolution of deceleration parameter across Redshift for the two diffu- +9 + +DiffusiveModel:Q2m=0.2678,h=0.6966,Q2r=0.00050.Am=0.00252A>=-0.00251 +200 +AcDM model using Planck 2o18 values +ACDM model based on MCMC data +180 +Diffusivemodelbased onMCMC data +1-g range forthe diffusive model +160 +140 +120 +100 +80 +60 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +Cosmological Redshift (z)DiffusiveModel:Q2m=0.3134.h=0.6955,Qr=0.00050.△m=0.12469,A^=-0.12440 +200 +AcDM model using Planck 2o18 values +ACDM model based on MCMC data +180 +Diffusivemodelbased onMCMC data +1-g range forthe diffusive model +160 +140 +120 +100 +80 +60 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +Cosmological Redshift (z)FIG. 9. The graph of deceleration parameter vs red- +shift for the diffusive model shown in Fig. 7. +FIG. 10. +The graph of deceleration parameter vs +redshift for the diffusive model shown in Fig. 8. +sive cosmological model cases discussed above. The blue curves represent the results obtained by +considering diffusive fluid and employing MCMC simulations, with 1-σ deviation results displayed in +yellowish shaded regions. The red curves represent ΛCDM cosmological results by using the MCMC +simulation data, where as the green curves represent those which are obtained directly by substitut- +ing the Planck 2018 data values. In Fig. 9 we observe a complete overlap between the two curves +which are obtained by using MCMC simulation data in the ΛCDM equation and the average value +of MCMC simulation data of the diffusive model, which is also observed in the Hubble parameter +vs Redshift plot of Fig. 7. Moreover, the 1-σ deviation result which is indicated in the yellowish +colored shaded region is observed to encompass all the curves for about (z ∼ 0.5) of the deceleration +parameter values given in Fig. 10. The diffusive model in this case has slightly larger values of +deceleration parameter in the present universe (z ∼ 0) compared to what is observed in the case of +Fig. 9. +The above two cases (Case I and Case II) were obtained with a positive ∆m and negative ∆Λ. +In what follows, let us provide the results corresponding to the cases of negative ∆m which can be +explained as being the situation when energy flows from dark matter sector to that of dark energy. +As shown in Fig. 11 we run MCMC simulation for the diffusive model, by combining Eqs. 14 and +15, we find on average the best-fitting parameter value for each free parameter to be h = 0.6967 for +the Hubble uncertainty parameter, Ωm0 = 0.2655 for the matter density parameter, Ωr0 = 0.00050 +for the radiation density parameter, along with a newly introduced parameters ∆m = −0.00251 and +∆Λ = 0.00246. We will call this diffusive model case hereafter Case III. +10 + +DiffusiveModel:Q2m=0.3134.h=0.6955,Qr=0.00050.△m=0.12469,A^=-0.12440 +0.4 +0.2 +Deccelartion Parameter q(z) +0.0 +-0.2 +-0.4 +ACDM model using Planck 2o18 values +-0.6 +ACDM modelbased on MCMC data +Diffusivemodelbased onMCMC data +l-o range forthe diffusive model +-0.8 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +Cosmological Redshift (z)DiffusiveModel:Q2m=0.2678.h=0.6966.Qr=0.00050,Am=0.00252.A^=-0.00251 +0.4 +0.2 +Deccelartion Parameter q(z) +0.0 +-0.2 +-0.4 +ACDM model using Planck 2o18 values +-0.6 +ACDM modelbased on MCMC data +Diffusivemodelbased onMCMC data +l-o range forthe diffusive model +-0.8 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +Cosmological Redshift (z)FIG. 11. The MCMC simulation results for the diffusive model’s which is given in Eq. 15 for cosmological +free parameters (Ωm0, ¯h, Ωr0, ∆m and ∆Λ), with the “true” values for Ωm0 = 0.315, ¯h = 0.674, and +Ωr0 = 2.47 × 10−5 /¯h2 provided by the Planck2018 collaboration data release [33]. We used 100 random +walkers and 10000 iterations. +In the following we will also provide one interesting case in which the best-fitting parameter value +for each free parameter to be h = 0.6976 for the Hubble uncertainty parameter, Ωm0 = 0.2283 for the +matter density parameter, and Ωr0 = 0.00050 for the radiation density parameter, along with a newly +introduced parameters ∆m = −0.10747 and ∆Λ = 0.10426. We will refer to this diffusive model case +as Case IV. +In Fig. 13 and Fig. 14 the above discussed two diffusive cosmological model cases were given +which clearly shows to fit the extremely well with the data. Even the corresponding 1σ-deviation do +11 + +Qm=0.26554±0.02498 +0.02404 +h=0.69676±0.00474 +0.00477 +0.71 +n +0.70 +0.68 +Qr=0.00050+0.00034 +0.00034 +0.0008 +0.0006 +0.0004 +0.0002 +-0.001/0 +T00'0- +0.004 +A=0.00246±0.00172 +0.00167 +0.003 +.69 +10 +15 +.30 +O. +O. +0.68 +O. +O. +O. +2m +h +AmFIG. 12. The MCMC simulation results for the diffusive model’s which is given in Eq. 15 for cosmological +free parameters (Ωm0, ¯h, Ωr0, ∆m and ∆Λ), with the “true” values for Ωm0 = 0.315, ¯h = 0.674, and +Ωr0 = 2.47 × 10−5 /¯h2 provided by the Planck2018 collaboration data release [33]. We used 100 random +walkers and 10000 iterations. +not really have an impact on the full range predicted by them. +Additionally, Fig. 15 and Fig. 16 displays the residuals obtained in the above two cases. It can +clear be seen that at no point does the models over-or under-estimate the resulting distance modulus +for each supernovae. We also note that the average off-set the model has, compared to the data, +is ¯xres = −0.0374 Mpc and ¯xres = −0.0386 Mpc, with a standard deviation of σres = 0.2147 and +σres = 0.2145, respectively. These results show that these are very strong relationships between the +models and data points. +12 + +Qm = 0.22830±0.03874 +0.03742 +h=0.69767±0.00477 +0.00475 +0.71 +n +0.70 +0.68 +Qr=0.00050+0.00034 +-0.00034 +0.0008 +0.0006 +0.0004 +0.0002 +0.10747 +±0.07165 +Mu +-0.04 +-0.08 +0.12 +A=0.10426±0.06691 +0.06971 +0.16 +m +0.12 +00 +36 +10 +6 +O. +.0 +O +O. +O +2m +h +Am +A^FIG. 13. +The diffusive model’s Eq. +15 for best- +fitting free parameters for the Supernovae Type +1A data with cosmological parameter values as +h = 0.6967+0.0047 +−0.0047, Ωm0 = 0.2655+0.0248 +−0.0237, Ωr0 = +0.0005+0.0003 +−0.0003, ∆m += +−0.0025+0.0169 +−0.0170 and ∆Λ += +0.0024+0.0172 +−0.0167 of the MCMC simulation result shown +in Fig. 11. +FIG. 14. +The diffusive model’s Eq. +15 for best- +fitting free parameters for the Supernovae Type +1A data with cosmological parameter values as +h = 0.6976+0.0047 +−0.0047, Ωm0 = 0.2283+0.0387 +−0.0374, Ωr0 = +0.0005+0.0003 +−0.0003, ∆m += +−0.1074+0.0716 +−0.0641 and ∆Λ = +0.1042+0.0660 +−0.0697 of the MCMC simulation result shown +in Fig. 12. +FIG. 15. This is the residuals distance in Mpc be- +tween the predicted model values and the data points +for the diffusive model’s Eq. 15 for best-fitting free +parameters shown in Fig. 13. +FIG. 16. This is the residuals distance in Mpc be- +tween the predicted model values and the data points +for the diffusive model’s Eq. 15 for best-fitting free +parameters shown in Fig. 14. +13 + +Theoreticalpredictionsforthediffusivemode +Model:Qm=0.2655,h=0.6968,Q,=0.00050, +46- +△m=-0.00251,^= 0.00246 +(odw) w +44 +modulus: m- +42 +40 +38 +Distance +36 +34 +10~2 +10~1 +100 +Redshift: zTheoreticalpredictionsforthediffusivemode +Model:Qm=0.2283,h=0.6977,Qr=0.00050 +46- +△m=-0.10747, ^= 0.10426 +(odw) w +44 +modulus: m- +42 +40 +38 +Distance +36 +34 +10~2 +10~1 +100 +Redshift: zResiduasforthediffusivemoderesultsonthedata +Ave = -0.0374 and o = 0.21475 +2 +10~2 +10~1 +100 +Redshift: zResiduasforthediffusivemoderesultsonthedata +Ave = -0.0386 and o = 0.21456 +2 +10~2 +10~1 +100 +Redshift: zFIG. 17. The Hubble parameter vs Redshift for the +Model displayed in Fig. 1. The blue curve repre- +sent the result obtained by considering diffusive fluid +and employing MCMC simulation, with 1-σ devia- +tion result displayed in yellowish shaded region. The +red curve represent ΛCDM cosmology result using +MCMC simulation where as the green curve repre- +sent one obtained directly by using the Planck 2018 +data for the purpose of comparison. +FIG. 18. The Hubble parameter vs Redshift for the +Model displayed in Fig. 2. The blue curve repre- +sent the result obtained by considering diffusive fluid +and employing MCMC simulation, with 1-σ devia- +tion result displayed in yellowish shaded region. The +red curve represent ΛCDM cosmology result using +MCMC simulation where as the green curve repre- +sent one obtained directly by using the Planck 2018 +data for the purpose of comparison. +Fig. 17 and 18 shows the evolution of the Hubble parameter across the Redshift for the two cos- +mological model cases discussed above. The blue curves represent the result obtained by considering +diffusive fluid and employing MCMC simulation, with 1-σ deviation results displayed in yellowish +shaded regions. The red curves represent ΛCDM cosmology result using MCMC simulation where +as the green curves represent those obtained directly by using the Planck 2018 data. An overlap is +observed in the result displayed in Fig. 17 between the two curves which are obtained by using the +MCMC simulation data in the ΛCDM model and that of the average value of MCMC simulation +data of the diffusive model. In contrast, a noticeable deviation emerges to be observed between the +the Hubble parameter values of the diffusive model result (the blue curve) and that of the ΛCDM +result based on MCMC simulation data (the red curve) in Fig. 18 from ∼ z of 0.75 onward. Even +though it is not expected to have a complete overlap between the ΛCDM model using Planck 2018 +values put directly in the cosmological equations (green curves) and the MCMC results (blue curves +and yellowish shaded regions), the difference between them is observed to be more prominent in the +14 + +DiffusiveModel:Q2m=0.2655.h=0.6968,Qr=0.00050.Am=-0.00251,A^=0.00246 +200 +AcDM model using Planck 2o18 values +ACDMmodelbasedonMCMC data +180 +Diffusivemodelbased onMCMC data +1-g range forthe diffusive model +160 +140 +120 +100 +80 +60 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +Cosmological Redshift (z)DiffusiveModel:Q2m=0.2283.h=0.6977.Qr=0.00050.△m=-0.10747.A^=0.10426 +200 +AcDM model using Planck 2o18 values +ACDM model based on MCMC data +180 +Diffusivemodelbased onMCMC data +1-g range forthe diffusive model +160 +140 +120 +100 +80 +60 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +Cosmological Redshift (z)FIG. 19. +The graph of deceleration parameter vs +redshift for the diffusive model shown in Fig. 17. +FIG. 20. +The graph of deceleration parameter vs +redshift for the diffusive model shown in Fig. 18. +case of Fig. 17 than that of Fig. 18. +Fig. 19 and 20 shows the evolution of the deceleration parameter across Redshift for the two cos- +mological model cases discussed above. The blue curves represent the result obtained by considering +diffusive fluid and employing MCMC simulation, with 1-σ deviation results displayed in yellowish +shaded regions. The red curves represent ΛCDM cosmology result using MCMC simulation where +as the green curves represent those obtained directly by using the Planck 2018 data. In Fig. 19 +we observe a complete overlap while using MCMC simulation data in the ΛCDM model and the +average value of MCMC simulation data of the diffusive model, which is also observed in the Hubble +parameter vs Redshift plot of Fig. 17. +As redshift increases, the Hubble parameter values of the diffusive model result (the blue curve) +and that of the ΛCDM result based on MCMC simulation data (the red curve), begin to acquire +similar values as can be seen in Fig. 20. The 1-σ deviation results indicated in yellow colored shaded +region is observed to encompass both diffusive (blue curve) and non-diffusive (red curve) cases of +the deceleration parameter values given in Fig. 20. In the current universe, the diffusive model has +slightly lower values of deceleration parameter compared to what is observed in the case of Fig. 19. +In Table I we provide some statistical result which allows us to determine the best diffusive model +case in comparisons to the ΛCDM model. The statistical analysis test that we have used is the Akaike +information criterion (AIC) and Bayesian/Schwarz information criterion (BIC) selections which were +used in a similar work in [32]. These information criteria evaluate the plausibility of an alternative +model explaining the data compared to an “accepted/true model”. In our case the ΛCDM model will +15 + +DiffusiveModel:Q2m=0.2655h=0.6968,Q2r=0.00050.Am=-0.00251,A^=0.00246 +0.4 +0.2 +Deccelartion Parameter q(z) +0.0 +-0.2 +-0.4 +ACDM model using Planck 2o18 values +-0.6 +ACDM modelbased on MCMC data +Diffusivemodelbased onMCMC data +l-o range forthe diffusive model +-0.8 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +Cosmological Redshift (z)DiffusiveModel:Q2m=0.2283.h=0.6977.Qr=0.00050.△m=-0.10747.A^=0.10426 +0.4 +0.2 +Deccelartion Parameter q(z) +0.0 +-0.2 +-0.4 +ACDM model using Planck 2o18 values +-0.6 +ACDM modelbased on MCMC data +Diffusivemodelbased onMCMC data +l-o range forthe diffusive model +-0.8 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +Cosmological Redshift (z)TABLE I. The best-fit for each tested model, including the ΛCDM model. The models are listed in the order +from the largest likelihood function value L(ˆθ|data) to the smallest likelihood of being viable. The reduced +χ2 -values are given as an indication of the goodness of fit for a particular model. The AIC and BIC values +are shown, as well as the ∆AIC and ∆BIC for each information criterion. The ΛCDM model is chosen as +the ”true model”. +Models +∆m ∆Λ L(ˆθ|data) +χ2 +Red.χ2 +AIC +|∆AIC| +BIC +|∆BIC| +Diffusive Case II +ve -ve -121.1677 242.3355 0.6845 252.3355 4.9405 271.7521 12.7072 +Diffusive Case I +ve -ve -120.7059 241.4118 0.6819 251.4118 4.0168 270.8285 11.7835 +ΛCDM +0 +0 +-120.6975 241.3950 0.6780 247.3950 +0 +259.0449 +0 +Diffusive Case III -ve +ve -120.6890 241.3781 0.6818 251.3781 3.9831 270.7947 11.7497 +Diffusive Case IV -ve +ve -120.3936 240.7872 0.6801 250.7872 3.3922 270.2039 11.1589 +be considered as the “true model”. Following the suggestion made in [32] as the calculated values for +the AIC and BIC can by very random, we will also use the difference in AIC (i.e., ∆AIC) and BIC +(i.e., ∆BIC) values of each model compared to the “true model’s AIC and BIC values, and we use the +Jeffrey’s scale in order to make conclusions about the viability of the various Diffusive model cases. +Moreover, the reduced χ2 -values are used as an indication of the goodness of fit for each model on the +supernovae data. It is observed that, the first two Diffusive model cases (shown in Fig. 1 and 2) have +obtained better likelihood function values than the ΛCDM model based on a Gaussian probability +distribution, with Case II obtaining the larger likelihood function value. However, in the reduced χ2 +-values in which the number of parameters are taken into account when determining the goodness of +fit, the ΛCDM model has the best value with the Diffusive model Case I (shown in Fig. 1) managing +to have a closer value to this accuracy. In order to find the better fitting model among these two +cases, we use AIC test, according to which the Diffusive model Cases I and II have obtained more +observational and less observational support, respectively. Case I is seen to have a value just missing +out on the substantial observational support category, but is still with a closer value to the boundary +for less observational support. Therefore, it can be concluded that Case I has some observational +support according to the AIC criterion, while Case II has less observational support. In terms of the +BIC criteria, we did not obtain one model to have some observational support category, but Case I +16 + +was the closer of being in one of the categories. Therefore, statistically, based on the likelihood, the +goodness of fit, the AIC and BIC criteria, Case I is the most likely to be an alternative model to the +ΛCDM model, with Case II not being ruled out, but will have to be tested on other datasets before +being accepted or rejected. +17 + +IV. +CONCLUSIONS +In this manuscript we considered diffusive cosmological models where dark matter and dark energy +interact by exchanging energy. The background cosmological parameters in particular the thermody- +namics parameters have been studied and compared against supernova cosmological data for different +Diffusive model cases using MCMC simulation results presented in the previous section. +For the two new parameters which arise in our Diffusive cosmological model, namely ∆m and +∆Λ, we have examined the Hubble and deceleration parameters results of Figs. 7 to 10 and that of +Figs. 17 to 20. Recalling the requirement that the sum of these two parameters need to be zero, +the magnitude of ∆m and ∆Λ of ≈ 0.0025 fit the parameter space very well. Following which we +investigated this deeply based on the statistical analysis made in the above section which is given +in Table I. From our analysis we observed that cases having positive values of ∆m were showing the +largest values of likelihood function. Based on the analysis of likelihood, goodness of fit, AIC and +BIC criteria, one can conclude that overall Case I is the most likely to be an alternative to the ΛCDM +model. +As we have highlighted in the discussion part our current work is to provide a viability test of the +different cases considered, but to reject or accept any of them more data and rigorous testing method +is needed. Moreover, our initial result such as the one shown in Figs. 7 and 17 suggest that one can +look for a potential explanation of the Hubble Tension in such models. +ACKNOWLEDGEMENTS +AA acknowledges that this work is based on the research supported in part by the National Re- +search Foundation (NRF) of South Africa (grant number 112131). This work was part of the research +programme “New Insights into Astrophysics and Cosmology with Theoretical Models confronting +Observational Data” of the National Institute for Theoretical and Computational Sciences of South +Africa. +[1] Adam G Riess, Alexei V Filippenko, Peter Challis, et al. Observational evidence from supernovae for +an accelerating universe and a cosmological constant. The Astronomical Journal, 116(3):1009, 1998. +[2] Saul Perlmutter et al. Supernovae, dark energy, and the accelerating universe. Physics today, 56(4):53– +62, 2003. +18 + +[3] Subhayan Maity, Pritikana Bhandari, and Subenoy Chakraborty. 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Planck 2018 results-vi. cosmological parameters. +Astronomy & Astrophysics, 641:A6, 2020. +20 + diff --git a/KNE1T4oBgHgl3EQfGQPs/content/tmp_files/load_file.txt b/KNE1T4oBgHgl3EQfGQPs/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b4bf13a01bef176ef986a230d57214b8012769a7 --- /dev/null +++ b/KNE1T4oBgHgl3EQfGQPs/content/tmp_files/load_file.txt @@ -0,0 +1,857 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf,len=856 +page_content='Observational constraints of diffusive dark-fluid cosmology Remudin Reshid Mekuria1a and Amare Abebe2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='3b 1 Faculty of Engineering and Computer Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' Ala-too International University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' Bishkek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' Kyrgyzstan 2 Centre for Space Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' North-West University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' Potchefstroom,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' South Africa and 3 National Institute for Theoretical and Computational Sciences (NITheCS),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' South Africa (Dated: January 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 2023) Abstract In this work,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' we consider an interacting dark-fluid cosmological model in which energy exchange between dark matter and dark energy occurs through diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' After solving the background expansion history for a late-time universe, we attempt to constrain the cosmological parameters by comparing simulated values of the model against Supernovae Type 1A data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' We consider four different cases and compare them against the ΛCDM model as the ”true model”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' Our results show that the diffusive model in which dark energy flows to dark matter is the most likely alternative to ΛCDM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' This model is not only in line with Planck 2018 observational results but can also give a potential explanation to the so-called Hubble tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' PACS numbers: 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='Kd, 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='Jk, 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='-k, 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='+x, 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='Cq a Remudin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='Mekuria@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='com b Amare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='Abebe@nithecs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='za 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='02913v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='CO] 7 Jan 2023 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' INTRODUCTION A lot has already been reported about the discrepancy between observational findings [1–6] and theo- retical predictions of the expansion history of the universe in standard cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The missing matter and energy in the universe, dubbed dark matter (DM) and dark energy (DE), respectively, account for a whopping 95% of the total content of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The nature of these dark components of the universe is not properly understood, but there are several candidates in the literature, including unified dark-fluid models, proposed to describe them and their effect on astrophysics and cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' On the DM side, most commonly studied candidates include Weakly Interacting Massive Particles (WIMPS) [7–10] or some astrophysical modification of gravity such as the Modified Newtonian Dy- namics (MOND)[11] among many others, whereas on the DE side, the cosmological constant Λ [12] is perhaps the simplest addition to the standard cosmological model needed to explain most of the observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' There are some serious issues associated with the cosmological constant, however, such as the eponymous cosmological constant problem [13] and the coincidence problem [14, 15] which make the choice less attractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' That is why there are currently a plethora of other alternatives to explain current cosmological observations, such as modifications to the gravitational theory it- self (see, for example, [16–19]), an evolving Λ [20–22], deviations from the standard homogeneous (see [23, 24] and references therein) and isotropic universe (such as the various Bianchi cosmological models) assumption, or some form of combination of these, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' Another aspect to consider, and one gaining much traction recently, is the interaction of dark matter and dark energy [3, 25–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' Such an approach is interesting because it has the potential to explain the cosmological and coincidence problems, the Hubble tension and/or the σ8 discrepancy [29–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' Our current work pursues the last aspect, and studies the cosmological viability of a model [3] of the dark-fluid interaction using Supernovae Type 1A data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' We organise the rest of the manuscript as follows: in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' II we give a covariant thermodynamics description of, and derive the field equa- tions for, the background universe involving the diffusive dark-fluid system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' III we give an observational-constraint analysis using MCMC simulations of Supernovae Type 1A and Planck 2018 data and give some predictions on the values of the defining parameters of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' Finally in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' IV we discuss the results and give conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 2 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' BACKGROUND THERMODYNAMICS The standard ΛCDM cosmology is a solution of the Einstein field equations (EFEs) derived from the action (From here onwards, we will work with units in which the speed of light c = 1): S = c4 16πG � d4x√−g [R + 2 (Lm − Λ)] , (1) where R, Lm and Λ are the Ricci scalar, the matter Lagrangian density and the cosmological constant, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The corresponding EFEs read: Gµν + Λgµν = 8πGTµν , (2) with the first (geometric) term represented by the Einstein tensor, and the RHS of the equation representing the total energy-momentum tensor (EMT) of matter fluid forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' Both Gµν and Tµν are covariantly conserved quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The EMT for perfect-fluid models is given by Tµν = (ρ + p)uµuν + pgµν , (3) where ρ and p are the energy density and isotropic pressure of matter, respectively, often related by the barotropic equation of state (EoS) p = wρ for a constant EoS parameter w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The normalised vector uα represents the four-velocity of fundamental observers comoving with the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The divergence-free EMT T µν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='µ = 0 leads to the fluid conservation equation ˙ρ + 3 ˙a a(1 + w)ρ = 0 , (4) where a(t) is the cosmological scale factor whose evolution is given by the Friedmann equation ˙a2 a2 = 8πG 3 ρ + Λ 3 − k a2 (5) where k is the normalised spatial curvature parameter with values −1 , 0 , 1 depending on an open, flat or closed spatial geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' In a multi-component fluid system, it is usually assumed that the energy density of each perfect-fluid component is assumed to evolve independently of the other fluids of the system: ˙ρi + 3 ˙a a(1 + w)ρi = 0 , (6) and in this case the EMT in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' (3) is the algebraic sum of the EMTS of each fluid, so are the total energy density and total pressure terms of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' (5) the algebraic sums of the individual components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 3 However, if we relax this assumption due to the presence of diffusion between the constituent com- ponents of the fluid, the individual components do not obey the matter conservation equation, but the total fluid still does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' For the the ith component fluid, the new conservation equation reads: T µν i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='µ = N ν i , (7) where N ν i corresponds to the current of diffusion term for that fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' One can then write the non- conservation equation for the fluid as: ˙ρi + 3 ˙a a(1 + w)ρi = γi a3 , (8) where γi is a constant for that fluid such that � i γi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' Integrating this equation gives ρi = a−3(1+wi) � ρi0 + γi � t t0 a3widt′ � , (9) with ρi0 representing the present-day (t = t0) value of the energy density of the ith fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' Using a late-time t − t0 ≪ t0 expansion and expressing a(t) = a0 [1 − (t0 − t)H0) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' ], we can write the last term of the above integrand as � t t0 a3widt= � t t0 a3wi [1 − (t0 − t)H0) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' ]3wi dt′ = − 1 1 + 3wi � (1 + (t0 − t)H0)1+3wi − (1 + (t0 − t)H0)1+3wi + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' � ≈ 1 1 + 3wi � 1 − (1 + (t0 − t)H0)1+3wi� = 1 (1 + 3wi)H0 � 1 − (2 − a)1+3wi� , (10) where in the last step, we have used normalised the scale factor to unity today: a0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' Thus the energy density of each diffusive fluid component is given according to the below relation: ρi = a−3(1+wi) � ρi0 + γi (1 + 3wi)H0 � 1 − (2 − a)1+3wi�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' (11) Assuming the well-known component of radiation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' dust-like matter (baryons and dark matter) and vacuum energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' the above diffusive solution leads to: ρr = a−4 � ρr0 + γr 2H0 � 1 − (2 − a)2�� ρm = a−3 � ρm0 + γm H0 [1 − (2 − a)] � ρΛ = ρΛ0 − γΛ 2H0 � 1 − (2 − a)−2� (12) 4 Let us now consider the Friedmann equation for the ΛCDM model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' assuming k = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' which can be given as: ˙a2 a2 = 8πG 3 � ρr0a−4 + ρm0 + γm H0 [1 − (2 − a)] a−3 + ρΛ0 − γΛ 2H0 � 1 − (2 − a)−2�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' (13) We are going to assume the diffusive interaction is limited between dark matter and dark energy for this work, and hence γr = 0 in the above equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' Let us now introduce the following dimensionless quantities: Ωi ≡ 8πG 3H2 0 ρi , ∆m ≡ 8πG 3H3 0 γm , ∆Λ ≡ 8πG 3H3 0 γΛ , 1 + z ≡ a−1 , h ≡ H H0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' (14) We can then show that the Friedmann equation can be recast as h2 = Ωr0(1 + z)4 + Ωm0(1 + z)3 + ΩΛ0 − ∆mz(1 + z)2 − ∆Λ � 1 2 − 1 2 �1 + 2z 1 + z �−2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' (15) Moreover, the deceleration parameter can be shown to be q≡ −¨aa ˙a2 = 4πG 3H2 � i ρi(1 + 3wi) = 1 2 � � � 2Ωr0(1 + z)4 + Ωm0(1 + z)3 − 2ΩΛ0 − ∆mz(1 + z)2 + ∆Λ � 1 − � 1+2z 1+z �−2� Ωr0(1 + z)4 + Ωm0(1 + z)3 + ΩΛ0 − ∆mz(1 + z)2 − ∆Λ � 1 2 − 1 2 � 1+2z 1+z �−2� � � � (16) These equations reduce to their respective ΛCDM limits when ∆m and ∆Λ both vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' OBSERVATIONAL CONSTRAINTS In the following we will provide the result for observational constraints for the diffused dark fluid models we have introduced in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' We have used the distance modulus equation which can be obtained by combining the different cosmological distance definitions to fit against the supernovae data in our MCMC simulation, which is presented in the work of [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 1 we run MCMC simulation for the diffusive model, by combining Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 14 and 15, we find on average the best-fitting parameter value for each free parameter to be h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='6966 for the Hubble uncertainty parameter, Ωm0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='2678 for the matter density parameter, and Ωr0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00050 for the radiation density parameter, along with a newly introduced parameters ∆m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00252 and ∆Λ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00251.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' We shall henceforth refer to this diffusive model case as Case I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' Among some of our optimum results we have also, as Case II, obtained the situation where Ωm0 result is much closer to the observational result of Planck 2018 (= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='315+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='555 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='111) as shown below which 5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The MCMC simulation results for the diffusive model’s which is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 15 for cosmological free parameters (Ωm0, ¯h, Ωr0, ∆m and ∆Λ), with the “true” values for Ωm0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='315, ¯h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='674, and Ωr0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='47 × 10−5 /¯h2 provided by the Planck2018 collaboration data release [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' We used 100 random walkers and 10000 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' are also obtained with MCMC simulation for the diffusive model, by combining Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 14 and 15, we find on average the best-fitting parameter value for each free parameter to be h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='6955 for the Hubble uncertainty parameter, Ωm0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='3134 for the matter density parameter, and Ωr0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00050 for the radiation density parameter, along with a newly introduced parameters ∆m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='1246 and ∆Λ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='1244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 3 and 4 the above discussed two diffusive cosmological cases were given which clearly showsto fit the extremely well with the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' Even the corresponding 1σ-deviation do not really have 6 Qm=0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 2m h Am A^FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The MCMC simulation results for the diffusive model’s which is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 15 for cosmological free parameters (Ωm0, ¯h, Ωr0, ∆m and ∆Λ), with the “true” values for Ωm0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='315, ¯h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='674, and Ωr0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='47 × 10−5 /¯h2 provided by the Planck2018 collaboration data release [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' We used 100 random walkers and 10000 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' an impact on the full range predicted by them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' Additionally, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 5 and 6 display the residuals obtained in the above two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' It can clearly be seen that at no point do the models over-or under-estimate the resulting distance modulus for each supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' We also note that the average off-set the model has, compared to the data, is ¯xres = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0374 Mpc in both cases with a standard deviation of σres = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='2148 and σres = 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='68 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='40 10 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' O O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 2m h Am AFIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The diffusive model’s Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 15 for best- fitting free parameters for the Supernovae Type 1A data with cosmological parameter values as h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='6966+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0047 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0047, Ωm0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='2678+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0248 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0237, Ωr0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0005+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0003 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0003, ∆m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0025+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0169 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0173 and ∆Λ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0025+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0172 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0169 of the MCMC simulation result shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The diffusive model’s Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 15 for best- fitting free parameters for the Supernovae Type 1A data with cosmological parameter values as h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='6955+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0047 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0047, Ωm0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='3134+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0531 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0478, Ωr0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0005+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0003 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0003, ∆m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='1246+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='1098 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0887 and ∆Λ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='1244+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0877 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='1096 of the MCMC simulation result shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' This is the residuals distance in Mpc between the predicted model values and the data points for the diffusive model’s Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 15 for best-fitting free pa- rameters shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' This is the residuals distance in Mpc between the predicted model values and the data points for the diffusive model’s Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 15 for best-fitting free pa- rameters shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 8 Theoreticalpredictionsforthediffusivemode Model:Qm=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='2678,h=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='6966,Qr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00050, 46- △m= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00252,A^=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00251 (odw) w 44 modulus: m- 42 40 38 Distance 36 34 10~2 10~1 100 Redshift: zTheoreticalpredictionsforthediffusivemode Model:Qm=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='3134,h=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='6955,Q,=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00050, 46- △m= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='12469, A^ = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='12440 (odw) w 44 modulus: m- 42 40 38 Distance 36 34 10~2 10~1 100 Redshift: zResiduasforthediffusivemoderesultsonthedata Ave = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0374 and o = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='21479 2 10~2 10-1 100 Redshift: zResiduasforthediffusivemoderesultsonthedata Ave = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0374 and o = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='21525 0 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='3 10~2 10~1 100 Redshift: zFIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The Hubble parameter vs Redshift for the Model displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The blue curve repre- sent the result obtained by considering diffusive fluid and employing MCMC simulation, with 1-σ devia- tion result displayed in yellowish shaded region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The red curve represent ΛCDM cosmology result using MCMC simulation where as the green curve repre- sent one obtained directly by using the Planck 2018 data for the purpose of comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The Hubble parameter vs Redshift for the Model displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The blue curve repre- sent the result obtained by considering diffusive fluid and employing MCMC simulation, with 1-σ devia- tion result displayed in yellowish shaded region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The red curve represent ΛCDM cosmology result using MCMC simulation where as the green curve repre- sent one obtained directly by using the Planck 2018 data for the purpose of comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' model cases discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The blue curves represent the results obtained by considering diffusive fluid and employing MCMC simulations, with 1-σ deviation results displayed in yellowish shaded regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The red curves represent ΛCDM cosmology result using the average values obtained from the MCMC simulations where as the green curves represent those which are obtained directly by using the Planck 2018 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 7 a complete overlap is observed between the two curves which are obtained by using the MCMC simulation data in the ΛCDM model and the average value of MCMC simulation data in diffusive model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' In contrast in the results of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 8 we begin to notice a deviation between the two curves from ∼ z of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='75 onward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' Even though it is not expected to have a complete overlap between the ΛCDM model using Planck 2018 values put directly in the cosmological equations (green curves) and the MCMC results (blue curves and yellowish shaded regions), the difference between them is observed to be more prominent in the case of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 8 than that of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 9 and 10 shows the evolution of deceleration parameter across Redshift for the two diffu- 9 DiffusiveModel:Q2m=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='2678,h=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='6966,Q2r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='Am=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00252A>=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00251 200 AcDM model using Planck 2o18 values ACDM model based on MCMC data 180 Diffusivemodelbased onMCMC data 1-g range forthe diffusive model 160 140 120 100 80 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00 Cosmological Redshift (z)DiffusiveModel:Q2m=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='3134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='h=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='6955,Qr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='△m=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='12469,A^=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='12440 200 AcDM model using Planck 2o18 values ACDM model based on MCMC data 180 Diffusivemodelbased onMCMC data 1-g range forthe diffusive model 160 140 120 100 80 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00 Cosmological Redshift (z)FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The graph of deceleration parameter vs red- shift for the diffusive model shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The graph of deceleration parameter vs redshift for the diffusive model shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' sive cosmological model cases discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The blue curves represent the results obtained by considering diffusive fluid and employing MCMC simulations, with 1-σ deviation results displayed in yellowish shaded regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The red curves represent ΛCDM cosmological results by using the MCMC simulation data, where as the green curves represent those which are obtained directly by substitut- ing the Planck 2018 data values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 9 we observe a complete overlap between the two curves which are obtained by using MCMC simulation data in the ΛCDM equation and the average value of MCMC simulation data of the diffusive model, which is also observed in the Hubble parameter vs Redshift plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' Moreover, the 1-σ deviation result which is indicated in the yellowish colored shaded region is observed to encompass all the curves for about (z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='5) of the deceleration parameter values given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The diffusive model in this case has slightly larger values of deceleration parameter in the present universe (z ∼ 0) compared to what is observed in the case of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The above two cases (Case I and Case II) were obtained with a positive ∆m and negative ∆Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' In what follows, let us provide the results corresponding to the cases of negative ∆m which can be explained as being the situation when energy flows from dark matter sector to that of dark energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 11 we run MCMC simulation for the diffusive model, by combining Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 14 and 15, we find on average the best-fitting parameter value for each free parameter to be h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='6967 for the Hubble uncertainty parameter, Ωm0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='2655 for the matter density parameter, Ωr0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00050 for the radiation density parameter, along with a newly introduced parameters ∆m = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00251 and ∆Λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' We will call this diffusive model case hereafter Case III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 10 DiffusiveModel:Q2m=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='3134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='h=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='6955,Qr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='△m=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='12469,A^=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='12440 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='2 Deccelartion Parameter q(z) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='4 ACDM model using Planck 2o18 values 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='6 ACDM modelbased on MCMC data Diffusivemodelbased onMCMC data l-o range forthe diffusive model 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00 Cosmological Redshift (z)DiffusiveModel:Q2m=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='2678.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='h=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='6966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='Qr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00050,Am=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='A^=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00251 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='2 Deccelartion Parameter q(z) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='4 ACDM model using Planck 2o18 values 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='6 ACDM modelbased on MCMC data Diffusivemodelbased onMCMC data l-o range forthe diffusive model 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00 Cosmological Redshift (z)FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The MCMC simulation results for the diffusive model’s which is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 15 for cosmological free parameters (Ωm0, ¯h, Ωr0, ∆m and ∆Λ), with the “true” values for Ωm0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='315, ¯h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='674, and Ωr0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='47 × 10−5 /¯h2 provided by the Planck2018 collaboration data release [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' We used 100 random walkers and 10000 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' In the following we will also provide one interesting case in which the best-fitting parameter value for each free parameter to be h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='6976 for the Hubble uncertainty parameter, Ωm0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='2283 for the matter density parameter, and Ωr0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00050 for the radiation density parameter, along with a newly introduced parameters ∆m = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='10747 and ∆Λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='10426.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' We will refer to this diffusive model case as Case IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 13 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 14 the above discussed two diffusive cosmological model cases were given which clearly shows to fit the extremely well with the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' Even the corresponding 1σ-deviation do 11 Qm=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='26554±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='02498 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='02404 h=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='69676±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00474 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00477 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='71 n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='68 Qr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00050+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00034 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00034 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content="001/0 T00'0- 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='004 A=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00246±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00172 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00167 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='003 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='69 10 15 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='30 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='68 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 2m h AmFIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The MCMC simulation results for the diffusive model’s which is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 15 for cosmological free parameters (Ωm0, ¯h, Ωr0, ∆m and ∆Λ), with the “true” values for Ωm0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='315, ¯h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='674, and Ωr0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='47 × 10−5 /¯h2 provided by the Planck2018 collaboration data release [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' We used 100 random walkers and 10000 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' not really have an impact on the full range predicted by them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' Additionally, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 15 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 16 displays the residuals obtained in the above two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' It can clear be seen that at no point does the models over-or under-estimate the resulting distance modulus for each supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' We also note that the average off-set the model has, compared to the data, is ¯xres = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0374 Mpc and ¯xres = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0386 Mpc, with a standard deviation of σres = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='2147 and σres = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='2145, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' These results show that these are very strong relationships between the models and data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 12 Qm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='22830±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='03874 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='03742 h=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='69767±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00477 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00475 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='71 n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='68 Qr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00050+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00034 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00034 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='10747 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='07165 Mu 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='12 A=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='10426±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='06691 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='06971 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='16 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='12 00 36 10 6 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0 O O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' O 2m h Am A^FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The diffusive model’s Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 15 for best- fitting free parameters for the Supernovae Type 1A data with cosmological parameter values as h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='6967+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0047 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0047, Ωm0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='2655+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0248 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0237, Ωr0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0005+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0003 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0003, ∆m = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0025+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0169 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0170 and ∆Λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0024+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0172 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0167 of the MCMC simulation result shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The diffusive model’s Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 15 for best- fitting free parameters for the Supernovae Type 1A data with cosmological parameter values as h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='6976+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0047 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0047, Ωm0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='2283+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0387 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0374, Ωr0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0005+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0003 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0003, ∆m = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='1074+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0716 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0641 and ∆Λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='1042+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0660 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0697 of the MCMC simulation result shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' This is the residuals distance in Mpc be- tween the predicted model values and the data points for the diffusive model’s Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 15 for best-fitting free parameters shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' This is the residuals distance in Mpc be- tween the predicted model values and the data points for the diffusive model’s Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 15 for best-fitting free parameters shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 13 Theoreticalpredictionsforthediffusivemode Model:Qm=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='2655,h=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='6968,Q,=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00050, 46- △m=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00251,^= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00246 (odw) w 44 modulus: m- 42 40 38 Distance 36 34 10~2 10~1 100 Redshift: zTheoreticalpredictionsforthediffusivemode Model:Qm=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='2283,h=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='6977,Qr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00050 46- △m=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='10747, ^= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='10426 (odw) w 44 modulus: m- 42 40 38 Distance 36 34 10~2 10~1 100 Redshift: zResiduasforthediffusivemoderesultsonthedata Ave = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0374 and o = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='21475 2 10~2 10~1 100 Redshift: zResiduasforthediffusivemoderesultsonthedata Ave = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0386 and o = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='21456 2 10~2 10~1 100 Redshift: zFIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The Hubble parameter vs Redshift for the Model displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The blue curve repre- sent the result obtained by considering diffusive fluid and employing MCMC simulation, with 1-σ devia- tion result displayed in yellowish shaded region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The red curve represent ΛCDM cosmology result using MCMC simulation where as the green curve repre- sent one obtained directly by using the Planck 2018 data for the purpose of comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The Hubble parameter vs Redshift for the Model displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The blue curve repre- sent the result obtained by considering diffusive fluid and employing MCMC simulation, with 1-σ devia- tion result displayed in yellowish shaded region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The red curve represent ΛCDM cosmology result using MCMC simulation where as the green curve repre- sent one obtained directly by using the Planck 2018 data for the purpose of comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 17 and 18 shows the evolution of the Hubble parameter across the Redshift for the two cos- mological model cases discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The blue curves represent the result obtained by considering diffusive fluid and employing MCMC simulation, with 1-σ deviation results displayed in yellowish shaded regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The red curves represent ΛCDM cosmology result using MCMC simulation where as the green curves represent those obtained directly by using the Planck 2018 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' An overlap is observed in the result displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 17 between the two curves which are obtained by using the MCMC simulation data in the ΛCDM model and that of the average value of MCMC simulation data of the diffusive model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' In contrast, a noticeable deviation emerges to be observed between the the Hubble parameter values of the diffusive model result (the blue curve) and that of the ΛCDM result based on MCMC simulation data (the red curve) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 18 from ∼ z of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='75 onward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' Even though it is not expected to have a complete overlap between the ΛCDM model using Planck 2018 values put directly in the cosmological equations (green curves) and the MCMC results (blue curves and yellowish shaded regions), the difference between them is observed to be more prominent in the 14 DiffusiveModel:Q2m=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='2655.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='h=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='6968,Qr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='Am=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00251,A^=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00246 200 AcDM model using Planck 2o18 values ACDMmodelbasedonMCMC data 180 Diffusivemodelbased onMCMC data 1-g range forthe diffusive model 160 140 120 100 80 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00 Cosmological Redshift (z)DiffusiveModel:Q2m=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='2283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='h=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='6977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='Qr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='△m=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='10747.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='A^=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='10426 200 AcDM model using Planck 2o18 values ACDM model based on MCMC data 180 Diffusivemodelbased onMCMC data 1-g range forthe diffusive model 160 140 120 100 80 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00 Cosmological Redshift (z)FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The graph of deceleration parameter vs redshift for the diffusive model shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The graph of deceleration parameter vs redshift for the diffusive model shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' case of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 17 than that of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 19 and 20 shows the evolution of the deceleration parameter across Redshift for the two cos- mological model cases discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The blue curves represent the result obtained by considering diffusive fluid and employing MCMC simulation, with 1-σ deviation results displayed in yellowish shaded regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The red curves represent ΛCDM cosmology result using MCMC simulation where as the green curves represent those obtained directly by using the Planck 2018 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 19 we observe a complete overlap while using MCMC simulation data in the ΛCDM model and the average value of MCMC simulation data of the diffusive model, which is also observed in the Hubble parameter vs Redshift plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' As redshift increases, the Hubble parameter values of the diffusive model result (the blue curve) and that of the ΛCDM result based on MCMC simulation data (the red curve), begin to acquire similar values as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The 1-σ deviation results indicated in yellow colored shaded region is observed to encompass both diffusive (blue curve) and non-diffusive (red curve) cases of the deceleration parameter values given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' In the current universe, the diffusive model has slightly lower values of deceleration parameter compared to what is observed in the case of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' In Table I we provide some statistical result which allows us to determine the best diffusive model case in comparisons to the ΛCDM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The statistical analysis test that we have used is the Akaike information criterion (AIC) and Bayesian/Schwarz information criterion (BIC) selections which were used in a similar work in [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' These information criteria evaluate the plausibility of an alternative model explaining the data compared to an “accepted/true model”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' In our case the ΛCDM model will 15 DiffusiveModel:Q2m=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='2655h=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='6968,Q2r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='Am=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00251,A^=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00246 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='2 Deccelartion Parameter q(z) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='4 ACDM model using Planck 2o18 values 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='6 ACDM modelbased on MCMC data Diffusivemodelbased onMCMC data l-o range forthe diffusive model 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00 Cosmological Redshift (z)DiffusiveModel:Q2m=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='2283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='h=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='6977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='Qr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='△m=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='10747.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='A^=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='10426 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='2 Deccelartion Parameter q(z) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='4 ACDM model using Planck 2o18 values 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='6 ACDM modelbased on MCMC data Diffusivemodelbased onMCMC data l-o range forthe diffusive model 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='00 Cosmological Redshift (z)TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The best-fit for each tested model, including the ΛCDM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The models are listed in the order from the largest likelihood function value L(ˆθ|data) to the smallest likelihood of being viable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The reduced χ2 -values are given as an indication of the goodness of fit for a particular model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The AIC and BIC values are shown, as well as the ∆AIC and ∆BIC for each information criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The ΛCDM model is chosen as the ”true model”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' Models ∆m ∆Λ L(ˆθ|data) χ2 Red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='χ2 AIC |∆AIC| BIC |∆BIC| Diffusive Case II +ve -ve -121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='1677 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='3355 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='6845 252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='3355 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='9405 271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='7521 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='7072 Diffusive Case I +ve -ve -120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='7059 241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='4118 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='6819 251.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='4118 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0168 270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='8285 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='7835 ΛCDM 0 0 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='6975 241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='3950 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='6780 247.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='3950 0 259.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0449 0 Diffusive Case III -ve +ve -120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='6890 241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='3781 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='6818 251.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='3781 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='9831 270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='7947 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='7497 Diffusive Case IV -ve +ve -120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='3936 240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='7872 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='6801 250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='7872 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='3922 270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='2039 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='1589 be considered as the “true model”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' Following the suggestion made in [32] as the calculated values for the AIC and BIC can by very random, we will also use the difference in AIC (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=', ∆AIC) and BIC (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=', ∆BIC) values of each model compared to the “true model’s AIC and BIC values, and we use the Jeffrey’s scale in order to make conclusions about the viability of the various Diffusive model cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' Moreover, the reduced χ2 -values are used as an indication of the goodness of fit for each model on the supernovae data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' It is observed that, the first two Diffusive model cases (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 1 and 2) have obtained better likelihood function values than the ΛCDM model based on a Gaussian probability distribution, with Case II obtaining the larger likelihood function value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' However, in the reduced χ2 values in which the number of parameters are taken into account when determining the goodness of fit, the ΛCDM model has the best value with the Diffusive model Case I (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 1) managing to have a closer value to this accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' In order to find the better fitting model among these two cases, we use AIC test, according to which the Diffusive model Cases I and II have obtained more observational and less observational support, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' Case I is seen to have a value just missing out on the substantial observational support category, but is still with a closer value to the boundary for less observational support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' Therefore, it can be concluded that Case I has some observational support according to the AIC criterion, while Case II has less observational support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' In terms of the BIC criteria, we did not obtain one model to have some observational support category, but Case I 16 was the closer of being in one of the categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' Therefore, statistically, based on the likelihood, the goodness of fit, the AIC and BIC criteria, Case I is the most likely to be an alternative model to the ΛCDM model, with Case II not being ruled out, but will have to be tested on other datasets before being accepted or rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 17 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' CONCLUSIONS In this manuscript we considered diffusive cosmological models where dark matter and dark energy interact by exchanging energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The background cosmological parameters in particular the thermody- namics parameters have been studied and compared against supernova cosmological data for different Diffusive model cases using MCMC simulation results presented in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' For the two new parameters which arise in our Diffusive cosmological model, namely ∆m and ∆Λ, we have examined the Hubble and deceleration parameters results of Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 7 to 10 and that of Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 17 to 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' Recalling the requirement that the sum of these two parameters need to be zero, the magnitude of ∆m and ∆Λ of ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content='0025 fit the parameter space very well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' Following which we investigated this deeply based on the statistical analysis made in the above section which is given in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' From our analysis we observed that cases having positive values of ∆m were showing the largest values of likelihood function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' Based on the analysis of likelihood, goodness of fit, AIC and BIC criteria, one can conclude that overall Case I is the most likely to be an alternative to the ΛCDM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' As we have highlighted in the discussion part our current work is to provide a viability test of the different cases considered, but to reject or accept any of them more data and rigorous testing method is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' Moreover, our initial result such as the one shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 7 and 17 suggest that one can look for a potential explanation of the Hubble Tension in such models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' ACKNOWLEDGEMENTS AA acknowledges that this work is based on the research supported in part by the National Re- search Foundation (NRF) of South Africa (grant number 112131).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' This work was part of the research programme “New Insights into Astrophysics and Cosmology with Theoretical Models confronting Observational Data” of the National Institute for Theoretical and Computational Sciences of South Africa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' [1] Adam G Riess, Alexei V Filippenko, Peter Challis, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' Observational evidence from supernovae for an accelerating universe and a cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' The Astronomical Journal, 116(3):1009, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' [2] Saul Perlmutter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' Supernovae, dark 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parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' Astronomy & Astrophysics, 641:A6, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} +page_content=' 20' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfGQPs/content/2301.02913v1.pdf'} diff --git a/KdFOT4oBgHgl3EQfzDRI/vector_store/index.pkl b/KdFOT4oBgHgl3EQfzDRI/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..1622dd4076dc04cb7560405941798f5f53212387 --- /dev/null +++ b/KdFOT4oBgHgl3EQfzDRI/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6adfac888b1ac3072a4afd518727c8bfad1646938290c8d9b3bf8cdd707782df +size 269330 diff --git a/LNE0T4oBgHgl3EQfSgB1/content/tmp_files/2301.02223v1.pdf.txt b/LNE0T4oBgHgl3EQfSgB1/content/tmp_files/2301.02223v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fb4ec7d7f826fbdc07356735ca3b7dea819ff73d --- /dev/null +++ b/LNE0T4oBgHgl3EQfSgB1/content/tmp_files/2301.02223v1.pdf.txt @@ -0,0 +1,2792 @@ +arXiv:2301.02223v1 [math.NT] 5 Jan 2023 +Measuring the Space of Metaplectic Whittaker Functions +Ilani Axelrod-Freed, Claire Frechette, and Veronica Lang +January 6, 2023 +Abstract +Whittaker functions are special functions that arise in p-adic number theory and representation theory. +They may be defined on representations of reductive groups as well as their metaplectic covering groups: +fascinatingly, many of their number theoretic applications survive the transition between the reductive +and metaplectic cases. However, one notable difference is that the space of Whittaker functions on a +reductive group over a nonarchimedean local field F is one-dimensional, whereas this is no longer true +in the metaplectic case. In a previous paper, the second author showed that the dimension of the space +of Whittaker functions on an arbitrary n-fold metaplectic cover of GLr(F) can be counted in terms of +the number of solutions to a particular system of linear Diophantine equations in terms of n and r. +In this paper, we calculate two precise formulae for dim(W), one inspired by viewing this system as a +homogenous specialization of an inhomogenous system and the other by the structure of the coroot lattice +of GLr(F). Then we use these formulae to investigate a homomorphism between W and a particular +quantum group module, built by the second author in a previous paper, and show precisely when this +map is well-defined for any choice of basis for W. +1 +Introduction +Whittaker functions arise in p-adic number theory and representation theory, specifically in the study of +automorphic forms over local fields and the study of principal series representations of reductive groups. They +can be written in many forms: as integrals over matrix groups, as generating functions over many different +combinatorial objects, as coefficients of automorphic forms, and in some cases as partition functions of lattice +models. In particular, when the lattice model is solvable, this viewpoint leads to a surprising connection +between the algebraic structures of the space of Whittaker functions and of modules for quantum groups. +One type of Whittaker functions of particular interest are metaplectic Whittaker functions, which are +Whittaker functions on the principal series representations of metaplectic covering groups, central extensions +of a reductive group by the n-th roots of unity. These groups are named after the first “Metaplectic Group,” +the unique double cover of the symplectic group Sp2n discovered by Weil [19]. However, the machinery +generating this particular cover can be applied in far greater generality and results in non-algebraic groups +that inherit much of the interesting representation theory and number theory of their algebraic base groups. +One reason for this phenomenon is that if G is a group that is also a topological space, the metaplectic cover +is a covering space in the topological sense as well: thus, the metaplectic covers of reductive groups, which are +equipped with a topological structure, are of particular interest. These groups have been studied in various +levels of generality by Kazhdan and Patterson [11], Matsumoto [13], Brylinski-Deligne [5], McNamara [15], +Gan, Gao, and Weissman [8, 9], and many others. For our purposes, a particularly useful description is that +of Brylinski-Deligne [5], who proved that metaplectic covers of reductive p-adic groups are in correspondence +with symmetric Weyl-group invariant bilinear forms on the cocharacter lattice. We will examine the structure +of these covering groups in more detail in Section 2, following the treatment of the second author in [7]. +The focus of this paper is the reductive group G = GLr(F), the general linear group of r × r matrices +over a nonarchimedean local field F containing µ2n. In this case, which was first studied by Matsumoto +[13], the bilinear forms prescribed by Brylinski-Deligne [5] recover a subset of the Kahzdan-Patterson covers +[11] and may be explicitly parametrized as in Frechette [7] as Bc,d in terms of two parameters c, d ∈ Z (see +Section 2 for the details of this construction). In general, metaplectic covers of G are denoted �G, so let +�Gc,d,r,n be the n-fold cover of GLr(F) corresponding to Bc,d. It is important to note that while there may +1 + +be multiple bilinear forms corresponding to a given cover, any such form will suffice for our purposes. We +refer the reader to [11] or [7] for more detailed descriptions of which forms give identical or similar covers. +One interesting difference between the algebraic (i.e., non-metaplectic) and metaplectic cases is the +dimension of the space of Whittaker functions. For a reductive algebraic group, the space of Whittaker +functions on any principal series representation is one-dimensional [18, 17, 10]. In the metaplectic case, +however, the construction of principal series representations becomes more complicated, due to the fact that +the metaplectic torus �T, the preimage in the metaplectic cover �G of the torus T (F), is no longer necessarily +abelian. Due to this phenomenon, the dimension of the space of Whittaker functions becomes dependent on +the choice of cover. As shown by McNamara [15], if W is the space of metaplectic Whittaker functions for +a principal series representation on �G and H is the maximal abelian subgroup of �T, then +dim(W) = +��� �T/H +��� , +and the basis vectors of W may be parametrized by the cosets in �T /H. Note that the space of Whittaker +functions is traditionally denoted Wz, where z = (z1, ..., zr) ∈ Cr lists the Satake parameters for the +principal series representation. Since the results in this paper largely do not depend on the choice of z, we +will generally drop it from the notation and write simply W. +Examining the group structures of �T and H for a non-archimedean local field F, we achieve an explicit +expression for the dimension, which we will prove in Section 2 as Theorem 2.4. +Theorem 1.1. For an n-fold metaplectic cover �G of GLr(F) corresponding to the bilinear form Bc,d, +��� �T/H +��� = +nr +|{x ∈ (Z/nZ)r : Bc,d(x, y) ≡ 0 +(mod n) for all y ∈ (Z/nZ)r}|. +Our main result is a closed formula for the order of the set in the denominator of Theorem 1.1. To this +end, let +Λfin := {x ∈ (Z/nZ)r : Bc,d(x, y) ≡ 0 +(mod n) for all y ∈ (Z/nZ)r} . +Then, using linear Diophantine equations to parametrize Λfin in two different ways, we arrive at the following +result, which is proven in two parts as Theorem 4.7 and Theorem 6.1, respectively. +Main Theorem 1. Given an n-fold metaplectic cover of GLr(F) corresponding to the bilinear form Bc,d, +|Λfin| = dr−1 +1 +gcd +� +d2, dn +d1 +� +, +where d1 = gcd(c − d, n) and d2 = gcd(c + (r − 1)d, n). Alternately, we also have that +|Λfin| = dr−1 +1 +d2 +n +gcd +� n +d1 +, n +d2 +, r +� +lcm +� +n +gcd(r, n), gcd +� +d2, dn +d1 +�� +, +where b = gcd(r, n). +The first formula arises from viewing the parametrizing Diophantine equations, which are generated by +the natural basis for the cocharacter lattice for GLr(F), as a homogeneous specialization of an inhomogenous +system. This viewpoint provides a more elegant formula and a more concrete description of the structure +of the space of Whittaker functions. On the other hand, while the second formula is more complicated, it +arises from the root structure of GLr(F), which provides a more direct path to extending this result to other +reductive groups. +For GLr(F), the space of Whittaker functions is also closely tied to a particular module for a quantum +group built from the Lie algebra gl. Despite the name, quantum groups are not groups at all, but rather +quasitriangular Hopf algebras. For this paper, we consider the quantum affine universal enveloping algebra +Uq(c, d, n) := Uq(�gl(n/d1)), where q is the cardinality of the residue field for F. This quantum group has a +n/d1-dimensional evaluation module V+(z) depending on a parameter z ∈ C, whose basis vectors may be +indexed using the elements of Z/(n/d1)Z. +2 + +In [1], Brubaker, Bump, and Buciumas prove that for the simplest n-fold metaplectic cover of GLr(F) +(the one where c = 1 and d = 0), the space of Whittaker functions is isomorphic to an r-fold tensor +product of evaluation modules and that after a Drinfeld twist (which changes the group action but does not +affect the module structure) this isomorphism matches the action of the quantum group to the action of +intertwining operators on the underlying principal series representation. The key ingredient in this proof is +a lattice model construction for metaplectic Whittaker functions in the case c = 1, d = 0 developed in [2] by +Brubaker, Bump, Chinta, Friedberg, and Gunnells. +In [7], the second author proves that both of these constructions are true in far greater generality, +constructing a Whittaker function lattice model and a map θz between Wz and an r-fold tensor product +V+(z1) ⊗ · · · ⊗ V+(zr) of evaluation modules for any metaplectic cover of GLr(F). (To match to the termi- +nology used in [7], set nQ := n/d1.) Moreover, passing through this map, the action of the quantum group +still matches exactly the action of intertwining operators on the Whittaker functions. +As c, d, r, n vary, the cost of dealing with more complicated covers is that this map shifts between being +an isomorphism, an injection, and a surjection, and the choice of representatives for H-cosets affects the +map. The lattice model construction used in [7] dictates a choice of coset representatives from �T/H giving +a basis for W on which θz is well-defined. However, the lattice model is not necessary for the connection +between W and the quantum module outside of this phenomenon. A natural question then arises: when is +the map θz well-defined for any choice of basis for W? +One of the main applications of our results is an answer for this question, using the characterization of +elements of Λfin from our proof of Main Theorem 1. Taking any basis for W, use Theorem 1.1 to express it +as a set of vectors in (Z/nZ)r. Then the map given in [7] is precisely +θz : Wz → V+(z1) ⊗ · · · ⊗ V+(zr) +ν +�→ +ρ − ν +(mod n/d1), +where ρ = (r−1, ..., 2, 1, 0) and the modulus is applied independently in each component of the vector. Using +our characterization to show how any particular coset νH sits within (Z/nZ)r, we arrive at the following +result, which will be proven as Theorem 8.2 and Corollary 8.3. +Main Theorem 2. For a vector z = (z1, ..., zr) ∈ Cr, the homomorphism θz given in [7] is well-defined for +any choice of basis for W if and only if +gcd +� +d2, dn +d1 +� += gcd(c, d, n). +Furthermore, if W is either of minimum or maximum dimension, θz is an isomorphism. +Understanding how this map is affected by the choice of cover is an important step to understanding +how we may extend these quantum connections to metaplectic covers over other reductive groups. While +the lattice model connection only exists in full for GLr(F) and SLr(F), the Whittaker function framework +exists for any reductive group, so we hope that further investigation of the structure of W will not only allow +us to develop analogues to Main Theorem 1 for other groups, but also to determine the precise quantum +group and module connected to the metaplectic Whittaker functions for other types. +Regarding the structure of this paper, in Section 2, we examine the construction of metaplectic covers of +GLr(F) and their Whittaker functions, culminating in a proof of Theorem 1.1. Section 3 introduces the first +set of Diophantine equations used to parametrize Λfin, which we then use in Section 4 to prove the first part +of Main Theorem 1 as Theorem 4.7. In Section 5, we introduce the second set of Diophantine equations for +Λfin, which facilitate the proof of the second part of Main Theorem 1 as Theorem 6.1 in Section 6. In Section +7, we examine some cases in which the formulae for dim(W) simplify dramatically and prove conditions for +certain dimensions of interest for W, including conditions for maximum and minimum dimension. Lastly, +in Section 8, we develop the quantum connection and use the structure of W to prove Main Theorem 2 as +Theorem 8.2 and Corollary 8.3. +Acknowledgements +This project was partially supported by NSF RTG grant DMS-1745638 and was supervised by the second +author as part of the University of Minnesota School of Mathematics Summer 2022 REU program. The +3 + +second author is also supported by NSF grant DMS-2203042. The authors would like to thank their TA +Carolyn Stephen for their guidance throughout the project, as well as Ben Brubaker and Darij Grinberg for +helpful comments. +2 +Spaces of Metaplectic Whittaker Functions +To understand the structure of the space of metaplectic Whittaker functions, we must first concretely describe +the metaplectic covers of GLr(F). We can then extend this explicit parametrization of all covers into a +description of the metaplectic torus and its maximal abelian subgroup. As mentioned in the introduction, +the quotient of these subgroups controls the dimension of the space of Whittaker functions: describing its +structure precisely in terms of the cover allows us to reduce a complicated representation theory question to +a straightforward linear algebra problem. +Suppose n is a natural number and F is a nonarchimedean local field containing 2n distinct 2n-th roots +of unity µ2n. Let o be the ring of integers of F and ̟ its uniformizing element. +Definition. Given a split reductive group G, an n-fold metaplectic cover or n-fold metaplectic covering +group �G is a non-algebraic central extension of G by the n-th roots of unity µn. That is, �G is defined by the +following short exact sequence: +1 → µn → �G +p−→ G → 1. +As a set, �G is the set of tuples (ζ, g) where ζ ∈ µn, g ∈ G. However, group multiplication is controlled +by a cocycle σ ∈ H2(G, µn); that is, for two elements (ζ1, g1), (ζ2, g2), their product in �G is +(ζ1, g1) · (ζ2, g2) = (ζ1ζ2σ(g1, g2), g1g2). +In the process of writing down an explicit form for cocycles for covers of GLr(F), we see that a slightly +more general case may be handled simultaneously. Set G = GLr(F) for the remainder of the paper. +Definition. More generally, a metaplectic covering group essentially of degree n is given by a short exact +sequence +1 → µm → �G +p−→ G → 1 +where n|m and the corresponding cocycle σ ∈ H2(G, µm) satisfies the property that [σn] is trivial in +H2(G, C×) under the inclusion induced by an embedding ε : µm → C×. +While it is slightly tedious to write down formulae for these cocycles on general elements of G, their +expressions over the torus T of diagonal matrices in G are quite elegant. In [7], the second author proves +that all metaplectic covers essentially of degree n over GLr(F) come from a cocycle of the form +σc,d(x, y) = (det(x), det(x))c +2n +� +i>j +(xi, yj)d−c +n +. +(1) +for c, d ∈ Z, where x, y ∈ T and (·, ·)k denotes the k-th Hilbert symbol (see Neukirch [16] for more details on +the construction of Hilbert symbols). Notably, making the shift to covers essentially of degree n rather than +“purely” of degree n is necessary to include the metaplectic cover corresponding to the cocycle σ1,0, which, +while only essentially of degree n, has been an integral example for this field (see for example [1, 2, 3, 4, 14].) +Remark 2.1. Since the 2n-th Hilbert symbol produces 2n-th roots of unity, it is necessary that F contain +µ2n for the group to be well defined. However, if we are considering a cocycle for which the parameter c is +even, we may relax this condition and require F to contain only µn. +In [5], Brylinski-Deligne prove that the set of metaplectic covers is in correspondence with the set of +symmetric Weyl-group invariant bilinear forms B : Y × Y → Z on the cocharacter lattice Y such that +B(α∨,α∨) +2 +∈ Z for all coroots α∨ ∈ Y . For G = GLr(F), a natural choice of basis for Y is the set of r +fundamental coweights ε∨ +i : F × → T , for i = 1, ..., r, in which ε∨ +i (a) := diag(1, ..., 1, a, 1, ..., 1), where a is in +the i-th entry. Note: while we will use the notation λ(a) for λ ∈ Y, a ∈ F ×, another common notation is aλ. +4 + +Under this basis, the cocharacter lattice Y is isomorphic to Zr; for instance, +(ε∨ +1 + 3ε∨ +2 )(a) = diag(a, a3, 1, ..., 1). +Using this basis, we represent a bilinear form on Y in terms of the corresponding matrix A such that for +x, y ∈ Y , +B(x, y) = xT Ay. +Each of the conditions from the Brylinski-Deligne correspondence translates into a condition for this matrix. +First, a symmetric bilinear form prescribes a symmetric matrix. Second, the Weyl group W is isomorphic +to the symmetric group Sr, and acts on Y by σ · ε∨ +i = ε∨ +σ(i). Thus, A must be invariant under conjugation +by permutation matrices, so for some (suggestively named) c, d ∈ Z, we have ai,i = c for all i and ai,j = d +for all i ̸= j. (See the matrix in (2) for an illustration of this requirement.) +There are r − 1 simple coroots, each of the form ε∨ +i − ε∨ +i+1. To check the integrality condition on the +coroot lattice, it suffices to show that it holds for simple coroots. However, for GLr(F), this condition is +satisfied already: for any simple coroot ε∨ +i , +Bc,d(ε∨ +i , ε∨ +i ) +2 += 2(c − d) +2 += c − d ∈ Z. +By Brylinski-Deligne, the metaplectic cover corresponding to this bilinear form satisfies the following +condition: if x, y ∈ �T = p−1(T ) such that p(x) = λ(x), p(y) = µ(y) for some x, y ∈ F × and λ, µ ∈ Y , then +the commutator of x and y is +[x, y] = (x, y)B(λ,µ) +n +. +Evaluating the commutator in terms of an explicit cocycle, we may identify the bilinear form correspond- +ing to a specific cocycle and vice versa. Note that this property illuminates one of the key reasons the +Brylinski-Deligne correspondence is not a bijection: since the cocycle in (1) depends on powers of Hilbert +symbols, there are many different cocycles which will give exactly the same cover, specifically any σc′,d′ such +that c′ ≡ c (mod 2n) and d′ − c′ ≡ d − c (mod n). +Theorem 2.2 (Frechette [7]). For c, d ∈ Z, the essentially n-fold metaplectic cover of GLr(F) with multi- +plication given by σc,d corresponds to the bilinear form Bc,d that acts on (x, y) ∈ Zr × Zr by +Bc,d(x, y) = xT · + + + + + + + +c +d +d +. . . +d +d +c +d +. . . +d +d +d +c +. . . +d +... +... +... +... +d +d +d +. . . +c + + + + + + + +· y. +(2) +Conflating the bilinear form with its corresponding matrix, we will denote both by Bc,d; we hope this +abuse of notation will not cause any confusion. Note: in [7], this bilinear form is parametrized slightly +differently as Bb,c, where b = c − d. +Now that we have an explicit description of our metaplectic covers, we investigate what this parametriza- +tion tells us about space of metaplectic Whittaker functions. For the purposes of this paper, we will not need +the constructions of the metaplectic Whittaker functions themselves, nor those of the metaplectic principal +series representations on which they are defined. Instead, we will use the following theorem of McNamara +to investigate the space of Whittaker functions through its connection to the metaplectic torus. For the +definitions of the metaplectic principal series representations and their Whittaker functions, we refer the +reader to Sections 6 and 8, respectively, of [15] as a convenient source. +Theorem 2.3 (McNamara [15]). Fix a metaplectic cover �G over a p-adic reductive group G and let W be +the space of metaplectic Whittaker functions for a principal series representation on �G. Let the metapletic +torus �T be the preimage in �G of the torus T (F), and let H be the maximal abelian subgroup of �T. Then +dim(W) = +��� �T/H +��� . +5 + +Note: the group T of diagonal matrices is denoted T because it is an abelian torus, that is, it is isomorphic +to (F ×)r. While we call �T the metaplectic torus, it is no longer abelian, nor is it technically a torus, as its +elements are (ζ, t) where ζ ∈ µn (where µn ⊊ F) and t ∈ T . Investigating the precise structure of �T, we +prove the following theorem, which is a restatement of Theorem 1.1. +Theorem 2.4. For a metaplectic cover �G of GLr(F) corresponding to the bilinear form Bc,d, +��� �T/H +��� = +nr +|{x ∈ (Z/nZ)r : Bc,d(x, y) ≡ 0 +(mod n) for all y ∈ (Z/nZ)r}|. +Proof. Using our description of the metaplectic covers, we can express the subgroups �T and H more explicitly: +using the Iwasawa decomposition of GLr(F), we have that �T = µn ×T (o)×Y as a set. That is, for (ζ, t) ∈ T, +we may write t = t0 · λ(̟) for some t0 ∈ T (o) and λ ∈ Y . +Since H is a subgroup of �T, its elements also look like (ζ, h) where ζ is an n-th root of unity and h is a +diagonal matrix with entries in F. Examining the group law on �G, we see that the root of unity does not +impede commutativity of elements, so it is the matrix component h we must examine further to obtain a +description of H. To do so, recall that o is the valuation ring of F and ̟ the uniformizing element. Then +by [15], as a set we have H = µn × T (o) × Λ, where Λ is the free abelian group +Λ := {λ ∈ Y : s(λ(̟)) ∈ H} +for s : G → �G the standard section s(g) = (1, g). Using the commutator relation and the fundamental +coweight basis for Y , an equivalent description for Λ is +Λ = {x ∈ Zr : Bc,d(x, y) ≡ 0 +(mod n) for all y ∈ Zr} . +(3) +It is useful to think of the group Λ as controlling the powers of ̟ in each entry on the diagonal of the +matrix h. That is, for any element (ζ, h) ∈ H, we have h = h0 · diag(̟λ1, ..., ̟λr) where h0 ∈ T (o) and +λ = λ1ε∨ +1 + · · · + λrε∨ +r is in Λ. +Then, combining our descriptions of �T and H to consider �T/H, we see that +| �T/H| = |Y/Λ| = |Zr/Λ| , +where the last description uses the embedding of Λ in Zr described in (3). Notice that if λi ∈ nZ for all i, then +B((λ1, ..., λr), y) will automatically be a multiple of n for any y ∈ Zr, and therefore λ = λ1ε∨ +1 + · · · + λrε∨ +r +will be in Λ. Therefore, it suffices to consider all coordinates λi mod n, and so +| �T/H| = |(Z/nZ)r/ (Λ ∩ (Z/nZ)r)| +which completes the proof. +Let Λfin := {x ∈ (Z/nZ)r : Bc,d(x, y) ≡ 0 (mod n) for all y ∈ (Z/nZ)r} . We will spend the next several +sections developing two related systems of linear Diophantine equations which allow us to describe the +elements in Λfin, each of which will give us a distinct formula for |Λfin|. We will then return to the broader +framework in Section 7 to what these different formulae tell us about the structure of | �T/H| and thus the +structure of W. +3 +Cocharacter Diophantine Equations and Phenomena +In this section, we use the natural basis for the cocharacter lattice Y of GLr(F) to develop a set of r +linear Diophantine equations in terms of c, d, and n that describe the set Λfin, which we call the cocharacter +equations. This perspective turns a representation theoretic question into a linear algebra one, where altering +each of the parameters c, d, r, and n has a different effect on the system. We also take time now to develop +a visual framework which illuminates this distinction in the roles of each of our parameters. +Examining the conditions for Λfin using the viewpoint of the fundamental coweight basis for Y (see +Section 2), we arrive at the following system of r equations. Let 0r = (0, 0, . . ., 0)T be the r × 1 column +vector with all entries equal to 0, and define 1r = (1, 1, . . . , 1)T similarly. Recall that Bc,d is both the bilinear +form given in Theorem 2.2 and its corresponding r × r matrix. +6 + +Definition. For natural numbers r, n ≥ 1 and constants c, d ∈ Z, we call the following system of equations +the cocharacter equations: +Bc,d · x = 0r +(mod n). +That is, for x = (x1, ..., xr)T , we have +cx1 + dx2 + · · · + dxr ≡ 0 +(mod n) +dx1 + cx2 + · · · + dxr ≡ 0 +(mod n) +... +dx1 + dx2 + · · · + cxr ≡ 0 +(mod n) +Here, the i-th equation arises from evaluating x ∈ Y against ε∨ +i in the bilinear form Bc,d for each +i ∈ {1, ..., r}. Thus, Lemma 3.1 follows directly. +Lemma 3.1. Let Scochar(c, d, r, n) be the number of solutions x ∈ (Z/nZ)r to the cocharacter equations. +Then, for the cover �Gc,d,r,n, we have Scochar(c, d, r, n) = |Λfin|. +Looking at the values of Scochar for a fixed r and n as we range over c and d, certain patterns emerge +which motivate defining constants which we call the diagonal numbers. These constants will be fundamental +in our formulas for Scochar, so we take the time to explore them now. +For a fixed r and n, note first that it suffices to understand Scochar for c, d (mod n), as Scochar(c, d, r, n) = +Scochar(c′, d, r, n) for c ≡ c′ (mod n) and likewise for d. It will be useful to visualize the values of Scochar as +a table ranging over c, d ∈ Z/nZ in the following manner: +d +0 +1 +2 +· · · (n − 1) +c +0 +1 +2 +... +(n − 1) +Examining Figure 1, which contains several examples of these tables, notice that the values of Scochar +on the marked diagonals in each picture are each divisible by common factors and that there are two sets of +diagonals in each picture. Motivated by this phenomena, we assign each entry a set of two diagonal numbers. +Definition. Let d1 = gcd(c − d, n) be the first diagonal number and define d2 = gcd(c + (r − 1)d, n) to be +the second diagonal number. +Note that for a specific entry in place c, d, its first diagonal number captures the column c − d where +its diagonal of slope −1 intersects the first row and similarly, the second diagonal number identifies the row +where its diagonal of slope r − 1 intersects the first column. +Example. When r and n are coprime, the table for Scochar depends solely on these diagonal numbers, which +we will later prove in Section 6 (see Corollary 7.4.) For instance, the table where n = 10, r = 3 is shown in +Figure 2, with diagonal numbers marked, and the value of every entry in this matrix is determined by its +two diagonal numbers. Specifically, we have Scochar(10, 3, c, d) = dr−1 +1 +d2 = d2 +1d2 for any c, d. +In general, given a random n and r, the value of Scochar will not depend nearly so simply on d1 and d2, +but they still play an important determining role. To find a closed formula for Scochar, we must look to an +inhomogenous generalization of the homogenous cocharacter equations with which we started. +Definition. Let a ∈ Z/nZ, and x ∈ (Z/nZ)r. Then the inhomogenous cocharacter equations for a ∈ Z are +defined by +Bc,d · x = a · 1r +(mod n). +(4) +Let Sinhom(c, d, r, n) be the number of total solutions to the inhomogenous cocharacter equations, ranging +over all values of a ∈ Z. +7 + +49 +1 +1 +1 +1 +1 +1 +1 +7 +1 +1 +1 +1 +7 +1 +1 +7 +1 +1 +7 +1 +1 +1 +1 +7 +7 +1 +1 +1 +1 +1 +7 +7 +1 +1 +1 +1 +7 +1 +1 +7 +1 +1 +7 +1 +1 +1 +1 +7 +r = 2, n = 7 +64 +1 +4 +1 +16 +1 +4 +1 +1 +8 +1 +8 +1 +8 +1 +8 +4 +1 +16 +1 +4 +1 +16 +1 +1 +8 +1 +8 +1 +8 +1 +8 +16 +1 +4 +1 +32 +1 +4 +1 +1 +8 +1 +8 +1 +8 +1 +8 +4 +1 +16 +1 +4 +1 +16 +1 +1 +8 +1 +8 +1 +8 +1 +8 +r = 2, n = 8 +81 +1 +1 +9 +1 +1 +9 +1 +1 +1 +9 +3 +1 +3 +3 +1 +3 +9 +1 +3 +9 +1 +3 +3 +1 +9 +3 +9 +1 +1 +27 +1 +1 +27 +1 +1 +1 +3 +3 +1 +9 +9 +1 +3 +3 +1 +3 +3 +1 +9 +9 +1 +3 +3 +9 +1 +1 +27 +1 +1 +27 +1 +1 +1 +3 +9 +1 +3 +3 +1 +9 +3 +1 +9 +3 +1 +3 +3 +1 +3 +9 +r = 2, n = 9 +729 +1 +1 +27 +1 +1 +27 +1 +1 +1 +81 +1 +1 +27 +1 +1 +27 +1 +1 +1 +81 +1 +1 +27 +1 +1 +27 +27 +1 +1 +243 +1 +1 +27 +1 +1 +1 +27 +1 +1 +81 +1 +1 +27 +1 +1 +1 +27 +1 +1 +81 +1 +1 +27 +27 +1 +1 +27 +1 +1 +243 +1 +1 +1 +27 +1 +1 +27 +1 +1 +81 +1 +1 +1 +27 +1 +1 +27 +1 +1 +81 +r = 3, n = 9 +4096 +1 +16 +1 +256 +1 +16 +1 +1 +512 +1 +16 +1 +128 +1 +16 +16 +1 +1024 +1 +16 +1 +256 +1 +1 +16 +1 +512 +1 +16 +1 +128 +256 +1 +16 +1 +2048 +1 +16 +1 +1 +128 +1 +16 +1 +512 +1 +16 +16 +1 +256 +1 +16 +1 +1024 +1 +1 +16 +1 +128 +1 +16 +1 +512 +r = 4, n = 8 +Figure 1: Examples of the Cocharacter Phenomena for different choices of r and n. In each example, notice +that there is one set of diagonals of slope -1 and one of slope r − 1: the former indicate the effect of the first +diagonal numbers d1 and the latter that of the second diagonal numbers d2. Diagonals for the same diagonal +numbers (greater than 1) are marked with the same color within each example. For instance, in the second +example (r = 2, n = 8) red marks diagonal numbers equal to 8, blue equal to 4, and green equal to 2. +In the next section, we will solve for Scochar(c, d, r, n) by characterizing the set of solutions to the inho- +mogenous cocharacter equations using straightforward linear algebra techniques and identifying the propor- +tion of solutions with a ≡ 0 (mod n). To do this, we will need to identify a precise formula for smallest +nonzero value of a for which (4) has a solution. +Definition. For a fixed c, d, r, n, let A(c, d, r, n) be the smallest positive integer value for a such that there +is a solution to the inhomogenous cocharacter equations (4). +4 +Proof of Main Theorem 1 Part 1 +For the entirety of this section, fix a set of parameters c, d, r, n. To find a formula for Scochar := Scochar(c, d, r, n), +we begin by showing that the solutions to the inhomogenous cocharacter equations fall into equally sized +equivalence classes defined by the values a ∈ Z, and that +Scochar(c, d, r, n) = A(c, d, r, n) +n +· Sinhom(c, d, r, n) +Characterizing the solutions to the inhomogenous cocharacter equations, we will then provide explicit +expressions for Sinhom(c, d, r, n) and A(c, d, r, n). +Lemma 4.1. The equation Bc,d · x ≡ a · 1r (mod n) has a solution if and only if a is a multiple of +A = A(c, d, r, n). Thus, A(c, d, r, n) divides n. +Proof. By definition, a solution xA to the equation Bc,d · x ≡ A · 1r (mod n) exists. If a = kA for some +k ∈ Z, then kxA is a solution to Bc,d · x ≡ a · 1r (mod n). For the other direction, suppose there exists a +8 + +1000 +2 +8 +2 +8 +250 +8 +2 +8 +2 +1 +100 +5 +4 +1 +4 +25 +20 +1 +4 +8 +2 +200 +2 +40 +2 +8 +50 +8 +10 +1 +20 +1 +100 +1 +4 +5 +4 +25 +4 +8 +2 +8 +10 +200 +2 +8 +2 +40 +50 +125 +4 +1 +4 +1 +500 +1 +4 +1 +4 +8 +50 +40 +2 +8 +2 +200 +10 +8 +2 +1 +4 +25 +4 +5 +4 +1 +100 +1 +20 +8 +10 +8 +50 +8 +2 +40 +2 +200 +2 +1 +4 +1 +20 +25 +4 +1 +4 +5 +100 +d1 =10 +1 +2 +1 +2 +5 +2 +1 +2 +1 +d2 = 10 +1 +2 +1 +2 +5 +2 +1 +2 +1 +d1 = 2, d2 = 10 +40 = 22 · 10 +Figure 2: The table showing Scochar(c, d, 3, 10) for all (c, d) ∈ Z10 × Z10 with diagonals for diagonal numbers +greater than 1 marked. Notice here that since r = 3 and n = 10 are coprime, every entry is equal to d2 +1 · d2. +In contrast, see the example in Figure 1 for r = 3 and n = 9, where this is not true. +positive integer g and solution xg ∈ (Z/nZ)r to the equation Bc,d · xg ≡ g · 1r (mod n), but that A does not +divide g. Then jA < g < (j + 1)A for some positive integer j. Therefore, +Bc,d · (xg − jxA) ≡ Bc,d · xg − jBc,d · xA +≡ (g − jA) · 1r, +which contradicts the minimality of A. Then, since x = 0r is a solution to Bc,d · x ≡ n · 1r ≡ 0r (mod n), +the second statement follows. +Splitting the solutions to (4) into equivalence classes based on a, we examine the number of solutions in +each class and characterize them more concretely. +Lemma 4.2. For k ∈ {1, ..., n +A}, let Wk be the set of solutions to Bc,d · x ≡ (kA) · 1r (mod n). Then +|Wk| = |W1| for all such k. +Proof. Consider any x ∈ W1. The function φx : W1 → Wk defined by y �→ y +(k −1)·x provides a bijection +between W1 and Wk. +Lemma 4.3. Let x = (x1, x2, . . . , xr)T . Then x solves the inhomogenous cocharacter equations if and only +if cx1 + dxj ≡ dx1 + cxj (mod n) for every 2 ≤ j ≤ r. +Proof. Let 2 ≤ j ≤ r. For a solution x, the first row of the equation Bc,d · x ≡ a · 1r (mod n) tells us that +cx1 + dxj + +� +2≤k≤r +k̸=j +dxk ≡ a +(mod n) +Subtracting the j-th row +dx1 + cxj + +� +2≤k≤r +k̸=j +dxk ≡ a +(mod n), +from the first, we obtain +cx1 + dxj +≡ +dx1 + cxj +(mod n). +For the other direction, suppose x satisfies cx1 + dxj ≡ dx1 + cxj (mod n) for all j ∈ {2, ..., r}. Then, x +satisfies the inhomogeneous cocharacter equations for the value a ≡ cx1 + dxj + +� +2≤k≤r +k̸=j +dxk (mod n). +9 + +Proposition 4.4. A vector x = (x1, x2, . . . , xr)T solves the inhomogenous cocharacter equations if and only +if for each j ∈ {2, ..., r} we have xj = x1 + vj +n +d1 +for some integer vj such that 1 ≤ vj ≤ d1. +Proof. By Lemma 4.3, it suffices to characterize the solutions x to the system of equations given by +cx1 + dxj ≡ dx1 + cxj +(mod n) +for every j ∈ {2, ..., r}, or equivalently, +(c − d)(x1 − xj) ≡ 0 +(mod n). +(5) +Recalling that d1 = gcd(c − d, n), a vector x satisfies (5) exactly when x1 − xj is a multiple of +n +d1 for all +j ∈ {2, ..., r}. Thus, the solutions to the inhomogeneous cocharacter equations are precisely the vectors of +the form +x = x1 · 1r + n +d1 +(0, v2, v3, ..., vr)T +where 0 ≤ x1 < n and vj ∈ Z such that 1 ≤ vj ≤ d1 for all j ∈ {2, ..., r}. +Now that we have precisely characterized the set of x which solve the inhomogeneous cocharacteristic +equations, we can count the size of this set by ranging over all distinct choices of tuples (x1, v2, ..., vr), which +each yield a distinct solution x. +Corollary 4.5. The number of solutions to the inhomogenous cocharacter equations is +Sinhom(c, d, r, n) = ndr−1 +1 +. +We are now prepared to identify a precise formula for A in terms of n, r, c, and d. +Proposition 4.6. The minimum positive integer A such that the inhomogenous cocharacter equations have +a solution is A = gcd +� +d2, dn +d1 +� +, recalling that d2 = gcd(c + (r − 1)d, n). +Proof. Substituting Proposition 4.4 into (4), we see that the left-hand side is + + + + + + + +c +d +d +. . . +d +d +c +d +. . . +d +d +d +c +. . . +d +... +... +... +... +d +d +d +. . . +c + + + + + + + + + + + + + + +x1 + + + + + + + +1 +1 +1 +... +1 + + + + + + + ++ n +d1 + + + + + + + +0 +v2 +v3 +... +vr + + + + + + + + + + + + + + +≡ x1(c + (r − 1)d) + + + + + + + +1 +1 +1 +... +1 + + + + + + + ++ n +d1 + + + + + + + +dv2 + dv3 + · · · + dvr +cv2 + dv3 + · · · + dvr +dv2 + cv3 + · · · + dvr +... +dv2 + dv3 + · · · + dvr−1 + cvr + + + + + + + +. +To have a solution, every row of this expression must must equal a constant A. Looking at the first row, +A ≡ x1(c + (r − 1)d) + dn +d1 +(v2 + v3 + · · · + vr) +(mod n). +From the proof of Proposition 4.4, x1 and v2 +v3 +· · ·+vr are both arbitrary constants. Thus, the minimum +value A can have is gcd +� +c + (r − 1)d, dn +d1 , n +� += gcd +� +d2, dn +d1 +� +. +Note, since d1 divides c − d, we can equivalently write A as A = gcd +� +d2, cn +d1 +� += gcd +� +d2, dn +d1 +� += +gcd +� +d2, n +d1 gcd(c, d, n) +� +. Thus, we arrive at a closed form for Scochar in terms of c, d, r, n. +Theorem 4.7. The number of solutions to the cocharacter equations is +Scochar(c, d, r, n) = dr−1 +1 +gcd +� +d2, dn +d1 +� +. +10 + +5 +Coroot Diophantine Equations +Inspired by the constants c + (r − 1)d and c − d showing up in the cocharacter equations, we define a second +system of related equations more closely tied to the root structure of GLr(F). +Definition. The coroot equations are the system of r equations: +(c − d)(xi − xr) ≡ 0 +(mod n) +for all i ∈ {1, ..., r − 1}, +(c + (r − 1)d)(x1 + · · · + xr) ≡ 0 +(mod n). +We call these the coroot equations because the i-th equation arises from evaluating x ∈ Y against the +coroot ε∨ +i − ε∨ +r in the bilinear form Bc,d for i ∈ {1, ..., r − 1}. We could similarly evaluate against the simple +coroots ε∨ +i − ε∨ +i+1, but this formulation will be more useful for our purposes. +In some cases, the coroot and cocharacter equations are equivalent, but in other cases they are not: +counting the solutions to the coroot equations and examining this connection will give us an alternate +formula for Scochar. +Remark 5.1. This system also illuminates the difference between metaplectic covers of SLr(F) and GLr(F). +For any cocharacter x for SLr(F), the last coroot equation is vacuously true, since x1+· · ·+xr ≡ 0 (mod n) is +necessary for the resulting matrices x(a) to have determinant one for any a ∈ F. In this case, the cocharacter +and coroot equations are equivalent, and they both give |Λfin ∩ SLr(F)| = dr−1 +1 +. +Theorem 5.2. The number of solutions to the coroot equations is +Scoroot(c, d, r, n) = dr−1 +1 +d2 gcd +� n +d1 +, n +d2 +, r +� +. +As in Section 4, we prove general properties about the solutions to the coroot equations. These descrip- +tions will allow us to directly relate Scochar to Scoroot in Section 6. +Proof. We start with a change of variables. Consider the coroot system in variables y1, ..., yr−1, z written as +(c − d)yi ≡ 0 +(mod n) +for all i ∈ {1, ..., r − 1}, +(c + (r − 1)d)z ≡ 0 +(mod n). +In terms of these variables, there are dr−1 +1 +· d2 tuples (y1, ..., yr−1, z) that solve the coroot equations: yi are +all multiples of +n +d1 and z is a multiple of +n +d2 . Let SY,Z be the set of such tuples. +We then classify x satisfying yi = xi − xr and z = x1 + · · · + xr such that (y1, .., yr−1, z) ∈ SY,Z: that is, +the set of x satisfying the original formulation of the coroot equations. Note that xi = yi+xr, so rearranging +the final coroot equation, we have +rxr ≡ z − (y1 + · · · + yr−1) +(mod n), +(6) +and thus the number of solutions in terms of x versus in terms of (y1, ..., yr−1, z) depends on whether r is +invertible mod n. Let b = gcd(n, r). Then xr has b solutions when z − (y1 + · · · + yr−1) is a multiple of b +and no solutions otherwise. A straightforward calculation verifies that there is no overlap between the sets +of x for distinct tuples (y1, .., yr−1, z) ∈ SY,Z. +Let Frb(d1, d2, n) be the proportion of (y1, . . . , yr, z) tuples that will yield a valid solution to the coroot +equations. In other words, +Frb := |{(y1, · · · , yr, z) ∈ SY,Z : z − y1 − · · · − yr is a multiple of b}| +|SY,Z| +. +Then Scoroot(c, d, r, n) = dr−1 +1 +d2·b·Frb(d1, d2, n), and it will suffice to develop a formula for Frb(d1, d2, n). +Remark 5.3. When n and r are relatively prime, r is invertible. Thus Frb evaluates to 1 because any tuple +we pick adds to a multiple of b = 1, so in this case the two sets of variables give equivalent conditions and +Scoroot = |SY,Z|. +11 + +Proposition 5.4. The function Frb evaluates to +Frb(d1, d2, n) = 1 +b · gcd +� n +d1 +, n +d2 +, b +� +. +Proof. We proceed by carefully considering the overlaps of factors of b with those of n +d1 , n +d2 . Let k1 = gcd( n +d1 , b) +and m1 ∈ Z such that b = m1k1. Similarly, let k2 = gcd( n +d2 , b) and m2 ∈ Z such that b = m2k2. +Since yi is a multiple of +n +d1 , it is also a multiple of k1: examining which multiples are possible modulo +b, we see that yi (mod b) can be any of the m1 multiples of k1 in Z/bZ with equal probability. Similarly, +considering the sum y = �r−1 +i=1 yi, we claim that the same is true for y. Let 1 ≤ g ≤ m1 and suppose y ≡ gk1 +(mod b): if we pick any arbitrary y1, y2, . . . , yr−2, we are left with +yr−1 ≡ gk1 − +r−2 +� +i=1 +yi +(mod b). +The right-hand side of this equation defines some equivalence class ℓk1 (mod b) from which we must choose +yr−1 to ensure that y ≡ gk1 (mod b). Exactly +1 +m1 of the possible values of yr−1 place us in the correct +equivalence class for a given g. +Thus, y falls into the equivalence classes k1, 2k1, . . . , m1k1 with equal +probability. +ks +s1 +s2 +m +c − d +c + (r − 1)d +b +n +n/d1 +n/d2 +Figure 3: A visualization of our factorization of b = gcd(r, n), where the overlap of any circle or shaded +region with the circle for b contains a factorization for their greatest common divisor. Note that the purple +region is the overlap of the red and blue regions. +Likewise, z (mod b) can be any of the m2 multiples of k2 modulo b with equal probability. To interface +between y and z, we must factor further: let ks = gcd(k1, k2) so that k1 = s1ks and k2 = s2ks, and +gcd(s1, s2) = 1. Letting m = gcd(m1, m2), factor b completely as b = ms1s2ks. The reader may find it +helpful to refer to Figure 3, which provides a visualization of how this factorization relates b, n +d1 , and +n +d2 . +We now identify the proportion of y- and z-values that satisfy z − y ≡ 0 (mod b). Since y, z, b all contain +a factor of ks, let y = αk1 = αs1ks and z = βk2 = βs2ks for some α, β ∈ Z. Then, equivalently, we seek the +proportion of (α, β) pairs such that +βs2 ≡ αs1 +(mod ms1s2). +Since s1, s2 are coprime, we must have α = as2 for some a ∈ Z. Exactly +1 +s2 of the possible α-values are +multiples of s2. Then +βs2 ≡ as1s2 +(mod ms1s2) +which has solutions only for β ≡ as1 (mod b). Out of the m2 = ms1 equivalence classes that z can fall into, +only the one defined by β = as1 works. Therefore, +Frb(d1, d2, n) = 1 +s2 +� 1 +ms1 +� += +1 +ms1s2 += ks +b = +gcd +� +n +d1 , n +d2 , b +� +b +. +12 + +Since b = gcd(r, n), we have gcd +� +n +d1 , n +d2 , b +� += gcd +� +n +d1 , n +d2 , r +� +, which completes proof of Theorem 5.2 and +allows us to express Scoroot solely in terms of c, d, r, and n. +6 +Proof of Main Theorem 1 Part 2 +We are now ready to prove the second part of our main result. In this section, we show how the coroot +equations are obtained from the cocharacter equations, and how this relates Scoroot and Scochar. +Theorem 6.1. The number of solutions to the cocharacter equations can also be defined as +Scochar = Scoroot · 1 +n · lcm +� +gcd +� +d2, dn +d1 +� +, +n +gcd(n, r) +� +. +Let M(c, d, r, n) := lcm +� +gcd +� +d2, dn +d1 +� +, +n +gcd(n,r) +� +. We will also prove the following formula for M(c, d, r, n), +which will be useful for our investigation into special dimensions for W in Section 7. +Proposition 6.2. Let r, n have prime factorizations r = pℓ1 +1 pℓ2 +2 . . . pℓj +j +and n = pm1 +1 pm2 +2 +. . . pmj +j +. For every +1 ≤ i ≤ j, let +(c − d) ≡ cipsi +i +(mod pmi +i ) and d ≡ dipti +i +(mod pmi +i +) for each 1 ≤ i ≤ j +so that 0 ≤ si, ti ≤ mi. Let µi = min(mi, ℓi) and ci, di are relatively prime to pi. Then +M(c, d, r, n) = +j� +i=1 +pmax(mi−µi,min(si,ti+mi−si)) +i +. +To obtain the coroot equations from the cocharacter equations, we can multiply the matrix Bc,d which +defines the cocharacter equations by the r × r matrix +Lr = + + + + + + + +1 +0 +. . . +0 +−1 +0 +1 +. . . +0 +−1 +... +... +... +... +0 +0 +1 +−1 +1 +1 +. . . +1 +1 + + + + + + + +. +That is, the new system of equations specified by this transformation is +LrBc,d · x ≡ 0r +(mod n) +which gives us precisely the coroot equations. Likewise, multiplying the coroot equations by +L′ +r = + + + + + + + +r − 1 +−1 +. . . +−1 +1 +−1 +r − 1 +. . . +−1 +1 +... +... +... +... +−1 +−1 +r − 1 +1 +−1 +−1 +. . . +−1 +1 + + + + + + + +obtains the cocharacter equations multiplied by r. That is, +L′ +r(LrBc,d) · x ≡ r · Bc,d · x ≡ 0r +(mod n). +As we discussed in Remark 5.3, if r and n are relatively prime, then r has an inverse r−1 in Z/nZ and +the cocharacter and coroot equations are equivalent. However, if r is not invertible modulo n, going back +from the coroot equations to the cocharacter equations is more complicated. +13 + +Recall that b = gcd(r, n). Then r · Bc,dx ≡ 0r (mod n) factors into +b +�r +b +� + + + + + + + +c +d +d +. . . +d +d +c +d +. . . +d +d +d +c +. . . +d +... +... +... +... +d +d +d +. . . +c + + + + + + + + + + + + + + +x1 +x2 +x3 +... +xr + + + + + + + +≡ + + + + + + + +0 +0 +0 +... +0 + + + + + + + +� +mod b · n +b +� +. +Since both sides of the equation and the modulus are multiples of b, this implies that +�r +b +� +Bc,d · x ≡ 0r +� +mod n +b +� +. +The number r +b is relatively prime to n +b and therefore invertible in Z/ n +b Z, so +Bc,d · x ≡ 0r +� +mod n +b +� +. +(7) +Remark 6.3. Again, (7) shows that the coroot and cocharacter equations are equivalent when r and n are +relatively prime, since then n = n/b and (7) recovers exactly the cocharacter equations. +If we are not in that case, i.e., if b ̸= 1, then for any coroot solution x, we get +Bc,dx ≡ n +b v +(mod n) +for some vector v ∈ Zr. We now show that each solution to the coroot equations also satisfies inhomogeneous +cocharacter equations for particular values of a. +Lemma 6.4. The coroot equations are equivalent to the inhomogeneous cocharacter equations with the con- +dition that a ∈ (n/b)Z. +Proof. Let x be a solution to the coroot equations and let the ith row of the left-hand side of Equation (7) +be +wi = cxi + +� +j̸=i +dxj. +Then by definition, +wi − wr = (c − d)(xi − xr) ≡ 0 +(mod n). +Thus, for some k ∈ {1, ..., b}, +Bc,dx ≡ n +b k · 1r +(mod n). +(8) +Similarly, suppose x satisfies (8). Then defining wi as above, +(c − d)(xi − xr) ≡ wi − wr ≡ k n +b − k n +b ≡ 0 +(mod n) +(c + (r − 1)d)(x1 + x2 + . . . xr) ≡ w1 + w2 + · · · + wr ≡ r +� +k n +b +� +≡ 0 +(mod n). +In Section 4 we showed that Equation (8) has solutions if and only if k n +b is a multiple of A(c, d, r, n), and +that each class of solutions (defined by having the same k) is of the same size. As in that section, we want +to find the smallest nonzero k for which (8) has a solution. +Definition. Let κ(c, d, r, n) be the smallest positive value of k such that there is a solution to Equation (8). +Again, it will often be clear from context that we are working with a particular fixed c, d, r, n in which +case we will write κ and M for brevity instead of κ(c, d, r, n) and M(c, d, r, n). +14 + +Proof of Theorem 6.1. We can relate the values of κ(c, d, r, n) and A(c, d, r, n) as follows: +κ · n +b = lcm +� +A, n +b +� += lcm +� +gcd +� +d2, dn +d1 +� +, n +b +� +. +Let M(c, d, r, n) := lcm +� +gcd +� +d2, dn +d1 +� +, n +b +� +. Then there are +n +M = b +κ equivalence classes of solutions to Equa- +tion (8). +Exactly one of these equivalence classes—the one given by k = b—gives the solutions to the +cocharacter equations. Therefore, +Scochar = Scoroot · M +n = Scoroot · κ +b . +Substituting in our earlier expressions for the values of Scoroot and M, we obtain +Scochar = dr−1 +1 +d2 +n +gcd +� n +d1 +, n +d2 +, r +� +lcm +� +n +gcd(r, n), gcd +� +d2, dn +d1 +�� +. +Although it is not immediately clear from looking at this equation, this formula is equivalent to the +one given in Theorem 4.7. One area of future work would be to simplify this expression and more directly +understand why it is equivalent to the statement of Theorem 4.7. Furthermore, while this formula appears +more complicated than that of Theorem 4.7, this approach is perhaps more suitable for extending past +GLr(F), as the metaplectic Whittaker functions developed in Section 2 can be defined over any reductive +group and this approach is more closely related to the root data structure of reductive groups. +Using the same visualization tables we used in Section 3 for Scochar shows more directly how M and κ +change as we vary c, d, r, and n. Here, for a fixed r, n, let the entry in position (c, d) be κ(c, d, r, n). (To +achieve a matching table for M, multiply the κ table by n/b.) +Example. For n = 8 = 23 and r = 2ℓ, the following tables show how κ changes as ℓ increases from 1 to 3. +Because M and κ depend on µ = min(ℓ, m) rather than on ℓ, any κ table for ℓ > 3 would be identical to the +table for ℓ = 3. +2 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +4 +1 +1 +1 +2 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +2 +1 +1 +1 +1 +1 +1 +1 +1 +1 +2 +1 +1 +1 +2 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +2 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +8 +1 +2 +1 +4 +1 +2 +1 +1 +1 +1 +2 +1 +2 +1 +2 +2 +1 +2 +1 +2 +1 +4 +1 +1 +2 +1 +1 +1 +2 +1 +2 +4 +1 +2 +1 +4 +1 +2 +1 +1 +2 +1 +2 +1 +1 +1 +2 +2 +1 +4 +1 +2 +1 +2 +1 +1 +2 +1 +2 +1 +2 +1 +1 +n = 8, r = 2 +n = 8, r = 4 +n = 8, r = 8 +The entries in these tables are determined by the main diagonals they lie on, which are described by s, and +the columns they lie in, which are described by t and index how far down the main diagonal an entry is. In +particular, notice that the only difference between the matrices for ℓ and ℓ + 1 is that a specific fraction of +the elements on each of the diagonals in the latter matrix have been multiplied by 2. For example, for ℓ = 2, +this fraction is 1/4 for the red diagonal and 1/2 for the blue. +These tables are also useful for visualizing the effect of combining distinct primes. +Example. When r = 22 and n = 22 · 3, we see that the table for κ is a 3 × 3 tessellation of that for +15 + +r = 22, n = 22: +4 +1 +2 +1 +1 +1 +1 +2 +2 +1 +2 +1 +1 +2 +1 +1 +r = 22, n = 22 +4 +1 +2 +1 +4 +1 +2 +1 +4 +1 +2 +1 +1 +1 +1 +2 +1 +1 +1 +2 +1 +1 +1 +2 +2 +1 +2 +1 +2 +1 +2 +1 +2 +1 +2 +1 +1 +2 +1 +1 +1 +2 +1 +1 +1 +2 +1 +1 +4 +1 +2 +1 +4 +1 +2 +1 +4 +1 +2 +1 +1 +1 +1 +2 +1 +1 +1 +2 +1 +1 +1 +2 +2 +1 +2 +1 +2 +1 +2 +1 +2 +1 +2 +1 +1 +2 +1 +1 +1 +2 +1 +1 +1 +2 +1 +1 +4 +1 +2 +1 +4 +1 +2 +1 +4 +1 +2 +1 +1 +1 +1 +2 +1 +1 +1 +2 +1 +1 +1 +2 +2 +1 +2 +1 +2 +1 +2 +1 +2 +1 +2 +1 +1 +2 +1 +1 +1 +2 +1 +1 +1 +2 +1 +1 +r = 22, n = 12 = 3 · 22 +Example. Below are the κ tables for n = 6 and r = 2, 3, 6, respectively. Note that the table for r = 6 is +obtained by multiplying the tables for r = 2 and r = 3 together elementwise. Upon proving Proposition 6.2, +we will see that this is true in greater generality. +2 +1 +2 +1 +2 +1 +1 +1 +1 +1 +1 +1 +2 +1 +2 +1 +2 +1 +1 +1 +1 +1 +1 +1 +2 +1 +2 +1 +2 +1 +1 +1 +1 +1 +1 +1 +r = 2, n = 6 +3 +1 +1 +3 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +3 +1 +1 +3 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +r = 3, n = 6 +6 +1 +2 +3 +2 +1 +1 +1 +1 +1 +1 +1 +2 +1 +2 +1 +2 +1 +3 +1 +1 +3 +1 +1 +2 +1 +2 +1 +2 +1 +1 +1 +1 +1 +1 +1 +r = 6, n = 6 +. +Expressing the values of M and κ directly in terms of the prime factors of n, r, c and d, we are ready to +prove Proposition 6.2. +Proof of Proposition 6.2. Let r = pℓ1 +1 pℓ2 +2 . . . pℓj +j +and n = pm1 +1 pm2 +2 +. . . pmj +j +, where some of ℓi or mi may be 0. +Recalling the definition of d2, we have that +M = lcm +� +gcd +� +c + (r − 1)d, n, dn +d1 +� +, n +b +� +. +Thus, the only prime factors of M are those that are prime factors of n, and M is multiplicative over these +prime powers. Considering only the power of pi arising in M for some i ∈ {1, j}, let +(c − d) ≡ cipsi +i +(mod pmi +i ) and d ≡ dipti +i +(mod pmi +i ) for each 1 ≤ i ≤ j +so that 0 ≤ si, ti ≤ mi, and ci, di are relatively prime to pi. Let µi = min(mi, ℓi). Then, n/b = bipmi−µi +i +, +where bi is relatively prime to pi. +Then consider the power of pi arising from gcd +� +c + (r − 1)d, n, dn +d1 +� +: recalling that d1 = gcd(c − d, n), +the power of pi in d1 is min{si, mi} = si. Then, the power of pi in dn/d1 is ti + mi − si. It remains to +consider the power arising in the first component of the gcd. +Consider c + (r − 1)d = c − d + r · d. Substituting in the factorizations, we have +c + (r − 1)d ≡ cipsi +i + ridipti+µi +i +(mod pmi +i ), +where ri is relatively prime to pi. +We consider three cases. If si < ti + µi, the power of pi in this component is si. Thus, the power of pi +in the gcd is min{si, ti + mi − si}, since si ≤ mi. The power of pi in M is then +max{mi − µi, min{si, ti + mi − si}}. +16 + +Next suppose that si > ti + µi. Then we have that the power of pi in the gcd is min{ti + µi, ti + mi − si}. +However, then the power of pi in M is +max{mi − µi, min{ti + µi, ti + mi − si}} = mi − µi, +since si > ti + µi, so ti + mi − si < mi − µi. +Lastly, if si = ti + µi, then +c − d + rd ≡ psi +i (ci + ridi) +(mod pmi +i ). +Note that ci + ridi may create an additional factor pτi +i +for some integer τi ≥ 0. +Then the power of pi +appearing in the gcd is +min{si + τi, mi, ti + mi − si} = min{si + τi, mi − µi} +Then the power of pi in M is +max{min{si + τi, mi − µi}, mi − µi} = mi − µi. +Collecting the three cases together, the expression +max{mi − µi, min{si, ti + mi − si}} +matches the power of pi in M in each case, completing the proof of the proposition. +Corollary 6.5. The quantity κ(c, d, r, n) is multiplicative over powers of distinct primes. +In the next section, we will see that the two different approaches for Scochar are each useful in different +ways. One potentially fruitful avenue for future exploration would be to see precisely why these two formulae +are equal, as it is not easily apparent. As the second approach relates more directly to the root structure of +GLr as a reductive group, but the first approach yields a simpler formula and proof, this connection would +illuminate a way to extend the simpler formula to general reductive groups. +7 +Structure of the Whittaker Space +These investigations into the structure of Λfin not only give us a method of calculating dim(W), they also +illuminate how the parameters c, d, r, and n affect the structure of W in different ways. In this section, we +start with a few natural corollaries to both parts of Main Theorem 1 (Theorems 4.7 and 6.1) and discuss +how they relate to the literature. We then develop necessary and sufficient conditions for dim(W) to be +of maximum and minimum dimension, as well as the conditions for several other desirable dimensions for +further connections. +Corollary 7.1. From Theorem 4.7, we have the following natural results about dim(W): +dim(W) = + + + +� +n +gcd(c,n) +�r +if d ≡ 0 +(mod n) +� +n +gcd(d,n) +�r−1 +· +n +gcd((r−1)d,n) +if c ≡ 0 +(mod n) +Proof. Recall that by Theorem 1.1, we have dim(W) = nr/|Scochar(c, d, r, n)|. Then if d ≡ 0 (mod n), +Scochar(c, d, r, n) = gcd(c − d, n)r−1 gcd +� +c + (r − 1)d, n, +dn +gcd(c − d, n) +� += gcd(c, n)r. +Likewise, if c ≡ 0 ≡ n, then +Scochar(c, d, r, n) = gcd(−d, n)r−1 gcd +� +(r − 1)d, n, +dn +gcd(−d, n) +� += gcd(d, n)r−1 gcd((r − 1)d, n). +17 + +As we can see from this corollary, the parameters c and d play significantly different roles in influencing +the structure of the Whittaker function space. In the simplest n-fold metaplectic cover (c = 1, d = 0), we see +| �T| = nr, which allowed Brubaker, Bump, and Buciumas to map W isomorphically to a quantum module +of dimension nr in [1] to explain the lattice model phenomena discovered by Brubaker, Bump, Chinta, +Friedberg, and Gunnells [2]. In the same spirit, the second author showed in [7] that this connection extends +quite naturally to an isomorphism for any cover coming from a diagonal matrix (i.e., d ≡ 0). However, +incorporating the parameter d adds complications, as the quantum module (which we will discuss later in +Section) does not see the factor of gcd +� +c + (r − 1)d, n, +dn +gcd(c−d,n) +� +appearing in dim(W). Thus, to understand +this connection, we will need additional information about the structure of W. +Corollary 7.2. We have dim(W) = 1 (that is, of minimum size) if and only if c ≡ d ≡ 0 (mod n). +Proof. The backward direction follows from Corollary 7.1. Now assume Scochar = nr. Since each of the r +factors in Theorem 4.7 are factors of n, we must have d1 = gcd(c − d, n) = n and so +Scochar = nr−1 gcd (c − (r + 1)d, n, c, d) . +So we must also have gcd(n, c, d) = n, which requires that c, d ≡ n (mod n). +Corollary 7.3. We have dim(W) = nr (that is, of maximum size) if and only if c − d and c + (r − 1)d are +coprime to n. +Proof. It suffices to show that Scochar = 1 if and only if d1 = d2 = 1. The backwards direction is easiest to +see from Theorem 6.1: if d1 = d2 = 1, then +Scochar = 1 +n gcd(r, n) · lcm +� +n +gcd(r, n), 1 +� += 1. +For the forward direction, we use Theorem 4.7. Here, Scochar = 1 implies both dr−1 +1 += 1 (and thus +d1 = 1) and gcd +� +d2, dn +d1 +� += 1. Since d1 = 1 we then have gcd (d2, dn) = 1 which tells us that d2 must be +relatively prime to n. But d2 = gcd(c + (r − 1)d, n) so thus d2 = 1. +We will later see that both maximizing and minimizing W result in very nice quantum connections. +It is also intriguing to ask when the diagonal number phenomenon developed in Section 3 matches the +dimension precisely: that is, when is |Λfin| = dr−1 +1 +d2? One case in which this is true is fairly straightforward. +Corollary 7.4. If n and r are relatively prime, dim(W) = nr/ +� +dr−1 +1 +d2 +� +. +Proof. Suppose gcd(r, n) = 1 and consult Theorem 6.1. Then, +Scochar = dr−1 +1 +d2 +n +· lcm +� +n, gcd +� +c + (r − 1)d, n, dn +d1 +�� += dr−1 +1 +d2 +as the gcd above is a factor of n. +However, the general conditions are a bit more complicated. +Proposition 7.5. Suppose n = pm1 +1 +· · · pmj +j +,, and for each pi, we have c − d ≡ cipsi +i +(mod pmi +i ), d ≡ dipti +i +(mod pmi +i ), and r ≡ ripµi +i +(mod pmi +i ), where ci, di, and ri are coprime to pi. Then we have dim(W) = +nr/ +� +dr−1 +1 +d2 +� +if and only if one of the following three conditions is true for every i: +• si < ti + µi and 2si ≤ ti + mi, +• si > ti + µi and si ≤ mi − µi, or +• si = ti + mi and 2si + τi ≤ ti + mi, where τi is the number of powers of pi in ci + ridi. +18 + +Proof. Using Theorem 4.7, Scochar = dr−1 +1 +d2 if and only if gcd(d2, dn +d1 ) = d2. That is, precisely when d2 +divides dn +d1 . Since the left side divides n, it suffices to check that for every prime factor pi of n, the power of +pi in d2 divides that in dn +d1 . Given any pi, by the proof of Proposition 6.2, we know that the power of pi on the +right hand side is ti+mi−si. Similarly, the power of pi appearing in c−d+rd is min{si, ti+µi}+τi·δsi=ti+µi, +where τi is the power of pi appearing in ci + ridi. Thus, it suffices to determine exactly when +min{si, ti + µi} + τi · δsi=ti+µi ≤ ti + mi − si. +(9) +To do so, we split into the same cases we used in the proof of Proposition 6.2, based on the power of pi +appearing in d2. First, suppose si < ti + µi. Then we wind up in the first condition, because (9) is true +precisely when +si ≤ ti + mi − si. +Then, suppose si > ti + µi. Then (9) is true if and only if +µi ≤ mi − si +satisfying the second condition. Lastly, suppose si = ti + µi. Then (9) is equivalent to the third condition +2si + τi ≤ ti + mi. +Using the same techniques, we can also describe all the cases when dim(W) = (n/d1)r. As we will see +later, the quantum module connected to W has dimension (n/d1)r, so this is a necessary condition for the +map to be an isomorphism. +Proposition 7.6. Suppose n = pm1 +1 +· · · pmj +j +,, and for each pi, we have c − d ≡ cipsi +i +(mod pmi +i ), d ≡ dipti +i +(mod pmi +i ), and r ≡ ripµi +i +(mod pmi +i +), where ci, di, and ri are coprime to pi. Then dim(W) = (n/d1)r if and +only if for every i, we have 2si ≤ mi + ti and at least one of the following conditions: +• si < ti + µi, +• si = ti + µi and 2si = ti + mi, or +• si = ti + µi and c + (r − 1)d contains no additional powers of pi. +Proof. Using Theorem 4.7, dim(W) = (n/d1)r if and only if gcd(d2, dn +d1 ) = d1. Using the machinery developed +in the proof of Proposition 6.2, notice that both sides are factors of n, so it suffices to check that the powers +of each prime pi appearing in the prime factorization of n match. +Let n = pm1 +1 +· · · pmj +j +and suppose that c − d ≡ cipsi +i +(mod pmi +i ) and d ≡ dipti +i +(mod pmi +i ), where ci and +di are coprime to pi. Also, note that r ≡ ripµi +i +(mod pmi +i ), where ri is also coprime to pi. Then the power +of pi appearing in d1 is si. From the proof of Proposition 6.2, recall that the power of pi appearing in dn +d1 is +ti + mi − si and the power of pi appearing in c − d + rd is min{si, ti + µi} + τi · δsi=ti+µi, where τi is the +power of pi appearing in ci + ridi. So gcd(d2, dn +d1 ) = d1 if and only if +min{min{si, ti + µi} + τi · δsi=ti+µi, ti + mi − si} = si. +(10) +As in the proof of Proposition 6.2, we split into three cases. If si < ti + µi, then (10) gives us +min{si, ti + mi − si} = si, +which is true precisely when ti + mi − si ≥ si, satisfying the first conditions. +If si > ti + µi, then we have a contradiction, since the minimum in (10) is already less than si, and vice +versa. +Finally, if si = ti + µi, then (10) is +min{si + τi, ti + mi − si} = si, +which is true exactly when 2si = ti + mi or τi = 0 and si ≤ ti + mi − si, satisfying the second and third +conditions, respectively. +19 + +We have seen in this section that the two different formulations of Theorems 4.7 and 6.1 are useful for +many different purposes. While the approach used to generate Theorem 6.1 provides a more natural path +for generalization beyond GLr(F), it would be interesting in future work to investigate whether there is an +analogous approach to that used in Theorem 4.7 for other reductive groups. In particular, understanding how +Theorems 4.7 and 6.1 are related for the case of GLr(F) will illuminate a path for extending this connection +further. +8 +Quantum Connections +Finally, we marshal together results from the previous sections to investigate how the space of Whittaker +functions is connected to quantum group modules, building the necessary quantum definitions along the way. +Let Uq(c, d, n) be the affine quantum group Uq(�gl(n/d1)), where q is the cardinality of the residue field +for our nonarchimedean local field F. For the results of this paper, we will not need the precise definition +here, so we refer the reader to Chari and Pressley [6] for the details of the construction and instead note +merely a few interesting facts about Uq(c, d, n). First, despite the name, Uq(c, d, n) is not a group, but rather +an algebra, specifically a quasitriangular Hopf algebra. That is, it is both an algebra and a coalgebra, so +it comes equipped with not only multiplication and a unit map but also comultiplication, a counit, and an +antipode map relating the algebra and coalgebra structures. Furthermore, this quantum group has a very +nice set of modules which we can model concretely. +Definition. For z ∈ C, let V+(z) be an evaluation module, or evaluation representation, for Uq(c, d, n). +Again, we will not need the full structure of this representation for this paper, but following Kojima [12] as +a convenient source, note that V+(z) is n/d1-dimensional and its basis may be parametrized by the elements +of Z/(n/d1)Z. +In addition, Uq(c, d, n) comes with an invertible element called a universal R-matrix R ∈ Uq(c, d, n) ⊗ +Uq(c, d, n), which acts on tensor products of Uq(c, d, n)-modules. Choosing a particular pair of modules and +their bases, R becomes an honest-to-goodness matrix. +It is this R-matrix that sparked the connection between Whittaker functions and quantum groups: R- +matrices are natural sources for solutions to Yang-Baxter equations, functional relations from statistical +mechanics that that arise, among other places, in the theory of lattice models. In [2], Brubaker, Bump, +Chinta, Friedberg, and Gunnells constructed a ice-type lattice model called Metaplectic Ice which computes +metaplectic Whittaker functions for the nicest cover (c = 1, d = 0, so d1 = d2 = 1). However, the Yang-Baxter +equation for this model was unknown until Brubaker, Bump, and Buciumas identified it as a Drinfeld twist +of the R-matrix for Uq(�gl(n)) in [1]. Using the lattice model as a bridge, they mapped the space of Whittaker +functions on this cover isomorphically into the tensor product V+(z1) ⊗ · · · ⊗ V+(zr), where zi ∈ C are the +Satake parameters for the principal series representation on which the Whittaker function space W = Wz +is built. Under this isomorphism, the action of the R-matrix on the components of V+(z1) ⊗ · · · ⊗ V+(zr) +matches precisely the action of intertwining operators on the principal series representation and thus the +Whittaker function space. +Fantastically, this connection extends for any metaplectic cover of GLr(F). In [7], the second author +built a generic lattice model for an arbitrary covering group, and used it to construct a map between the +space of Whittaker functions and a quantum group module for the quantum group Uq(c, d, n) = Uq(�gl(n/d1)). +However, as we saw already from the formulae for dim(W) and the structure theory in Section 7, changing the +parameters c and d results in a significantly more complicated function space. These complications extend +to the map, as the quantum space changes differently than W does. In spite of this, the map prescribed by +the lattice model in the fully general case is still a homomorphism and it matches exactly the actions of the +R-matrix on the right side to those of the intertwining operators on the left. +Consider the tensor product V+(z1) ⊗ · · · ⊗ V+(zr) of quantum group evaluation modules for U. As a +vector space, we have +dim +� +V+(z1) ⊗ · · · ⊗ V+(zr) +� += +� n +d1 +�r +. +Note that unlike either of the formulae for dim(W) in Main Theorem 1, this formula is not affected by d2. +20 + +Now we come to the connection precisely. Using Theorem 1.1, take representatives for the cosets �T/H +from the set (Z/nZ)r. Using Theorem 2.2, use these representatives to construct a basis for W. +Theorem 8.1 (Frechette [7], Theorem 1.1). Let ρ = (r − 1, ..., 2, 1, 0). For z ∈ Cr, the map +θz : Wz → V+(z1) ⊗ · · · ⊗ V+(zr) +ν +�→ +ρ − ν +(mod n/d1), +where the modulus is taken in each component, is a homomorphism compatible with the actions of intertwining +operators on W and the R-matrix on the quantum tensor product. +One of the difficulties that arose in extending from the nicest cover to generic covers is that the lattice +model specifies a choice of basis for W that makes this map a homomorphism, but the lattice model itself is +not necessary for the proof and serves as a removable bridge between the Whittaker function space and the +quantum group model. Without the lattice model, however, there is no canonical choice of basis for W, so +we ask: when is this map well-defined regardless of the choice of representative for each coset in �T/H? +Using the structure of W developed in Section 7, we can investigate this map more precisely, and arrive +at the following theorem, which is a restatement of the first part of Main Theorem 2. +Theorem 8.2. For the metaplectic cover �Gc,d,r,n, the map θz : W → V+(z1) ⊗ · · · ⊗ V+(zr) from Theorem +8.1 is well-defined independent of choice of coset representatives for �T/H if and only if +gcd +� +d2, dn +d1 +� += gcd(c, d, n). +Proof. Using the characterization developed in Theorem 6.1, θz is well-defined if and only if all the elements +in Λfin map to the same element in the module. +Using the description of Proposition 4.4, write x = +x1 · 1r + n +d1 (0, v2, v3, ..., vr) and y = y1 · 1r + n +d1 (0, v′ +2, v′ +3, ..., v′ +r), for x1, y1, vi, v′ +i ∈ Z for all i. Thus, +θz(x) − θz(y) = (y1 − x1) · 1r +(mod n/d1). +Since x, y ∈ Λfin, we have y − x ∈ Λfin as well, so the defining cocharacter equations give more +information about the possible values of y1 − x1. Using the first cocharacter equation, there exists k ∈ Z +such that +(c + (r − 1)d) · (y1 − x1) ≡ dn +d1 +· k +(mod n). +Varying over all x, y ∈ Λfin, the possible values for the right hand side of this equation are precisely the +integer multiples of gcd +� +n, dn +d1 +� +. Then, both sides must be a multiple of lcm +� +c + (r − 1)d, gcd +� +n, dn +d1 +�� +. +Using the fact that lcm(A, B) = (A · B)/ gcd(A, B), the possible values for y1 − x1 are all the integer +multiples of +gcd +� +n, dn +d1 +� +gcd +� +c + (r − 1)d, n, dn +d1 +� = +n +d1 gcd (d1, d) +gcd +� +d2, dn +d1 +� = n +d1 +· gcd (c, d, n) +gcd +� +d2, dn +d1 +�. +(11) +Going back to the map, θz(x) − θz(y) = 0 if and only if y1 − x1 ≡ 0 (mod n/d1). Since gcd(c, d, n) +divides both d2 and dn +d1 , we have that the expression in (11) is a multiple of +n +d1 if and only if gcd(c, d, n) = +gcd +� +d2, dn +d1 +� +. Therefore the map θz is well-defined for any choice of coset representatives of �T/H if and only +if gcd(c, d, n) = gcd +� +d2, dn +d1 +� +. +Corollary 8.3. When W is either maximum or minimum size, θz is an isomorphism. +21 + +Proof. If W is of maximum size nr, then by Corollary 7.3, we have d1 = d2 = 1. Thus, gcd +� +d2, dn +d1 +� += 1, +which forces gcd(c, d, n) = 1, so θz is well-defined. In this case �T/H is parametrized by all of (Z/nZ)r, and +since n/d1 = n, so is the quantum module. Looking at the description of θz in Theorem 8.1, we see that θz +is an isomorphism by definition, flipping W and shifting by ρ. +If W is of minimum size 1, then Corollary 7.2 shows that d1 = d2 = n. Thus, gcd(c, d, n) = n, which +forces gcd +� +d2, dn +d1 +� += n and makes θz well-defined. Here, �T /H is a single element, which maps to the single +element 0 in (Z/(n/d1)Z)r, since n/d1 = 1. Thus the map is vacuously an isomorphism. +Note that the first case of Corollary 8.3 includes the nicest cover c = 1, d = 0 originally treated by [2] +and [1], explaining why the quantum map on W for this case is an isomorphism. +Using our description of Λfin, we intend in the future to come up with a precise description of the +structure of W in the style of Corollary 8.3 for more general cases, which will allow us to characterize the +precise behavior of θz. In particular, we are interested in providing a companion to Proposition 7.6 by finding +a sufficient condition for all cases when θz is an isomorphism. Extending our methods and results for W from +GLr(F) to arbitrary reductive groups will then give us more information about what the quantum objects +connected to other types of reductive groups should be. While some solvable lattice models for other types +exist, they have not yet been linked to modules for quantum groups or other quantum algebraic objects, +so we believe that investigating the dimension and description of W for other types will illuminate likely +candidates for broader quantum connections. +References +[1] Ben Brubaker, Valentin Buciumas, Daniel Bump, and Nathan Gray, A Yang-Baxter equation for meta- +plectic ice, Commun. Number Theory Phys. 13 (2019), no. 1, 101–148. MR 3951106 +[2] Ben Brubaker, Daniel Bump, Gautam Chinta, Solomon Friedberg, and Paul E. Gunnells, Meta- +plectic ice, Multiple Dirichlet series, L-functions and automorphic forms, Progr. Math., vol. 300, +Birkh¨auser/Springer, New York, 2012, pp. 65–92. MR 2952572 +[3] +, Metaplectic ice, Multiple Dirichlet series, L-functions and automorphic forms, Progr. Math., +vol. 300, Birkh¨auser/Springer, New York, 2012, pp. 65–92. MR 2952572 +[4] Ben Brubaker, Daniel Bump, and Solomon Friedberg, Weyl group multiple Dirichlet series: type A +combinatorial theory, Annals of Mathematics Studies, vol. 175, Princeton University Press, Princeton, +NJ, 2011. MR 2791904 +[5] Jean-Luc Brylinski and Pierre Deligne, Central extensions of reductive groups by K2, Publ. Math. Inst. +Hautes ´Etudes Sci. (2001), no. 94, 5–85. MR 1896177 +[6] Vyjayanthi Chari and Andrew Pressley, A guide to quantum groups, Cambridge University Press, Cam- +bridge, 1994. MR 1300632 +[7] Claire Frechette, Yang-baxter equations for general metaplectic ice, arxiv:2009.13669, 2020. +[8] Wee Teck Gan and Fan Gao, The Langlands-Weissman program for Brylinski-Deligne extensions, no. +398, 2018, L-groups and the Langlands program for covering groups, pp. 187–275. MR 3802419 +[9] Wee Teck Gan, Fan Gao, and Martin H. Weissman, L-groups and the Langlands program for covering +groups: a historical introduction, no. 398, 2018, L-groups and the Langlands program for covering +groups, pp. 1–31. MR 3802417 +[10] I. M. Gel’fand and D. A. Kajdan, Representations of the group GL(n, K) where K is a local field, Lie +groups and their representations (Proc. Summer School, Bolyai J´anos Math. Soc., Budapest, 1971), +Halsted, New York, 1975, pp. 95–118. MR 0404534 +[11] D. A. Kazhdan and S. J. Patterson, Metaplectic forms, Inst. Hautes ´Etudes Sci. Publ. Math. (1984), +no. 59, 35–142. MR 743816 +22 + +[12] Takeo Kojima, Diagonalization of transfer matrix of supersymmetry Uq(�sl(M + 1|N + 1)) chain with a +boundary, J. Math. Phys. 54 (2013), no. 4, 043507, 40. MR 3088809 +[13] Hideya Matsumoto, Sur les sous-groupes arithm´etiques des groupes semi-simples d´eploy´es, Ann. Sci. +´Ecole Norm. Sup. (4) 2 (1969), 1–62. MR 0240214 +[14] Peter J. McNamara, Metaplectic Whittaker functions and crystal bases, Duke Math. J. 156 (2011), +no. 1, 1–31. MR 2746386 +[15] +, Principal series representations of metaplectic groups over local fields, Multiple Dirichlet series, +L-functions and automorphic forms, Progr. Math., vol. 300, Birkh¨auser/Springer, New York, 2012, +pp. 299–327. MR 2963537 +[16] J¨urgen Neukirch, Algebraic number theory, Grundlehren der Mathematischen Wissenschaften [Funda- +mental Principles of Mathematical Sciences], vol. 322, Springer-Verlag, Berlin, 1999, Translated from +the 1992 German original and with a note by Norbert Schappacher, With a foreword by G. Harder. MR +1697859 +[17] I. I. Pjateckij-ˇSapiro, Euler subgroups, Lie groups and their representations (Proc. Summer School, +Bolyai J´anos Math. Soc., Budapest, 1971), Halsted, New York, 1975, pp. 597–620. MR 0406935 +[18] J. A. Shalika, The multiplicity one theorem for GLn, Ann. of Math. (2) 100 (1974), 171–193. MR 348047 +[19] Andr´e Weil, Sur certains groupes d’op´erateurs unitaires, Acta Math. 111 (1964), 143–211. MR 165033 +23 + diff --git a/LNE0T4oBgHgl3EQfSgB1/content/tmp_files/load_file.txt b/LNE0T4oBgHgl3EQfSgB1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..53548a7eaf740de4a6f5243aee6c6b56095dea24 --- /dev/null +++ b/LNE0T4oBgHgl3EQfSgB1/content/tmp_files/load_file.txt @@ -0,0 +1,1134 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf,len=1133 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='02223v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='NT] 5 Jan 2023 Measuring the Space of Metaplectic Whittaker Functions Ilani Axelrod-Freed, Claire Frechette, and Veronica Lang January 6, 2023 Abstract Whittaker functions are special functions that arise in p-adic number theory and representation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' They may be defined on representations of reductive groups as well as their metaplectic covering groups: fascinatingly, many of their number theoretic applications survive the transition between the reductive and metaplectic cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' However, one notable difference is that the space of Whittaker functions on a reductive group over a nonarchimedean local field F is one-dimensional, whereas this is no longer true in the metaplectic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' In a previous paper, the second author showed that the dimension of the space of Whittaker functions on an arbitrary n-fold metaplectic cover of GLr(F) can be counted in terms of the number of solutions to a particular system of linear Diophantine equations in terms of n and r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' In this paper, we calculate two precise formulae for dim(W), one inspired by viewing this system as a homogenous specialization of an inhomogenous system and the other by the structure of the coroot lattice of GLr(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Then we use these formulae to investigate a homomorphism between W and a particular quantum group module, built by the second author in a previous paper, and show precisely when this map is well-defined for any choice of basis for W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' 1 Introduction Whittaker functions arise in p-adic number theory and representation theory, specifically in the study of automorphic forms over local fields and the study of principal series representations of reductive groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' They can be written in many forms: as integrals over matrix groups, as generating functions over many different combinatorial objects, as coefficients of automorphic forms, and in some cases as partition functions of lattice models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' In particular, when the lattice model is solvable, this viewpoint leads to a surprising connection between the algebraic structures of the space of Whittaker functions and of modules for quantum groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' One type of Whittaker functions of particular interest are metaplectic Whittaker functions, which are Whittaker functions on the principal series representations of metaplectic covering groups, central extensions of a reductive group by the n-th roots of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' These groups are named after the first “Metaplectic Group,” the unique double cover of the symplectic group Sp2n discovered by Weil [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' However, the machinery generating this particular cover can be applied in far greater generality and results in non-algebraic groups that inherit much of the interesting representation theory and number theory of their algebraic base groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' One reason for this phenomenon is that if G is a group that is also a topological space, the metaplectic cover is a covering space in the topological sense as well: thus, the metaplectic covers of reductive groups, which are equipped with a topological structure, are of particular interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' These groups have been studied in various levels of generality by Kazhdan and Patterson [11], Matsumoto [13], Brylinski-Deligne [5], McNamara [15], Gan, Gao, and Weissman [8, 9], and many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' For our purposes, a particularly useful description is that of Brylinski-Deligne [5], who proved that metaplectic covers of reductive p-adic groups are in correspondence with symmetric Weyl-group invariant bilinear forms on the cocharacter lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' We will examine the structure of these covering groups in more detail in Section 2, following the treatment of the second author in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' The focus of this paper is the reductive group G = GLr(F), the general linear group of r × r matrices over a nonarchimedean local field F containing µ2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' In this case, which was first studied by Matsumoto [13], the bilinear forms prescribed by Brylinski-Deligne [5] recover a subset of the Kahzdan-Patterson covers [11] and may be explicitly parametrized as in Frechette [7] as Bc,d in terms of two parameters c, d ∈ Z (see Section 2 for the details of this construction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' In general, metaplectic covers of G are denoted �G, so let �Gc,d,r,n be the n-fold cover of GLr(F) corresponding to Bc,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' It is important to note that while there may 1 be multiple bilinear forms corresponding to a given cover, any such form will suffice for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' We refer the reader to [11] or [7] for more detailed descriptions of which forms give identical or similar covers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' One interesting difference between the algebraic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=', non-metaplectic) and metaplectic cases is the dimension of the space of Whittaker functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' For a reductive algebraic group, the space of Whittaker functions on any principal series representation is one-dimensional [18, 17, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' In the metaplectic case, however, the construction of principal series representations becomes more complicated, due to the fact that the metaplectic torus �T, the preimage in the metaplectic cover �G of the torus T (F), is no longer necessarily abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Due to this phenomenon, the dimension of the space of Whittaker functions becomes dependent on the choice of cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' As shown by McNamara [15], if W is the space of metaplectic Whittaker functions for a principal series representation on �G and H is the maximal abelian subgroup of �T, then dim(W) = ��� �T/H ��� , and the basis vectors of W may be parametrized by the cosets in �T /H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Note that the space of Whittaker functions is traditionally denoted Wz, where z = (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=', zr) ∈ Cr lists the Satake parameters for the principal series representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Since the results in this paper largely do not depend on the choice of z, we will generally drop it from the notation and write simply W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Examining the group structures of �T and H for a non-archimedean local field F, we achieve an explicit expression for the dimension, which we will prove in Section 2 as Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' For an n-fold metaplectic cover �G of GLr(F) corresponding to the bilinear form Bc,d, ��� �T/H ��� = nr |{x ∈ (Z/nZ)r : Bc,d(x, y) ≡ 0 (mod n) for all y ∈ (Z/nZ)r}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Our main result is a closed formula for the order of the set in the denominator of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' To this end, let Λfin := {x ∈ (Z/nZ)r : Bc,d(x, y) ≡ 0 (mod n) for all y ∈ (Z/nZ)r} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Then, using linear Diophantine equations to parametrize Λfin in two different ways, we arrive at the following result, which is proven in two parts as Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='7 and Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Main Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Given an n-fold metaplectic cover of GLr(F) corresponding to the bilinear form Bc,d, |Λfin| = dr−1 1 gcd � d2, dn d1 � , where d1 = gcd(c − d, n) and d2 = gcd(c + (r − 1)d, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Alternately, we also have that |Λfin| = dr−1 1 d2 n gcd � n d1 , n d2 , r � lcm � n gcd(r, n), gcd � d2, dn d1 �� , where b = gcd(r, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' The first formula arises from viewing the parametrizing Diophantine equations, which are generated by the natural basis for the cocharacter lattice for GLr(F), as a homogeneous specialization of an inhomogenous system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' This viewpoint provides a more elegant formula and a more concrete description of the structure of the space of Whittaker functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' On the other hand, while the second formula is more complicated, it arises from the root structure of GLr(F), which provides a more direct path to extending this result to other reductive groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' For GLr(F), the space of Whittaker functions is also closely tied to a particular module for a quantum group built from the Lie algebra gl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Despite the name, quantum groups are not groups at all, but rather quasitriangular Hopf algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' For this paper, we consider the quantum affine universal enveloping algebra Uq(c, d, n) := Uq(�gl(n/d1)), where q is the cardinality of the residue field for F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' This quantum group has a n/d1-dimensional evaluation module V+(z) depending on a parameter z ∈ C, whose basis vectors may be indexed using the elements of Z/(n/d1)Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' 2 In [1], Brubaker, Bump, and Buciumas prove that for the simplest n-fold metaplectic cover of GLr(F) (the one where c = 1 and d = 0), the space of Whittaker functions is isomorphic to an r-fold tensor product of evaluation modules and that after a Drinfeld twist (which changes the group action but does not affect the module structure) this isomorphism matches the action of the quantum group to the action of intertwining operators on the underlying principal series representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' The key ingredient in this proof is a lattice model construction for metaplectic Whittaker functions in the case c = 1, d = 0 developed in [2] by Brubaker, Bump, Chinta, Friedberg, and Gunnells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' In [7], the second author proves that both of these constructions are true in far greater generality, constructing a Whittaker function lattice model and a map θz between Wz and an r-fold tensor product V+(z1) ⊗ · · · ⊗ V+(zr) of evaluation modules for any metaplectic cover of GLr(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' (To match to the termi- nology used in [7], set nQ := n/d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=') Moreover, passing through this map, the action of the quantum group still matches exactly the action of intertwining operators on the Whittaker functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' As c, d, r, n vary, the cost of dealing with more complicated covers is that this map shifts between being an isomorphism, an injection, and a surjection, and the choice of representatives for H-cosets affects the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' The lattice model construction used in [7] dictates a choice of coset representatives from �T/H giving a basis for W on which θz is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' However, the lattice model is not necessary for the connection between W and the quantum module outside of this phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' A natural question then arises: when is the map θz well-defined for any choice of basis for W?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' One of the main applications of our results is an answer for this question, using the characterization of elements of Λfin from our proof of Main Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Taking any basis for W, use Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='1 to express it as a set of vectors in (Z/nZ)r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Then the map given in [7] is precisely θz : Wz → V+(z1) ⊗ · · · ⊗ V+(zr) ν �→ ρ − ν (mod n/d1), where ρ = (r−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=', 2, 1, 0) and the modulus is applied independently in each component of the vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Using our characterization to show how any particular coset νH sits within (Z/nZ)r, we arrive at the following result, which will be proven as Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='2 and Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Main Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' For a vector z = (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=', zr) ∈ Cr, the homomorphism θz given in [7] is well-defined for any choice of basis for W if and only if gcd � d2, dn d1 � = gcd(c, d, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Furthermore, if W is either of minimum or maximum dimension, θz is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Understanding how this map is affected by the choice of cover is an important step to understanding how we may extend these quantum connections to metaplectic covers over other reductive groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' While the lattice model connection only exists in full for GLr(F) and SLr(F), the Whittaker function framework exists for any reductive group, so we hope that further investigation of the structure of W will not only allow us to develop analogues to Main Theorem 1 for other groups, but also to determine the precise quantum group and module connected to the metaplectic Whittaker functions for other types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Regarding the structure of this paper, in Section 2, we examine the construction of metaplectic covers of GLr(F) and their Whittaker functions, culminating in a proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Section 3 introduces the first set of Diophantine equations used to parametrize Λfin, which we then use in Section 4 to prove the first part of Main Theorem 1 as Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' In Section 5, we introduce the second set of Diophantine equations for Λfin, which facilitate the proof of the second part of Main Theorem 1 as Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='1 in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' In Section 7, we examine some cases in which the formulae for dim(W) simplify dramatically and prove conditions for certain dimensions of interest for W, including conditions for maximum and minimum dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Lastly, in Section 8, we develop the quantum connection and use the structure of W to prove Main Theorem 2 as Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='2 and Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Acknowledgements This project was partially supported by NSF RTG grant DMS-1745638 and was supervised by the second author as part of the University of Minnesota School of Mathematics Summer 2022 REU program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' The 3 second author is also supported by NSF grant DMS-2203042.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' The authors would like to thank their TA Carolyn Stephen for their guidance throughout the project, as well as Ben Brubaker and Darij Grinberg for helpful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' 2 Spaces of Metaplectic Whittaker Functions To understand the structure of the space of metaplectic Whittaker functions, we must first concretely describe the metaplectic covers of GLr(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' We can then extend this explicit parametrization of all covers into a description of the metaplectic torus and its maximal abelian subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' As mentioned in the introduction, the quotient of these subgroups controls the dimension of the space of Whittaker functions: describing its structure precisely in terms of the cover allows us to reduce a complicated representation theory question to a straightforward linear algebra problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Suppose n is a natural number and F is a nonarchimedean local field containing 2n distinct 2n-th roots of unity µ2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Let o be the ring of integers of F and ̟ its uniformizing element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Given a split reductive group G, an n-fold metaplectic cover or n-fold metaplectic covering group �G is a non-algebraic central extension of G by the n-th roots of unity µn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' That is, �G is defined by the following short exact sequence: 1 → µn → �G p−→ G → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' As a set, �G is the set of tuples (ζ, g) where ζ ∈ µn, g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' However, group multiplication is controlled by a cocycle σ ∈ H2(G, µn);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' that is, for two elements (ζ1, g1), (ζ2, g2), their product in �G is (ζ1, g1) · (ζ2, g2) = (ζ1ζ2σ(g1, g2), g1g2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' In the process of writing down an explicit form for cocycles for covers of GLr(F), we see that a slightly more general case may be handled simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Set G = GLr(F) for the remainder of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' More generally, a metaplectic covering group essentially of degree n is given by a short exact sequence 1 → µm → �G p−→ G → 1 where n|m and the corresponding cocycle σ ∈ H2(G, µm) satisfies the property that [σn] is trivial in H2(G, C×) under the inclusion induced by an embedding ε : µm → C×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' While it is slightly tedious to write down formulae for these cocycles on general elements of G, their expressions over the torus T of diagonal matrices in G are quite elegant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' In [7], the second author proves that all metaplectic covers essentially of degree n over GLr(F) come from a cocycle of the form σc,d(x, y) = (det(x), det(x))c 2n � i>j (xi, yj)d−c n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' (1) for c, d ∈ Z, where x, y ∈ T and (·, ·)k denotes the k-th Hilbert symbol (see Neukirch [16] for more details on the construction of Hilbert symbols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Notably, making the shift to covers essentially of degree n rather than “purely” of degree n is necessary to include the metaplectic cover corresponding to the cocycle σ1,0, which, while only essentially of degree n, has been an integral example for this field (see for example [1, 2, 3, 4, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=') Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Since the 2n-th Hilbert symbol produces 2n-th roots of unity, it is necessary that F contain µ2n for the group to be well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' However, if we are considering a cocycle for which the parameter c is even, we may relax this condition and require F to contain only µn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' In [5], Brylinski-Deligne prove that the set of metaplectic covers is in correspondence with the set of symmetric Weyl-group invariant bilinear forms B : Y × Y → Z on the cocharacter lattice Y such that B(α∨,α∨) 2 ∈ Z for all coroots α∨ ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' For G = GLr(F), a natural choice of basis for Y is the set of r fundamental coweights ε∨ i : F × → T , for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=', r, in which ε∨ i (a) := diag(1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=', 1, a, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=', 1), where a is in the i-th entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Note: while we will use the notation λ(a) for λ ∈ Y, a ∈ F ×, another common notation is aλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' 4 Under this basis, the cocharacter lattice Y is isomorphic to Zr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' for instance, (ε∨ 1 + 3ε∨ 2 )(a) = diag(a, a3, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=', 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Using this basis, we represent a bilinear form on Y in terms of the corresponding matrix A such that for x, y ∈ Y , B(x, y) = xT Ay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Each of the conditions from the Brylinski-Deligne correspondence translates into a condition for this matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' First, a symmetric bilinear form prescribes a symmetric matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Second, the Weyl group W is isomorphic to the symmetric group Sr, and acts on Y by σ · ε∨ i = ε∨ σ(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Thus, A must be invariant under conjugation by permutation matrices, so for some (suggestively named) c, d ∈ Z, we have ai,i = c for all i and ai,j = d for all i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' (See the matrix in (2) for an illustration of this requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=') There are r − 1 simple coroots, each of the form ε∨ i − ε∨ i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' To check the integrality condition on the coroot lattice, it suffices to show that it holds for simple coroots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' However, for GLr(F), this condition is satisfied already: for any simple coroot ε∨ i , Bc,d(ε∨ i , ε∨ i ) 2 = 2(c − d) 2 = c − d ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' By Brylinski-Deligne, the metaplectic cover corresponding to this bilinear form satisfies the following condition: if x, y ∈ �T = p−1(T ) such that p(x) = λ(x), p(y) = µ(y) for some x, y ∈ F × and λ, µ ∈ Y , then the commutator of x and y is [x, y] = (x, y)B(λ,µ) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Evaluating the commutator in terms of an explicit cocycle, we may identify the bilinear form correspond- ing to a specific cocycle and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Note that this property illuminates one of the key reasons the Brylinski-Deligne correspondence is not a bijection: since the cocycle in (1) depends on powers of Hilbert symbols, there are many different cocycles which will give exactly the same cover, specifically any σc′,d′ such that c′ ≡ c (mod 2n) and d′ − c′ ≡ d − c (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='2 (Frechette [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' For c, d ∈ Z, the essentially n-fold metaplectic cover of GLr(F) with multi- plication given by σc,d corresponds to the bilinear form Bc,d that acts on (x, y) ∈ Zr × Zr by Bc,d(x, y) = xT · \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed c d d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' d d c d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' d d d c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' d d d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' c \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' (2) Conflating the bilinear form with its corresponding matrix, we will denote both by Bc,d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' we hope this abuse of notation will not cause any confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Note: in [7], this bilinear form is parametrized slightly differently as Bb,c, where b = c − d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Now that we have an explicit description of our metaplectic covers, we investigate what this parametriza- tion tells us about space of metaplectic Whittaker functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' For the purposes of this paper, we will not need the constructions of the metaplectic Whittaker functions themselves, nor those of the metaplectic principal series representations on which they are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Instead, we will use the following theorem of McNamara to investigate the space of Whittaker functions through its connection to the metaplectic torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' For the definitions of the metaplectic principal series representations and their Whittaker functions, we refer the reader to Sections 6 and 8, respectively, of [15] as a convenient source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='3 (McNamara [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Fix a metaplectic cover �G over a p-adic reductive group G and let W be the space of metaplectic Whittaker functions for a principal series representation on �G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Let the metapletic torus �T be the preimage in �G of the torus T (F), and let H be the maximal abelian subgroup of �T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Then dim(W) = ��� �T/H ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' 5 Note: the group T of diagonal matrices is denoted T because it is an abelian torus, that is, it is isomorphic to (F ×)r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' While we call �T the metaplectic torus, it is no longer abelian, nor is it technically a torus, as its elements are (ζ, t) where ζ ∈ µn (where µn ⊊ F) and t ∈ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Investigating the precise structure of �T, we prove the following theorem, which is a restatement of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' For a metaplectic cover �G of GLr(F) corresponding to the bilinear form Bc,d, ��� �T/H ��� = nr |{x ∈ (Z/nZ)r : Bc,d(x, y) ≡ 0 (mod n) for all y ∈ (Z/nZ)r}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Using our description of the metaplectic covers, we can express the subgroups �T and H more explicitly: using the Iwasawa decomposition of GLr(F), we have that �T = µn ×T (o)×Y as a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' That is, for (ζ, t) ∈ T, we may write t = t0 · λ(̟) for some t0 ∈ T (o) and λ ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Since H is a subgroup of �T, its elements also look like (ζ, h) where ζ is an n-th root of unity and h is a diagonal matrix with entries in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Examining the group law on �G, we see that the root of unity does not impede commutativity of elements, so it is the matrix component h we must examine further to obtain a description of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' To do so, recall that o is the valuation ring of F and ̟ the uniformizing element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Then by [15], as a set we have H = µn × T (o) × Λ, where Λ is the free abelian group Λ := {λ ∈ Y : s(λ(̟)) ∈ H} for s : G → �G the standard section s(g) = (1, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Using the commutator relation and the fundamental coweight basis for Y , an equivalent description for Λ is Λ = {x ∈ Zr : Bc,d(x, y) ≡ 0 (mod n) for all y ∈ Zr} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' (3) It is useful to think of the group Λ as controlling the powers of ̟ in each entry on the diagonal of the matrix h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' That is, for any element (ζ, h) ∈ H, we have h = h0 · diag(̟λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=', ̟λr) where h0 ∈ T (o) and λ = λ1ε∨ 1 + · · · + λrε∨ r is in Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Then, combining our descriptions of �T and H to consider �T/H, we see that | �T/H| = |Y/Λ| = |Zr/Λ| , where the last description uses the embedding of Λ in Zr described in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Notice that if λi ∈ nZ for all i, then B((λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=', λr), y) will automatically be a multiple of n for any y ∈ Zr, and therefore λ = λ1ε∨ 1 + · · · + λrε∨ r will be in Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Therefore, it suffices to consider all coordinates λi mod n, and so | �T/H| = |(Z/nZ)r/ (Λ ∩ (Z/nZ)r)| which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Let Λfin := {x ∈ (Z/nZ)r : Bc,d(x, y) ≡ 0 (mod n) for all y ∈ (Z/nZ)r} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' We will spend the next several sections developing two related systems of linear Diophantine equations which allow us to describe the elements in Λfin, each of which will give us a distinct formula for |Λfin|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' We will then return to the broader framework in Section 7 to what these different formulae tell us about the structure of | �T/H| and thus the structure of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' 3 Cocharacter Diophantine Equations and Phenomena In this section, we use the natural basis for the cocharacter lattice Y of GLr(F) to develop a set of r linear Diophantine equations in terms of c, d, and n that describe the set Λfin, which we call the cocharacter equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' This perspective turns a representation theoretic question into a linear algebra one, where altering each of the parameters c, d, r, and n has a different effect on the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' We also take time now to develop a visual framework which illuminates this distinction in the roles of each of our parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Examining the conditions for Λfin using the viewpoint of the fundamental coweight basis for Y (see Section 2), we arrive at the following system of r equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Let 0r = (0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=', 0)T be the r × 1 column vector with all entries equal to 0, and define 1r = (1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' , 1)T similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Recall that Bc,d is both the bilinear form given in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='2 and its corresponding r × r matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' 6 Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' For natural numbers r, n ≥ 1 and constants c, d ∈ Z, we call the following system of equations the cocharacter equations: Bc,d · x = 0r (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' That is, for x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=', xr)T , we have cx1 + dx2 + · · · + dxr ≡ 0 (mod n) dx1 + cx2 + · · · + dxr ≡ 0 (mod n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' dx1 + dx2 + · · · + cxr ≡ 0 (mod n) Here, the i-th equation arises from evaluating x ∈ Y against ε∨ i in the bilinear form Bc,d for each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=', r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Thus, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='1 follows directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Let Scochar(c, d, r, n) be the number of solutions x ∈ (Z/nZ)r to the cocharacter equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Then, for the cover �Gc,d,r,n, we have Scochar(c, d, r, n) = |Λfin|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Looking at the values of Scochar for a fixed r and n as we range over c and d, certain patterns emerge which motivate defining constants which we call the diagonal numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' These constants will be fundamental in our formulas for Scochar, so we take the time to explore them now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' For a fixed r and n, note first that it suffices to understand Scochar for c, d (mod n), as Scochar(c, d, r, n) = Scochar(c′, d, r, n) for c ≡ c′ (mod n) and likewise for d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' It will be useful to visualize the values of Scochar as a table ranging over c, d ∈ Z/nZ in the following manner: d 0 1 2 · · (n − 1) c 0 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' (n − 1) Examining Figure 1, which contains several examples of these tables, notice that the values of Scochar on the marked diagonals in each picture are each divisible by common factors and that there are two sets of diagonals in each picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Motivated by this phenomena, we assign each entry a set of two diagonal numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Let d1 = gcd(c − d, n) be the first diagonal number and define d2 = gcd(c + (r − 1)d, n) to be the second diagonal number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Note that for a specific entry in place c, d, its first diagonal number captures the column c − d where its diagonal of slope −1 intersects the first row and similarly, the second diagonal number identifies the row where its diagonal of slope r − 1 intersects the first column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' When r and n are coprime, the table for Scochar depends solely on these diagonal numbers, which we will later prove in Section 6 (see Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=') For instance, the table where n = 10, r = 3 is shown in Figure 2, with diagonal numbers marked, and the value of every entry in this matrix is determined by its two diagonal numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Specifically, we have Scochar(10, 3, c, d) = dr−1 1 d2 = d2 1d2 for any c, d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' In general, given a random n and r, the value of Scochar will not depend nearly so simply on d1 and d2, but they still play an important determining role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' To find a closed formula for Scochar, we must look to an inhomogenous generalization of the homogenous cocharacter equations with which we started.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Let a ∈ Z/nZ, and x ∈ (Z/nZ)r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Then the inhomogenous cocharacter equations for a ∈ Z are defined by Bc,d · x = a · 1r (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' (4) Let Sinhom(c, d, r, n) be the number of total solutions to the inhomogenous cocharacter equations, ranging over all values of a ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' 7 49 1 1 1 1 1 1 1 7 1 1 1 1 7 1 1 7 1 1 7 1 1 1 1 7 7 1 1 1 1 1 7 7 1 1 1 1 7 1 1 7 1 1 7 1 1 1 1 7 r = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' n = 7 64 1 4 1 16 1 4 1 1 8 1 8 1 8 1 8 4 1 16 1 4 1 16 1 1 8 1 8 1 8 1 8 16 1 4 1 32 1 4 1 1 8 1 8 1 8 1 8 4 1 16 1 4 1 16 1 1 8 1 8 1 8 1 8 r = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' n = 8 81 1 1 9 1 1 9 1 1 1 9 3 1 3 3 1 3 9 1 3 9 1 3 3 1 9 3 9 1 1 27 1 1 27 1 1 1 3 3 1 9 9 1 3 3 1 3 3 1 9 9 1 3 3 9 1 1 27 1 1 27 1 1 1 3 9 1 3 3 1 9 3 1 9 3 1 3 3 1 3 9 r = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' n = 9 729 1 1 27 1 1 27 1 1 1 81 1 1 27 1 1 27 1 1 1 81 1 1 27 1 1 27 27 1 1 243 1 1 27 1 1 1 27 1 1 81 1 1 27 1 1 1 27 1 1 81 1 1 27 27 1 1 27 1 1 243 1 1 1 27 1 1 27 1 1 81 1 1 1 27 1 1 27 1 1 81 r = 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' n = 9 4096 1 16 1 256 1 16 1 1 512 1 16 1 128 1 16 16 1 1024 1 16 1 256 1 1 16 1 512 1 16 1 128 256 1 16 1 2048 1 16 1 1 128 1 16 1 512 1 16 16 1 256 1 16 1 1024 1 1 16 1 128 1 16 1 512 r = 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' n = 8 Figure 1: Examples of the Cocharacter Phenomena for different choices of r and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' In each example, notice that there is one set of diagonals of slope -1 and one of slope r − 1: the former indicate the effect of the first diagonal numbers d1 and the latter that of the second diagonal numbers d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Diagonals for the same diagonal numbers (greater than 1) are marked with the same color within each example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' For instance, in the second example (r = 2, n = 8) red marks diagonal numbers equal to 8, blue equal to 4, and green equal to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' In the next section, we will solve for Scochar(c, d, r, n) by characterizing the set of solutions to the inho- mogenous cocharacter equations using straightforward linear algebra techniques and identifying the propor- tion of solutions with a ≡ 0 (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' To do this, we will need to identify a precise formula for smallest nonzero value of a for which (4) has a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' For a fixed c, d, r, n, let A(c, d, r, n) be the smallest positive integer value for a such that there is a solution to the inhomogenous cocharacter equations (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' 4 Proof of Main Theorem 1 Part 1 For the entirety of this section, fix a set of parameters c, d, r, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' To find a formula for Scochar := Scochar(c, d, r, n), we begin by showing that the solutions to the inhomogenous cocharacter equations fall into equally sized equivalence classes defined by the values a ∈ Z, and that Scochar(c, d, r, n) = A(c, d, r, n) n Sinhom(c, d, r, n) Characterizing the solutions to the inhomogenous cocharacter equations, we will then provide explicit expressions for Sinhom(c, d, r, n) and A(c, d, r, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' The equation Bc,d · x ≡ a · 1r (mod n) has a solution if and only if a is a multiple of A = A(c, d, r, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Thus, A(c, d, r, n) divides n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' By definition, a solution xA to the equation Bc,d · x ≡ A · 1r (mod n) exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' If a = kA for some k ∈ Z, then kxA is a solution to Bc,d · x ≡ a · 1r (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' For the other direction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' suppose there exists a 8 1000 2 8 2 8 250 8 2 8 2 1 100 5 4 1 4 25 20 1 4 8 2 200 2 40 2 8 50 8 10 1 20 1 100 1 4 5 4 25 4 8 2 8 10 200 2 8 2 40 50 125 4 1 4 1 500 1 4 1 4 8 50 40 2 8 2 200 10 8 2 1 4 25 4 5 4 1 100 1 20 8 10 8 50 8 2 40 2 200 2 1 4 1 20 25 4 1 4 5 100 d1 =10 1 2 1 2 5 2 1 2 1 d2 = 10 1 2 1 2 5 2 1 2 1 d1 = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' d2 = 10 40 = 22 · 10 Figure 2: The table showing Scochar(c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' 10) for all (c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' d) ∈ Z10 × Z10 with diagonals for diagonal numbers greater than 1 marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Notice here that since r = 3 and n = 10 are coprime, every entry is equal to d2 1 · d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' In contrast, see the example in Figure 1 for r = 3 and n = 9, where this is not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' positive integer g and solution xg ∈ (Z/nZ)r to the equation Bc,d · xg ≡ g · 1r (mod n), but that A does not divide g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Then jA < g < (j + 1)A for some positive integer j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Therefore, Bc,d · (xg − jxA) ≡ Bc,d · xg − jBc,d · xA ≡ (g − jA) · 1r, which contradicts the minimality of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Then, since x = 0r is a solution to Bc,d · x ≡ n · 1r ≡ 0r (mod n), the second statement follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Splitting the solutions to (4) into equivalence classes based on a, we examine the number of solutions in each class and characterize them more concretely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' For k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=', n A}, let Wk be the set of solutions to Bc,d · x ≡ (kA) · 1r (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Then |Wk| = |W1| for all such k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Consider any x ∈ W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' The function φx : W1 → Wk defined by y �→ y +(k −1)·x provides a bijection between W1 and Wk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Let x = (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' , xr)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Then x solves the inhomogenous cocharacter equations if and only if cx1 + dxj ≡ dx1 + cxj (mod n) for every 2 ≤ j ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Let 2 ≤ j ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' For a solution x, the first row of the equation Bc,d · x ≡ a · 1r (mod n) tells us that cx1 + dxj + � 2≤k≤r k̸=j dxk ≡ a (mod n) Subtracting the j-th row dx1 + cxj + � 2≤k≤r k̸=j dxk ≡ a (mod n), from the first, we obtain cx1 + dxj ≡ dx1 + cxj (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' For the other direction, suppose x satisfies cx1 + dxj ≡ dx1 + cxj (mod n) for all j ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=', r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Then, x satisfies the inhomogeneous cocharacter equations for the value a ≡ cx1 + dxj + � 2≤k≤r k̸=j dxk (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' 9 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' A vector x = (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' , xr)T solves the inhomogenous cocharacter equations if and only if for each j ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=', r} we have xj = x1 + vj n d1 for some integer vj such that 1 ≤ vj ≤ d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='3, it suffices to characterize the solutions x to the system of equations given by cx1 + dxj ≡ dx1 + cxj (mod n) for every j ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=', r}, or equivalently, (c − d)(x1 − xj) ≡ 0 (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' (5) Recalling that d1 = gcd(c − d, n), a vector x satisfies (5) exactly when x1 − xj is a multiple of n d1 for all j ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=', r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Thus, the solutions to the inhomogeneous cocharacter equations are precisely the vectors of the form x = x1 · 1r + n d1 (0, v2, v3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=', vr)T where 0 ≤ x1 < n and vj ∈ Z such that 1 ≤ vj ≤ d1 for all j ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=', r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Now that we have precisely characterized the set of x which solve the inhomogeneous cocharacteristic equations, we can count the size of this set by ranging over all distinct choices of tuples (x1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=', vr), which each yield a distinct solution x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' The number of solutions to the inhomogenous cocharacter equations is Sinhom(c, d, r, n) = ndr−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' We are now prepared to identify a precise formula for A in terms of n, r, c, and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' The minimum positive integer A such that the inhomogenous cocharacter equations have a solution is A = gcd � d2, dn d1 � , recalling that d2 = gcd(c + (r − 1)d, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Substituting Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='4 into (4), we see that the left-hand side is \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed c d d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' d d c d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' d d d c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' d d d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' c \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed x1 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' 1 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 + n d1 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 0 v2 v3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' vr \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 ≡ x1(c + (r − 1)d) \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' 1 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 + n d1 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed dv2 + dv3 + · · · + dvr cv2 + dv3 + · · · + dvr dv2 + cv3 + · · · + dvr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' dv2 + dv3 + · · · + dvr−1 + cvr \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' To have a solution, every row of this expression must must equal a constant A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Looking at the first row, A ≡ x1(c + (r − 1)d) + dn d1 (v2 + v3 + · · · + vr) (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' From the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='4, x1 and v2 +v3 +· · ·+vr are both arbitrary constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Thus, the minimum value A can have is gcd � c + (r − 1)d, dn d1 , n � = gcd � d2, dn d1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Note, since d1 divides c − d, we can equivalently write A as A = gcd � d2, cn d1 � = gcd � d2, dn d1 � = gcd � d2, n d1 gcd(c, d, n) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Thus, we arrive at a closed form for Scochar in terms of c, d, r, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' The number of solutions to the cocharacter equations is Scochar(c, d, r, n) = dr−1 1 gcd � d2, dn d1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' 10 5 Coroot Diophantine Equations Inspired by the constants c + (r − 1)d and c − d showing up in the cocharacter equations, we define a second system of related equations more closely tied to the root structure of GLr(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' The coroot equations are the system of r equations: (c − d)(xi − xr) ≡ 0 (mod n) for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=', r − 1}, (c + (r − 1)d)(x1 + · · · + xr) ≡ 0 (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' We call these the coroot equations because the i-th equation arises from evaluating x ∈ Y against the coroot ε∨ i − ε∨ r in the bilinear form Bc,d for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=', r − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' We could similarly evaluate against the simple coroots ε∨ i − ε∨ i+1, but this formulation will be more useful for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' In some cases, the coroot and cocharacter equations are equivalent, but in other cases they are not: counting the solutions to the coroot equations and examining this connection will give us an alternate formula for Scochar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' This system also illuminates the difference between metaplectic covers of SLr(F) and GLr(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' For any cocharacter x for SLr(F), the last coroot equation is vacuously true, since x1+· · ·+xr ≡ 0 (mod n) is necessary for the resulting matrices x(a) to have determinant one for any a ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' In this case, the cocharacter and coroot equations are equivalent, and they both give |Λfin ∩ SLr(F)| = dr−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' The number of solutions to the coroot equations is Scoroot(c, d, r, n) = dr−1 1 d2 gcd � n d1 , n d2 , r � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' As in Section 4, we prove general properties about the solutions to the coroot equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' These descrip- tions will allow us to directly relate Scochar to Scoroot in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' We start with a change of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Consider the coroot system in variables y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=', yr−1, z written as (c − d)yi ≡ 0 (mod n) for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=', r − 1}, (c + (r − 1)d)z ≡ 0 (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' In terms of these variables, there are dr−1 1 d2 tuples (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=', yr−1, z) that solve the coroot equations: yi are all multiples of n d1 and z is a multiple of n d2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Let SY,Z be the set of such tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' We then classify x satisfying yi = xi − xr and z = x1 + · · · + xr such that (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='., yr−1, z) ∈ SY,Z: that is, the set of x satisfying the original formulation of the coroot equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Note that xi = yi+xr, so rearranging the final coroot equation, we have rxr ≡ z − (y1 + · · · + yr−1) (mod n), (6) and thus the number of solutions in terms of x versus in terms of (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=', yr−1, z) depends on whether r is invertible mod n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Let b = gcd(n, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Then xr has b solutions when z − (y1 + · · · + yr−1) is a multiple of b and no solutions otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' A straightforward calculation verifies that there is no overlap between the sets of x for distinct tuples (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='., yr−1, z) ∈ SY,Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Let Frb(d1, d2, n) be the proportion of (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' , yr, z) tuples that will yield a valid solution to the coroot equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' In other words, Frb := |{(y1, · · · , yr, z) ∈ SY,Z : z − y1 − · · · − yr is a multiple of b}| |SY,Z| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Then Scoroot(c, d, r, n) = dr−1 1 d2·b·Frb(d1, d2, n), and it will suffice to develop a formula for Frb(d1, d2, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' When n and r are relatively prime, r is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Thus Frb evaluates to 1 because any tuple we pick adds to a multiple of b = 1, so in this case the two sets of variables give equivalent conditions and Scoroot = |SY,Z|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' 11 Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' The function Frb evaluates to Frb(d1, d2, n) = 1 b · gcd � n d1 , n d2 , b � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' We proceed by carefully considering the overlaps of factors of b with those of n d1 , n d2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Let k1 = gcd( n d1 , b) and m1 ∈ Z such that b = m1k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Similarly, let k2 = gcd( n d2 , b) and m2 ∈ Z such that b = m2k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Since yi is a multiple of n d1 , it is also a multiple of k1: examining which multiples are possible modulo b, we see that yi (mod b) can be any of the m1 multiples of k1 in Z/bZ with equal probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Similarly, considering the sum y = �r−1 i=1 yi, we claim that the same is true for y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Let 1 ≤ g ≤ m1 and suppose y ≡ gk1 (mod b): if we pick any arbitrary y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' , yr−2, we are left with yr−1 ≡ gk1 − r−2 � i=1 yi (mod b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' The right-hand side of this equation defines some equivalence class ℓk1 (mod b) from which we must choose yr−1 to ensure that y ≡ gk1 (mod b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Exactly 1 m1 of the possible values of yr−1 place us in the correct equivalence class for a given g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Thus, y falls into the equivalence classes k1, 2k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' , m1k1 with equal probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' ks s1 s2 m c − d c + (r − 1)d b n n/d1 n/d2 Figure 3: A visualization of our factorization of b = gcd(r, n), where the overlap of any circle or shaded region with the circle for b contains a factorization for their greatest common divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Note that the purple region is the overlap of the red and blue regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Likewise, z (mod b) can be any of the m2 multiples of k2 modulo b with equal probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' To interface between y and z, we must factor further: let ks = gcd(k1, k2) so that k1 = s1ks and k2 = s2ks, and gcd(s1, s2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Letting m = gcd(m1, m2), factor b completely as b = ms1s2ks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' The reader may find it helpful to refer to Figure 3, which provides a visualization of how this factorization relates b, n d1 , and n d2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' We now identify the proportion of y- and z-values that satisfy z − y ≡ 0 (mod b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Since y, z, b all contain a factor of ks, let y = αk1 = αs1ks and z = βk2 = βs2ks for some α, β ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Then, equivalently, we seek the proportion of (α, β) pairs such that βs2 ≡ αs1 (mod ms1s2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Since s1, s2 are coprime, we must have α = as2 for some a ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Exactly 1 s2 of the possible α-values are multiples of s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Then βs2 ≡ as1s2 (mod ms1s2) which has solutions only for β ≡ as1 (mod b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Out of the m2 = ms1 equivalence classes that z can fall into, only the one defined by β = as1 works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Therefore, Frb(d1, d2, n) = 1 s2 � 1 ms1 � = 1 ms1s2 = ks b = gcd � n d1 , n d2 , b � b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' 12 Since b = gcd(r, n), we have gcd � n d1 , n d2 , b � = gcd � n d1 , n d2 , r � , which completes proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='2 and allows us to express Scoroot solely in terms of c, d, r, and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' 6 Proof of Main Theorem 1 Part 2 We are now ready to prove the second part of our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' In this section, we show how the coroot equations are obtained from the cocharacter equations, and how this relates Scoroot and Scochar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' The number of solutions to the cocharacter equations can also be defined as Scochar = Scoroot · 1 n · lcm � gcd � d2, dn d1 � , n gcd(n, r) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Let M(c, d, r, n) := lcm � gcd � d2, dn d1 � , n gcd(n,r) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' We will also prove the following formula for M(c, d, r, n), which will be useful for our investigation into special dimensions for W in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Let r, n have prime factorizations r = pℓ1 1 pℓ2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' pℓj j and n = pm1 1 pm2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' pmj j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' For every 1 ≤ i ≤ j, let (c − d) ≡ cipsi i (mod pmi i ) and d ≡ dipti i (mod pmi i ) for each 1 ≤ i ≤ j so that 0 ≤ si, ti ≤ mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Let µi = min(mi, ℓi) and ci, di are relatively prime to pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Then M(c, d, r, n) = j� i=1 pmax(mi−µi,min(si,ti+mi−si)) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' To obtain the coroot equations from the cocharacter equations, we can multiply the matrix Bc,d which defines the cocharacter equations by the r × r matrix Lr = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' 0 −1 0 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' 0 −1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' 0 0 1 −1 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' 1 1 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' That is, the new system of equations specified by this transformation is LrBc,d · x ≡ 0r (mod n) which gives us precisely the coroot equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Likewise, multiplying the coroot equations by L′ r = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed r − 1 −1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' −1 1 −1 r − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' −1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' −1 −1 r − 1 1 −1 −1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' −1 1 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 obtains the cocharacter equations multiplied by r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' That is, L′ r(LrBc,d) · x ≡ r · Bc,d · x ≡ 0r (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' As we discussed in Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='3, if r and n are relatively prime, then r has an inverse r−1 in Z/nZ and the cocharacter and coroot equations are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' However, if r is not invertible modulo n, going back from the coroot equations to the cocharacter equations is more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' 13 Recall that b = gcd(r, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Then r · Bc,dx ≡ 0r (mod n) factors into b �r b � \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed c d d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' d d c d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' d d d c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' d d d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' c \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed x1 x2 x3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' xr \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 ≡ \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' 0 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 � mod b · n b � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Since both sides of the equation and the modulus are multiples of b, this implies that �r b � Bc,d · x ≡ 0r � mod n b � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' The number r b is relatively prime to n b and therefore invertible in Z/ n b Z, so Bc,d · x ≡ 0r � mod n b � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' (7) Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Again, (7) shows that the coroot and cocharacter equations are equivalent when r and n are relatively prime, since then n = n/b and (7) recovers exactly the cocharacter equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' If we are not in that case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=', if b ̸= 1, then for any coroot solution x, we get Bc,dx ≡ n b v (mod n) for some vector v ∈ Zr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' We now show that each solution to the coroot equations also satisfies inhomogeneous cocharacter equations for particular values of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' The coroot equations are equivalent to the inhomogeneous cocharacter equations with the con- dition that a ∈ (n/b)Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Let x be a solution to the coroot equations and let the ith row of the left-hand side of Equation (7) be wi = cxi + � j̸=i dxj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Then by definition, wi − wr = (c − d)(xi − xr) ≡ 0 (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Thus, for some k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=', b}, Bc,dx ≡ n b k · 1r (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' (8) Similarly, suppose x satisfies (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Then defining wi as above, (c − d)(xi − xr) ≡ wi − wr ≡ k n b − k n b ≡ 0 (mod n) (c + (r − 1)d)(x1 + x2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' xr) ≡ w1 + w2 + · · · + wr ≡ r � k n b � ≡ 0 (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' In Section 4 we showed that Equation (8) has solutions if and only if k n b is a multiple of A(c, d, r, n), and that each class of solutions (defined by having the same k) is of the same size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' As in that section, we want to find the smallest nonzero k for which (8) has a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Let κ(c, d, r, n) be the smallest positive value of k such that there is a solution to Equation (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Again, it will often be clear from context that we are working with a particular fixed c, d, r, n in which case we will write κ and M for brevity instead of κ(c, d, r, n) and M(c, d, r, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' 14 Proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' We can relate the values of κ(c, d, r, n) and A(c, d, r, n) as follows: κ · n b = lcm � A, n b � = lcm � gcd � d2, dn d1 � , n b � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Let M(c, d, r, n) := lcm � gcd � d2, dn d1 � , n b � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Then there are n M = b κ equivalence classes of solutions to Equa- tion (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Exactly one of these equivalence classes—the one given by k = b—gives the solutions to the cocharacter equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Therefore, Scochar = Scoroot · M n = Scoroot · κ b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Substituting in our earlier expressions for the values of Scoroot and M, we obtain Scochar = dr−1 1 d2 n gcd � n d1 , n d2 , r � lcm � n gcd(r, n), gcd � d2, dn d1 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Although it is not immediately clear from looking at this equation, this formula is equivalent to the one given in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' One area of future work would be to simplify this expression and more directly understand why it is equivalent to the statement of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Furthermore, while this formula appears more complicated than that of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='7, this approach is perhaps more suitable for extending past GLr(F), as the metaplectic Whittaker functions developed in Section 2 can be defined over any reductive group and this approach is more closely related to the root data structure of reductive groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Using the same visualization tables we used in Section 3 for Scochar shows more directly how M and κ change as we vary c, d, r, and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Here, for a fixed r, n, let the entry in position (c, d) be κ(c, d, r, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' (To achieve a matching table for M, multiply the κ table by n/b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=') Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' For n = 8 = 23 and r = 2ℓ, the following tables show how κ changes as ℓ increases from 1 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Because M and κ depend on µ = min(ℓ, m) rather than on ℓ, any κ table for ℓ > 3 would be identical to the table for ℓ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='1 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='n = 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' r = 2 n = 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' r = 4 n = 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' r = 8 The entries in these tables are determined by the main diagonals they lie on,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' which are described by s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' and the columns they lie in,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' which are described by t and index how far down the main diagonal an entry is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' In particular, notice that the only difference between the matrices for ℓ and ℓ + 1 is that a specific fraction of the elements on each of the diagonals in the latter matrix have been multiplied by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' For example, for ℓ = 2, this fraction is 1/4 for the red diagonal and 1/2 for the blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' These tables are also useful for visualizing the effect of combining distinct primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' When r = 22 and n = 22 · 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' we see that the table for κ is a 3 × 3 tessellation of that for 15 r = 22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' n = 22: 4 1 2 1 1 1 1 2 2 1 2 1 1 2 1 1 r = 22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' n = 22 4 1 2 1 4 1 2 1 4 1 2 1 1 1 1 2 1 1 1 2 1 1 1 2 2 1 2 1 2 1 2 1 2 1 2 1 1 2 1 1 1 2 1 1 1 2 1 1 4 1 2 1 4 1 2 1 4 1 2 1 1 1 1 2 1 1 1 2 1 1 1 2 2 1 2 1 2 1 2 1 2 1 2 1 1 2 1 1 1 2 1 1 1 2 1 1 4 1 2 1 4 1 2 1 4 1 2 1 1 1 1 2 1 1 1 2 1 1 1 2 2 1 2 1 2 1 2 1 2 1 2 1 1 2 1 1 1 2 1 1 1 2 1 1 r = 22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' n = 12 = 3 · 22 Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Below are the κ tables for n = 6 and r = 2, 3, 6, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Note that the table for r = 6 is obtained by multiplying the tables for r = 2 and r = 3 together elementwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Upon proving Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='2, we will see that this is true in greater generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' 2 1 2 1 2 1 1 1 1 1 1 1 2 1 2 1 2 1 1 1 1 1 1 1 2 1 2 1 2 1 1 1 1 1 1 1 r = 2, n = 6 3 1 1 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 1 1 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r = 3, n = 6 6 1 2 3 2 1 1 1 1 1 1 1 2 1 2 1 2 1 3 1 1 3 1 1 2 1 2 1 2 1 1 1 1 1 1 1 r = 6, n = 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Expressing the values of M and κ directly in terms of the prime factors of n, r, c and d, we are ready to prove Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Let r = pℓ1 1 pℓ2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' pℓj j and n = pm1 1 pm2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' pmj j , where some of ℓi or mi may be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Recalling the definition of d2, we have that M = lcm � gcd � c + (r − 1)d, n, dn d1 � , n b � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Thus, the only prime factors of M are those that are prime factors of n, and M is multiplicative over these prime powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Considering only the power of pi arising in M for some i ∈ {1, j}, let (c − d) ≡ cipsi i (mod pmi i ) and d ≡ dipti i (mod pmi i ) for each 1 ≤ i ≤ j so that 0 ≤ si, ti ≤ mi, and ci, di are relatively prime to pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Let µi = min(mi, ℓi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Then, n/b = bipmi−µi i , where bi is relatively prime to pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Then consider the power of pi arising from gcd � c + (r − 1)d, n, dn d1 � : recalling that d1 = gcd(c − d, n), the power of pi in d1 is min{si, mi} = si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Then, the power of pi in dn/d1 is ti + mi − si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' It remains to consider the power arising in the first component of the gcd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Consider c + (r − 1)d = c − d + r · d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Substituting in the factorizations, we have c + (r − 1)d ≡ cipsi i + ridipti+µi i (mod pmi i ), where ri is relatively prime to pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' We consider three cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' If si < ti + µi, the power of pi in this component is si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Thus, the power of pi in the gcd is min{si, ti + mi − si}, since si ≤ mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' The power of pi in M is then max{mi − µi, min{si, ti + mi − si}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' 16 Next suppose that si > ti + µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Then we have that the power of pi in the gcd is min{ti + µi, ti + mi − si}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' However, then the power of pi in M is max{mi − µi, min{ti + µi, ti + mi − si}} = mi − µi, since si > ti + µi, so ti + mi − si < mi − µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Lastly, if si = ti + µi, then c − d + rd ≡ psi i (ci + ridi) (mod pmi i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Note that ci + ridi may create an additional factor pτi i for some integer τi ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Then the power of pi appearing in the gcd is min{si + τi, mi, ti + mi − si} = min{si + τi, mi − µi} Then the power of pi in M is max{min{si + τi, mi − µi}, mi − µi} = mi − µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Collecting the three cases together, the expression max{mi − µi, min{si, ti + mi − si}} matches the power of pi in M in each case, completing the proof of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' The quantity κ(c, d, r, n) is multiplicative over powers of distinct primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' In the next section, we will see that the two different approaches for Scochar are each useful in different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' One potentially fruitful avenue for future exploration would be to see precisely why these two formulae are equal, as it is not easily apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' As the second approach relates more directly to the root structure of GLr as a reductive group, but the first approach yields a simpler formula and proof, this connection would illuminate a way to extend the simpler formula to general reductive groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' 7 Structure of the Whittaker Space These investigations into the structure of Λfin not only give us a method of calculating dim(W), they also illuminate how the parameters c, d, r, and n affect the structure of W in different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' In this section, we start with a few natural corollaries to both parts of Main Theorem 1 (Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='7 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='1) and discuss how they relate to the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' We then develop necessary and sufficient conditions for dim(W) to be of maximum and minimum dimension, as well as the conditions for several other desirable dimensions for further connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' From Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='7, we have the following natural results about dim(W): dim(W) = \uf8f1 \uf8f2 \uf8f3 � n gcd(c,n) �r if d ≡ 0 (mod n) � n gcd(d,n) �r−1 n gcd((r−1)d,n) if c ≡ 0 (mod n) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Recall that by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='1, we have dim(W) = nr/|Scochar(c, d, r, n)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Then if d ≡ 0 (mod n), Scochar(c, d, r, n) = gcd(c − d, n)r−1 gcd � c + (r − 1)d, n, dn gcd(c − d, n) � = gcd(c, n)r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Likewise, if c ≡ 0 ≡ n, then Scochar(c, d, r, n) = gcd(−d, n)r−1 gcd � (r − 1)d, n, dn gcd(−d, n) � = gcd(d, n)r−1 gcd((r − 1)d, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' 17 As we can see from this corollary, the parameters c and d play significantly different roles in influencing the structure of the Whittaker function space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' In the simplest n-fold metaplectic cover (c = 1, d = 0), we see | �T| = nr, which allowed Brubaker, Bump, and Buciumas to map W isomorphically to a quantum module of dimension nr in [1] to explain the lattice model phenomena discovered by Brubaker, Bump, Chinta, Friedberg, and Gunnells [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' In the same spirit, the second author showed in [7] that this connection extends quite naturally to an isomorphism for any cover coming from a diagonal matrix (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=', d ≡ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' However, incorporating the parameter d adds complications, as the quantum module (which we will discuss later in Section) does not see the factor of gcd � c + (r − 1)d, n, dn gcd(c−d,n) � appearing in dim(W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Thus, to understand this connection, we will need additional information about the structure of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' We have dim(W) = 1 (that is, of minimum size) if and only if c ≡ d ≡ 0 (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' The backward direction follows from Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Now assume Scochar = nr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Since each of the r factors in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='7 are factors of n, we must have d1 = gcd(c − d, n) = n and so Scochar = nr−1 gcd (c − (r + 1)d, n, c, d) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' So we must also have gcd(n, c, d) = n, which requires that c, d ≡ n (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' We have dim(W) = nr (that is, of maximum size) if and only if c − d and c + (r − 1)d are coprime to n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' It suffices to show that Scochar = 1 if and only if d1 = d2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' The backwards direction is easiest to see from Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='1: if d1 = d2 = 1, then Scochar = 1 n gcd(r, n) · lcm � n gcd(r, n), 1 � = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' For the forward direction, we use Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Here, Scochar = 1 implies both dr−1 1 = 1 (and thus d1 = 1) and gcd � d2, dn d1 � = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Since d1 = 1 we then have gcd (d2, dn) = 1 which tells us that d2 must be relatively prime to n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' But d2 = gcd(c + (r − 1)d, n) so thus d2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' We will later see that both maximizing and minimizing W result in very nice quantum connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' It is also intriguing to ask when the diagonal number phenomenon developed in Section 3 matches the dimension precisely: that is, when is |Λfin| = dr−1 1 d2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' One case in which this is true is fairly straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' If n and r are relatively prime, dim(W) = nr/ � dr−1 1 d2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Suppose gcd(r, n) = 1 and consult Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Then, Scochar = dr−1 1 d2 n lcm � n, gcd � c + (r − 1)d, n, dn d1 �� = dr−1 1 d2 as the gcd above is a factor of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' However, the general conditions are a bit more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Suppose n = pm1 1 · · pmj j ,, and for each pi, we have c − d ≡ cipsi i (mod pmi i ), d ≡ dipti i (mod pmi i ), and r ≡ ripµi i (mod pmi i ), where ci, di, and ri are coprime to pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Then we have dim(W) = nr/ � dr−1 1 d2 � if and only if one of the following three conditions is true for every i: si < ti + µi and 2si ≤ ti + mi, si > ti + µi and si ≤ mi − µi, or si = ti + mi and 2si + τi ≤ ti + mi, where τi is the number of powers of pi in ci + ridi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' 18 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Using Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='7, Scochar = dr−1 1 d2 if and only if gcd(d2, dn d1 ) = d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' That is, precisely when d2 divides dn d1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Since the left side divides n, it suffices to check that for every prime factor pi of n, the power of pi in d2 divides that in dn d1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Given any pi, by the proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='2, we know that the power of pi on the right hand side is ti+mi−si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Similarly, the power of pi appearing in c−d+rd is min{si, ti+µi}+τi·δsi=ti+µi, where τi is the power of pi appearing in ci + ridi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Thus, it suffices to determine exactly when min{si, ti + µi} + τi · δsi=ti+µi ≤ ti + mi − si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' (9) To do so, we split into the same cases we used in the proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='2, based on the power of pi appearing in d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' First, suppose si < ti + µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Then we wind up in the first condition, because (9) is true precisely when si ≤ ti + mi − si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Then, suppose si > ti + µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Then (9) is true if and only if µi ≤ mi − si satisfying the second condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Lastly, suppose si = ti + µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Then (9) is equivalent to the third condition 2si + τi ≤ ti + mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Using the same techniques, we can also describe all the cases when dim(W) = (n/d1)r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' As we will see later, the quantum module connected to W has dimension (n/d1)r, so this is a necessary condition for the map to be an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Suppose n = pm1 1 · · pmj j ,, and for each pi, we have c − d ≡ cipsi i (mod pmi i ), d ≡ dipti i (mod pmi i ), and r ≡ ripµi i (mod pmi i ), where ci, di, and ri are coprime to pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Then dim(W) = (n/d1)r if and only if for every i, we have 2si ≤ mi + ti and at least one of the following conditions: si < ti + µi, si = ti + µi and 2si = ti + mi, or si = ti + µi and c + (r − 1)d contains no additional powers of pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Using Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='7, dim(W) = (n/d1)r if and only if gcd(d2, dn d1 ) = d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Using the machinery developed in the proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='2, notice that both sides are factors of n, so it suffices to check that the powers of each prime pi appearing in the prime factorization of n match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Let n = pm1 1 · · pmj j and suppose that c − d ≡ cipsi i (mod pmi i ) and d ≡ dipti i (mod pmi i ), where ci and di are coprime to pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Also, note that r ≡ ripµi i (mod pmi i ), where ri is also coprime to pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Then the power of pi appearing in d1 is si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' From the proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='2, recall that the power of pi appearing in dn d1 is ti + mi − si and the power of pi appearing in c − d + rd is min{si, ti + µi} + τi · δsi=ti+µi, where τi is the power of pi appearing in ci + ridi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' So gcd(d2, dn d1 ) = d1 if and only if min{min{si, ti + µi} + τi · δsi=ti+µi, ti + mi − si} = si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' (10) As in the proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='2, we split into three cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' If si < ti + µi, then (10) gives us min{si, ti + mi − si} = si, which is true precisely when ti + mi − si ≥ si, satisfying the first conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' If si > ti + µi, then we have a contradiction, since the minimum in (10) is already less than si, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Finally, if si = ti + µi, then (10) is min{si + τi, ti + mi − si} = si, which is true exactly when 2si = ti + mi or τi = 0 and si ≤ ti + mi − si, satisfying the second and third conditions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' 19 We have seen in this section that the two different formulations of Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='7 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='1 are useful for many different purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' While the approach used to generate Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='1 provides a more natural path for generalization beyond GLr(F), it would be interesting in future work to investigate whether there is an analogous approach to that used in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='7 for other reductive groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' In particular, understanding how Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='7 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='1 are related for the case of GLr(F) will illuminate a path for extending this connection further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' 8 Quantum Connections Finally, we marshal together results from the previous sections to investigate how the space of Whittaker functions is connected to quantum group modules, building the necessary quantum definitions along the way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Let Uq(c, d, n) be the affine quantum group Uq(�gl(n/d1)), where q is the cardinality of the residue field for our nonarchimedean local field F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' For the results of this paper, we will not need the precise definition here, so we refer the reader to Chari and Pressley [6] for the details of the construction and instead note merely a few interesting facts about Uq(c, d, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' First, despite the name, Uq(c, d, n) is not a group, but rather an algebra, specifically a quasitriangular Hopf algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' That is, it is both an algebra and a coalgebra, so it comes equipped with not only multiplication and a unit map but also comultiplication, a counit, and an antipode map relating the algebra and coalgebra structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Furthermore, this quantum group has a very nice set of modules which we can model concretely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' For z ∈ C, let V+(z) be an evaluation module, or evaluation representation, for Uq(c, d, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Again, we will not need the full structure of this representation for this paper, but following Kojima [12] as a convenient source, note that V+(z) is n/d1-dimensional and its basis may be parametrized by the elements of Z/(n/d1)Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' In addition, Uq(c, d, n) comes with an invertible element called a universal R-matrix R ∈ Uq(c, d, n) ⊗ Uq(c, d, n), which acts on tensor products of Uq(c, d, n)-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Choosing a particular pair of modules and their bases, R becomes an honest-to-goodness matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' It is this R-matrix that sparked the connection between Whittaker functions and quantum groups: R- matrices are natural sources for solutions to Yang-Baxter equations, functional relations from statistical mechanics that that arise, among other places, in the theory of lattice models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' In [2], Brubaker, Bump, Chinta, Friedberg, and Gunnells constructed a ice-type lattice model called Metaplectic Ice which computes metaplectic Whittaker functions for the nicest cover (c = 1, d = 0, so d1 = d2 = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' However, the Yang-Baxter equation for this model was unknown until Brubaker, Bump, and Buciumas identified it as a Drinfeld twist of the R-matrix for Uq(�gl(n)) in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Using the lattice model as a bridge, they mapped the space of Whittaker functions on this cover isomorphically into the tensor product V+(z1) ⊗ · · · ⊗ V+(zr), where zi ∈ C are the Satake parameters for the principal series representation on which the Whittaker function space W = Wz is built.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Under this isomorphism, the action of the R-matrix on the components of V+(z1) ⊗ · · · ⊗ V+(zr) matches precisely the action of intertwining operators on the principal series representation and thus the Whittaker function space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Fantastically, this connection extends for any metaplectic cover of GLr(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' In [7], the second author built a generic lattice model for an arbitrary covering group, and used it to construct a map between the space of Whittaker functions and a quantum group module for the quantum group Uq(c, d, n) = Uq(�gl(n/d1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' However, as we saw already from the formulae for dim(W) and the structure theory in Section 7, changing the parameters c and d results in a significantly more complicated function space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' These complications extend to the map, as the quantum space changes differently than W does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' In spite of this, the map prescribed by the lattice model in the fully general case is still a homomorphism and it matches exactly the actions of the R-matrix on the right side to those of the intertwining operators on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Consider the tensor product V+(z1) ⊗ · · · ⊗ V+(zr) of quantum group evaluation modules for U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' As a vector space, we have dim � V+(z1) ⊗ · · · ⊗ V+(zr) � = � n d1 �r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Note that unlike either of the formulae for dim(W) in Main Theorem 1, this formula is not affected by d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' 20 Now we come to the connection precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Using Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='1, take representatives for the cosets �T/H from the set (Z/nZ)r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='2, use these representatives to construct a basis for W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='1 (Frechette [7], Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Let ρ = (r − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=', 2, 1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' For z ∈ Cr, the map θz : Wz → V+(z1) ⊗ · · · ⊗ V+(zr) ν �→ ρ − ν (mod n/d1), where the modulus is taken in each component, is a homomorphism compatible with the actions of intertwining operators on W and the R-matrix on the quantum tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' One of the difficulties that arose in extending from the nicest cover to generic covers is that the lattice model specifies a choice of basis for W that makes this map a homomorphism, but the lattice model itself is not necessary for the proof and serves as a removable bridge between the Whittaker function space and the quantum group model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Without the lattice model, however, there is no canonical choice of basis for W, so we ask: when is this map well-defined regardless of the choice of representative for each coset in �T/H?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Using the structure of W developed in Section 7, we can investigate this map more precisely, and arrive at the following theorem, which is a restatement of the first part of Main Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' For the metaplectic cover �Gc,d,r,n, the map θz : W → V+(z1) ⊗ · · · ⊗ V+(zr) from Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='1 is well-defined independent of choice of coset representatives for �T/H if and only if gcd � d2, dn d1 � = gcd(c, d, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Using the characterization developed in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='1, θz is well-defined if and only if all the elements in Λfin map to the same element in the module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Using the description of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='4, write x = x1 · 1r + n d1 (0, v2, v3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=', vr) and y = y1 · 1r + n d1 (0, v′ 2, v′ 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=', v′ r), for x1, y1, vi, v′ i ∈ Z for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Thus, θz(x) − θz(y) = (y1 − x1) · 1r (mod n/d1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Since x, y ∈ Λfin, we have y − x ∈ Λfin as well, so the defining cocharacter equations give more information about the possible values of y1 − x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Using the first cocharacter equation, there exists k ∈ Z such that (c + (r − 1)d) · (y1 − x1) ≡ dn d1 k (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Varying over all x, y ∈ Λfin, the possible values for the right hand side of this equation are precisely the integer multiples of gcd � n, dn d1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Then, both sides must be a multiple of lcm � c + (r − 1)d, gcd � n, dn d1 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Using the fact that lcm(A, B) = (A · B)/ gcd(A, B), the possible values for y1 − x1 are all the integer multiples of gcd � n, dn d1 � gcd � c + (r − 1)d, n, dn d1 � = n d1 gcd (d1, d) gcd � d2, dn d1 � = n d1 gcd (c, d, n) gcd � d2, dn d1 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' (11) Going back to the map, θz(x) − θz(y) = 0 if and only if y1 − x1 ≡ 0 (mod n/d1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Since gcd(c, d, n) divides both d2 and dn d1 , we have that the expression in (11) is a multiple of n d1 if and only if gcd(c, d, n) = gcd � d2, dn d1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Therefore the map θz is well-defined for any choice of coset representatives of �T/H if and only if gcd(c, d, n) = gcd � d2, dn d1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' When W is either maximum or minimum size, θz is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' 21 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' If W is of maximum size nr, then by Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='3, we have d1 = d2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Thus, gcd � d2, dn d1 � = 1, which forces gcd(c, d, n) = 1, so θz is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' In this case �T/H is parametrized by all of (Z/nZ)r, and since n/d1 = n, so is the quantum module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Looking at the description of θz in Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='1, we see that θz is an isomorphism by definition, flipping W and shifting by ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' If W is of minimum size 1, then Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='2 shows that d1 = d2 = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Thus, gcd(c, d, n) = n, which forces gcd � d2, dn d1 � = n and makes θz well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Here, �T /H is a single element, which maps to the single element 0 in (Z/(n/d1)Z)r, since n/d1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Thus the map is vacuously an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Note that the first case of Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='3 includes the nicest cover c = 1, d = 0 originally treated by [2] and [1], explaining why the quantum map on W for this case is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Using our description of Λfin, we intend in the future to come up with a precise description of the structure of W in the style of Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='3 for more general cases, which will allow us to characterize the precise behavior of θz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' In particular, we are interested in providing a companion to Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content='6 by finding a sufficient condition for all cases when θz is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Extending our methods and results for W from GLr(F) to arbitrary reductive groups will then give us more information about what the quantum objects connected to other types of reductive groups should be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' While some solvable lattice models for other types exist, they have not yet been linked to modules for quantum groups or other quantum algebraic objects, so we believe that investigating the dimension and description of W for other types will illuminate likely candidates for broader quantum connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' References [1] Ben Brubaker, Valentin Buciumas, Daniel Bump, and Nathan Gray, A Yang-Baxter equation for meta- plectic ice, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' Number Theory Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' 13 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfSgB1/content/2301.02223v1.pdf'} +page_content=' 1, 101–148.' metadata={'source': 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b/NdE2T4oBgHgl3EQfqwhx/content/tmp_files/2301.04042v1.pdf.txt @@ -0,0 +1,1247 @@ +arXiv:2301.04042v1 [astro-ph.GA] 10 Jan 2023 +Research in Astronomy and Astrophysics manuscript no. +(LATEX: ms2022-0209.tex; printed on January 11, 2023; 1:30) +Modeling the vertical distribution of the Milky Way’s flat subsystem +objects +Igor’ I. Nikiforov, Vadim A. Usik and Angelina V. Veselova +Saint Petersburg State University, Universitetskij Prospekt 28, Staryj Peterhof, Saint Petersburg 198504, +Russia; i.nikiforov@spbu.ru (IIN) +Received 20XX Month Day; accepted 20XX Month Day +Abstract This paper is an initial stage of consideration of the general problem of joint mod- +eling of the vertical structure of a Galactic flat subsystem and the average surface of the disk +of the Galaxy, taking into account the natural and measurement dispersions. We approximate +the average surface of the Galactic disk in the region covered by the data with a general +(polynomial) model and determine its parameters by minimizing the squared deviations of +objects along the normal to the model surface. The smoothness of the model, i.e., its order n, +is optimized. An outlier elimination algorithm is applied. The developed method allows us to +simultaneously identify significant details of the Galactic warping and estimate the offset z⊙ +of the Sun relative to the average (in general, non-flat) surface of the Galactic disk and the +vertical scale of the object system under consideration for an arbitrary area of the disk covered +by data. The method is applied to data on classical Cepheids (Berdnikov et al., Mel’nik et al.). +Significant local extremes of the average disk surface model were found based on Cepheid +data: the minimum in the first Galactic quadrant and the maximum in the second. A well- +known warp (lowering of the disk surface) in the third quadrant has been confirmed. The opti- +mal order of the model describing all these warping details was found to be no = 4. The local +(for a small neighborhood of the Sun, no = 0) estimate of z⊙ = 28.1± 6.1|stat. ± 1.3|cal. pc is +close to the non-local (taking into account warping, no = 4) z⊙ = 27.1 ± 8.8|stat. ++1.3 +−1.2 +�� +cal. pc +(statistical and calibration uncertainties are indicated), which suggests that the proposed mod- +eling method eliminates the influence of warping on the z⊙ estimate. However, the non-local +estimate of the vertical standard deviation of Cepheids σρ = 132.0 ± 3.7|stat. ++6.3 +−5.9 +�� +cal. pc dif- +fers significantly from the local σρ = 76.5 ± 4.4|stat. ++3.6 +−3.4 +�� +cal. pc, which means the need to +introduce more complex models for the vertical distribution outside the Sun’s vicinity. +Key words: Galaxy: disk — Galaxy: structure — Galaxy: fundamental parameters — meth- +ods: data analysis + +2 +Nikiforov et al. +1 INTRODUCTION +The vertical distribution of objects in various Galactic subsystems contains valuable information about the +origin, evolution and dynamics of our Galaxy, so determining the characteristics of this distribution is an +important task of Galactic Astronomy. The study of the vertical distribution may include consideration +of many phenomena, but one of them should be taken into account necessarily—this is the offset z⊙ of +the Sun relative to the plane of the Milky Way’s disk towards the North Galactic Pole. Therefore, in the +simplest case, modeling of the vertical distribution is reduced to determining the value of z⊙ and some +dispersion parameter (standard deviation, scale height, etc.) that characterizes the scattering of subsystem +objects relative to the average plane of the Galactic disk (usually relative to the midplane of this subsystem). +The first estimate of z⊙ = 13.5 ± 1.7 pc was obtained by van Tulder (1942) from the analysis of nearby +stars. Subsequently, in many papers, the solar offset was determined by different methods for various objects +and Galactic subsystems. Bland-Hawthorn & Gerhard (2016) adopted as the best (local) estimate the result +of Juri´c et al. (2008) from the complete SDSS photometric survey, z⊙ = 25 ± 5 pc, which covers many +other estimates. +However, the solar offset relative to the differently defined midplane of the disk does not seem to be +described by a single value of z⊙. For example, Bobylev & Bajkova (2016b) obtained significantly different +results for reference objects (tracers) of different types: z⊙ = 5.7±0.5 pc for a sample of methanol masers, +z⊙ = 7.6 ± 0.4 pc for data on H II regions and z⊙ = 10.1 ± 0.5 pc for data on giant molecular clouds; +at the same time, Ferguson et al. (2017) derived values of z⊙ = 14.9 ± 0.5 pc for a uniform selection +of SDSS K and M dwarf stars and z⊙ = 15.3 ± 0.4 pc for an expanded selection, Buckner & Froebrich +(2014) found an estimate z⊙ = 18.5 ± 1.2 pc for open clusters, and Majaess et al. (2009) obtained values +of z⊙ = 26 ± 3 pc for Cepheids. A comparison of these and other z⊙ estimates obtained in various studies +(see, e.g., summaries in Yao et al. 2017; Skowron et al. 2019a) shows that the differences between these +estimates cannot be explained only by statistical errors, with some estimates varing significantly, even for +objects of the same type (e.g., for open clusters and Cepheids). This shows that the discrepancies reflect not +only the possible objective difference in the values of z⊙ between different types of objects (subsystems of +the Galaxy), but also other factors in the problem. +In addition to the Z-offset of the Sun, the number of already established or potential factors affecting +the results of modeling the vertical distribution of objects includes: 1) the warp of the Galactic disk, 2) the +dependence of the values of the characteristics of the vertical distribution on the position on the disk for +the selected Galactic subsystem (e.g., the flare of the Galactic disk), 3) the possible (and in the case of +vertical dispersion, real) dependence of these characteristics on the type of Galactic subsystem, 4) the need +to establish the functional type of the vertical distribution and its possible variations with the position on +the disk and with the type of subsystem, as well as 5) taking into account the random uncertainty of helio- +centric distances, systematically distorting the true vertical distribution. The problem in general (taking into +account all these factors) has not yet been solved. Meanwhile, different combinations of these factors may +be responsible for discrepancy of the results (in particular, of z⊙ estimates) in different papers. Subjective +factors can also lead to this: the choice of the general appearance of the model of the average surface of + +Modeling the vertical distribution of disk objects +3 +the disk, possible mismatch of the distance scales used in different works, the dependence of the results of +modeling on the size and configuration of the disk area under consideration (the area covered by the data). +Despite the lack of a solution to the problem in general, some of these factors and their combinations +were considered. The most important factor is the presence of a warp of the Milky Way’s disk. The warp was +noticed as soon as the observation data in the 21-cm line of neutral hydrogen appeared for the southern hemi- +sphere (Burke 1957; Kerr 1957). Subsequent studies (Oort et al. 1958; see, e.g., Binney & Merrifield 1998 +and Bland-Hawthorn & Gerhard 2016 reviews and references therein; Skowron et al. 2019a; Chrob´akov´a +et al. 2020, among others) have shown that a significant stellar/gas warp begins outside the solar circle, and +in the inner Galaxy the disk is very close to flat, including on the far side of the disk (Minniti et al. 2021). +Various data indicate that one part of the warped disk deviates from the plane of the inner disk towards the +North Galactic Pole, the other deviates in the opposite direction. +Not taking into account the large-scale warp (if a plane parallel to the equator of the Galactic coordinate +system b = 0◦ is taken as a model of the average surface of the Galactic disk) can significantly affect +the estimates of the solar offset z⊙ and the vertical scales of flat subsystems (see, e.g., the dependence of +these characteristics for planetary nebulae on the size of considered near-solar region in Bobylev & Bajkova +2017). One way to avoid this is to exclude the warp zone from consideration under the assumption that in +the remaining area of the disk its average surface is flat: restrictions are imposed on the selection of tracers, +for example, by the heliocentric distances r (e.g., r ≲ 4 kpc in Bobylev & Bajkova 2016a; r ≲ 4.5 kpc +in Bobylev & Bajkova 2016b), by the predicted maximum warp offsets (<10 pc in Yao et al. 2017), by +the distance R to the axis of rotation of the Galaxy (R < 7.0 kpc in Reid et al. 2019). However, the +exclusion of the warp zone requires the adoption of a specific warp model, and it is often taken simple for +this and other applications: the disk in the inner Galaxy (R ≤ Rw) is considered undisturbed, and in the +outer one (R > Rw) it is usually represented by a combination of a power dependence on R and a simple +trigonometric function of the azimuthal coordinate (e.g., Binney & Merrifield 1998; Pohl et al. 2008; Xu +et al. 2015; Yao et al. 2017; Romero-G´omez et al. 2019; Cheng et al. 2020; Mosenkov et al. 2021). At the +same time, to describe the warp in the outer Galaxy, in most of its morphological studies, simple symmetric +models with a limited set of parameters are used—the radius Rw at which the disk starts bending, the phase +angle of the line-of-nodes and the maximum amplitude of the warp (see, e.g., Romero-G´omez et al. 2019 +and references therein). +However, the warp is clearly more complicated. Firstly, the inner part of the disk is not perfectly flat— +there are corrugations on the scale of ∼30 pc (Oort et al. 1958, fig. 3; Spicker & Feitzinger 1986; Binney +& Merrifield 1998, fig. 9.22). N-body simulations of the Milky Way interacting with a satellite similar to +the Sagittarius dwarf galaxy show that repeated satellite passes can generate local ripples, including in the +inner disk (Poggio et al. 2020, fig. 2). According to kinematics, the onset of the warp occurs at a guiding +radius inside the Solar circle, Rg ≲ 7 kpc (Schoenrich & Dehnen 2018), or even in the center of the Galaxy +(Li et al. 2020). Secondly, the outer part of the warp is also not described by a simple model—there are +manifestations of lopsidedness of the warp and twisting of its line-of-nodes (Romero-G´omez et al. 2019; +Chrob´akov´a et al. 2020); Xu et al. (2015) detected an oscillating asymmetry in the SDSS main-sequence star +counts on either side of the Galactic plane in the anticenter region, between longitudes of 110◦ < l < 229◦. + +4 +Nikiforov et al. +In addition, the morphology and kinematics of the warp depend on the type/age of the tracers (e.g., Romero- +G´omez et al. 2019; Chrob´akov´a et al. 2020). Moreover, hydrodynamic modeling of the evolution of an +ensemble of stars formed in the warp shows that only younger populations trace the warp detected by HI +(Khachaturyants et al. 2021) and that the influence of the bending waves excited by irregular gas inflow is +most strongly manifested in the young populations (Khachaturyants et al. 2022). This means that the warp +model, universal for all disk subsystems of the Galaxy, can hardly be accepted. +Kinematic manifestations of the warp also indicate its asymmetry and complexity in general, as well as +the dependence of its characteristics on the age of tracers (e.g., Romero-G´omez et al. 2019; Li et al. 2020; +Cheng et al. 2020 and references in these works). +Based on the above, the exclusion of the warp zone as a method of eliminating biases in the vertical dis- +tribution parameters can only give a partial (local) solution to the problem, the accuracy of which depends +on the details of the accepted warp model and on its realism in the case of the tracers under consideration +and on assumptions about the boundaries of the warp-distorted area. All these assumptions can be sources +of systematic errors. That is why it seems important to us to abandon simplified warp models and consider +the most general analytical warp model describing all the significant structural features of the middle sur- +face of the disk identified by the tracers under consideration. The method of excluding the warp zone is also +unsuccessful due to the presence of the disk flaring, which begins at R ≳ R0, where R0 is the Galactic cen- +ter distance (e.g., Reid et al. 2019; Mosenkov et al. 2021), since the dependence of the dispersion parameter +on the accepted boundaries of the area “undisturbed” by the warp appears. +The warp is currently being actively explored in many ways. In particular, warp precession is actively +discussed (e.g., Cheng et al. 2020). However, as noted by Poggio et al. (2020), the precession parameters +depend on our knowledge of the shape of the warp and its differences for different stellar populations. In ad- +dition, Chrob´akov´a & L´opez-Corredoira (2021) even raise the question of the very existence of precession, +since the application of a warp model inconsistent with the tracers used leads to a fictitious precession. +Detailed warp models are also important both for studying the dependence of z⊙ and vertical dispersion +characteristics on the type/age of tracers, and for identifying the cause and dynamic nature of the warp of +our Galaxy, which remain unclear (Binney & Merrifield 1998; Poggio et al. 2020; Khachaturyants et al. +2021). +Note also that in the framework of an alternative approach applied by Mosenkov et al. (2021)— pho- +tometric 3D decomposition of the Milky Way taking into account flaring and warp—the parameters of the +warp disk are poorly determined, since only a 2D map is considered, whereas for creating a reliable 3D +model of the warp one needs to have a 3D distribution of stars in the Galaxy. +Despite the fact that the best solution would be to model the Z-distribution of objects taking into account +all the factors mentioned at once, due to the complexity of the overall task we focus in this paper on the task +of constructing a detailed warp model with a minimum of assumptions. We will not consider the influence +of random errors in the distance here (the selected data catalog allows this, see Sect. 3), as well as the +disk flaring, since without taking into account errors in distances, the flaring parameters may turn out to be +strongly biased. + +Modeling the vertical distribution of disk objects +5 +2 METHOD +We will study the spatial distribution of objects in the heliocentric Cartesian coordinate system, which does +not require taking any value of R0: X-axis is directed towards the Galactic center, Y -axis is towards the +rotation of the Galaxy, Z-axis is towards the North Galactic Pole. +In order to free the warp model as much as possible from pre-accepted assumptions, we will consider as +models the ζn(X, Y ) polynomials, each of which is a Maclaurin series expansion in the solar neighborhood +up to the n-th order of the average Z-coordinate of the Galactic disk as a function of position on the XY +plane: +ζn(X, Y ) = +n +� +i=0 +i +� +j=0 +zi−j, jXi−jY j += z00 + z10X + z01Y + z20X2 + z11XY + z02Y 2 + . . . + zn0Xn + . . . + z0nY n, +(1) +zζ = +� +zij +�n i +0 0 , +z⊙ = −z00. +Here zζ is the vector of M = (n+2)(n+1)/2 parameters of the model. Function (1) represents the average +surface of the Galactic disk, defined by the spatial distribution of objects (in the general case of matter) of +the selected Galactic subsystem. The distance ρ from the object to the surface ζn(R, Z) along the normal +to this surface will be considered as the value of the deviation of the object from the model average surface +of the disk. +To obtain an estimate of the vector zζ, we generally use the maximum likelihood method. In this paper, +to simplify, we assume that ρ as a random variable is distributed according to a normal law with zero mean, +that is, that the probability density of ρ has the form +f(ρ) = +1 +σρ +√ +2π e +− ρ2 +2σ2ρ , +(2) +where σρ is the standard deviation. We assume that the value of σρ is the same for all objects in the sample. +However, in future work we propose to investigate and take into account the dependence of σρ and/or other +dispersion parameters on the Galactocentric distance. With probability density (2), the likelihood function L +and logarithmic likelihood function L are +L = +N +� +i=1 +1 +σρ +√ +2π e +− +ρ2 +i +2σ2ρ , +L ≡ − ln L = N +2 ln (2π) + N ln σρ + 1 +2 +N +� +i=1 +ρ2 +i +σ2ρ +, +(3) +ρi = ρi(ri, li, bi; zζ), +where N is the number of objects in the sample; ri, li and bi are the heliocentric distance, galactic longitude +and latitude of i-th object, correspondingly. Being a dispersion parameter under parametrization (2), σρ +is included in the general vector of the problem parameters z = (σρ, zζ) = (z1, z2, . . . , zK), where +K = M + 1. The minimum of the function L gives estimates of the parameters, including the value of the +standard deviation σρ of objects across the disk, which represents the contributions of both the true (natural) +vertical dispersion of objects and the random uncertainty of distance estimates (the latter contribution is +negligible, see Section 5). Thus, under assumption (2), the maximum likelihood method was reduced to + +6 +Nikiforov et al. +the nonlinear least squares method (Eq. (3)). The resulting value of σρ was multiplied by the coefficient +� +N/(N − M) to obtain an unbiased estimate. The vector zerr of mean parameter errors and the mean +model prediction error σζn(X, Y ) were calculated based on the Hessian H(L) with elements hij (Hudson +1964; Wall & Jenkins 2012): +hij = +∂L2 +∂zi∂zj +, i = 1, . . . , K, j = 1, . . . , K, +C = [H(L)]−1, +zerr = {√cii}K +i=1 , +σ2 +ζn(X, Y ) = {grad[ζn(X, Y ; zζ)]}T C′ grad[ζn(X, Y ; zζ)], +(4) +where cii are diagonal elements of the covariance matrix C, C′ is a submatrix of C that does not contain +covariances involving σρ; here, an estimate of the vector z obtained by minimizing L is substituted for all +values. +After finding the parameters, an outlier exclusion algorithm described in Nikiforov (2012) was applied +to the sample of objects under consideration. This algorithm differs from the usual 3σ criterion in that it +uses a variable exclusion limit, which increases as the number of objects increases. +In order to check how well the observed distribution of sample objects by deviations ρ agrees with the +model probability density function (2), we use Pearson’s chi-square test. +A priori choice of the order n of expansion (1) would lead to significant errors in all parameters, includ- +ing z00 (see Sect. 4), so the value of n was also optimized using a simple algorithm. Models of the order +n = 0, 1, 2 and so on are built sequentially. For each model of order n, the number of parameters zij of +order n (i + j = n) whose estimates differ from zero at the significance level ≥2σ (σzij/|zij| ≤ 0.5) is +calculated. Then we find a model of the highest order such that it has at least one 2σ-significant parameter +of the same order as the model. If the total number of significant parameters of the selected model is greater +than the corresponding number for any lower-order model, then the selected model of order n is assumed +to be optimal. Otherwise, a model of order n − 1 is considered as possibly optimal and is compared in the +same way with models of lower orders in terms of the number of significant parameters. Then either the +order n − 1 is assumed to be optimal, or a transition is made to the order n − 2, and so on until some order +n ≥ 0 is accepted as optimal, no. The importance of choosing the correct model order will be illustrated in +Section 4. +3 DATA +We use the catalog of classical Cepheids by Berdnikov et al. (2000) in the version of Mel’nik et al. (2015), +which provides data for 674 Cepheids from the General Catalogue of Variable Stars (e.g., Samus et al. +2017). The catalog is an updated version of the catalog of Cepheid parameters by Berdnikov et al. (2000). +Observational data were obtained with 0.4–1 m telescopes of the Maidanak Observatory (Republic of +Uzbekistan), Cerro Tololo and Las Campanas observatories (Chile), Cerro Armazones Observatory of the +Catholic University (Chile) and South African Astronomical Observatory (see Dambis et al. 2015 for de- +tails). In particular, in order to reliably determine the distances, Cepheids discovered during the CCD moni- +toring of the southern sky performed as a part of the All Sky Automated Survey (ASAS) project (Pojmanski +2002) were observed, therefore, a survey of Cepheids across the entire sky was performed. The peak of the +distribution of Cepheids by the mean apparent magnitudes in the V band falls on the values of ⟨V ⟩ = 12– +13 mag, the limiting magnitude of the catalog is ⟨V ⟩ = 15 mag. The distances were obtained based on the + +Modeling the vertical distribution of disk objects +7 +period–luminosity relation in the K infrared band and interstellar-extinction law using the period – normal +color (B − V ) relation derived earlier (see Dambis et al. 2015). +The authors of the catalog did not specify the possible value of the distance modulus error. However, the +analysis of these data in Veselova & Nikiforov (2020) showed that the nominal random errors of distance +estimates given in Mel’nik et al. (2015) are small (the mean error of distance moduli σd < 0.14m). Using +this distance catalog gives us the opportunity to apply a simpler method that does not take into account ran- +dom distance errors. In the future, we intend to use newer data covering more extensive areas where taking +into account the uncertainty of distances within a more complex method is necessary. Recent versions of +Berdnikov et al.’s catalog have been successfully used to study the Galactic structure. Based on the catalog, +Mel’nik et al. (2015) identified signs of ring formations in the Galaxy, Dambis et al. (2015) and Veselova +& Nikiforov (2020) performed spatial modeling to determine the parameters of spiral arm segments. +According to the original distance scale calibration of the catalog the distance to the Large Magellanic +Cloud (LMC) is d∗ +LMC = 18.25 ± 0.05 mag (Berdnikov et al. 2000). Modern LMC calibration is dLMC = +18.49 ± 0.09 mag (de Grijs et al. 2014), which leads to a correction factor c for the distances of the catalog +used: +c = r + ∆r +r += 100.2∆d = 1.117+0.053 +−0.050, +∆d = dLMC − d∗ +LMC. +(5) +We analyzed the original catalog estimates of distances, and then adjusted the main results for the factor c. +Spatially isolated objects were manually excluded from the initial sample, i.e., those that are nominally +located clearly far from the main group of catalog objects. The remaining objects, the totality of which +we called the working sample, were used for calculations. After applying the outlier elimination algorithm +(Nikiforov 2012) during the analysis of the working sample, we obtained the final working sample contain- +ing 615 objects (Fig. 1). +We also identified a local sample, which is part of the working sample representing the largest neigh- +borhood of the Sun with the property of relative completeness of the identification of objects of this type +(classical Cepheids). Assuming a homogeneous distribution of objects projected onto the XY -plane (the +Galactic plane) in a vicinity of the Sun, the number of objects projected onto a section of the XY -plane +should be proportional to the area of this section. Usually, a set of concentric rings is used to select a local, +close to complete, subsample, but the working sample has an asymmetry with respect to the origin of co- +ordinates on the XY plane. For this reason, it was decided to use a set of rectangular frames with borders +homothetic to the rectangular border of the working sample: the lengths of each side of the frame vary +according to a given scale factor, and the ratio of distances from the origin to the edges of the frames remain +constant (Fig. 2, left panel). In the case of a small frame width, the area of the frame will be proportional to +the outer perimeter, so for a complete sample, the number of objects in the frame should increase linearly +with the size of the frame (scale factor). On the right panel of Fig. 2 the linear growth is observed within +the first three bins (frames), which are marked in red; objects within these frames were taken as a local +sample. The boundaries of this sample are shown in Fig. 1 (purple rectangle). The sample size is 154. After +excluding an outlier, the final local sample of 153 objects was obtained. + +8 +Nikiforov et al. +Fig. 1: Objects of the final working sample and outliers projected onto the XY plane. The purple rectangle +marks the boundary of the local sample (see text). X-axis is directed towards the Galactic center, Y -axis is +towards the rotation of the Galaxy. The Sun is placed at X = 0 kpc, Y = 0 kpc. +4 RESULTS +The influence of the accepted order of the model ζn on the results can be illustrated by the example of +the dependence of z00 on n for the final working sample (Fig. 3). It can be seen that for small orders the +estimates of z00 strongly depend on n, while for larger orders the estimates vary insignificantly. +The optimal order of the model ζn for the final working sample turned out to be no = 4. Estimates of +model parameters are presented in Table 1, significant estimates are given in bold. For the local and final +local samples, no is 0. Final results for z⊙ and σρ without and with the correction of distance scale (5) for +the final working and final local samples are listed in Table 2. Comparison with Figure 3 shows that the +choice of an underestimated order (n < no) of the model can lead to an obviously incorrect estimate of z⊙ +(here at n = 0, 2). +The model surface ζ4(X, Y ) for the final working sample is shown in Figure 4. Here the level +ζ4(X, Y ) = z00 +(6) +is represented by a black line. This line is the intersection of the model of the average surface of the disk with +the nominal plane of the Galaxy XY . Line (6) corresponds to the line-of-nodes in simple models. Figure 4 +shows that in reality the line (6), skirting the areas of local extrema, is quite curved. By analogy with the + +2 +-2 +kpc +-6 +-10 +Final working sample +-14 +Outliers +-4 +0 +4 +8 +X, kpcModeling the vertical distribution of disk objects +9 +Fig. 2: Left: Homothetic boundaries of rectangular frames used to construct the local sample (see text), +superimposed on the distribution of objects of the working sample in projection on the XY plane. Right: +Histogram of the distribution of objects of the working sample in rectangular frames. The red color cor- +responds to the frames within which the sample (local sample) can be considered complete; the dashed +straight line shows the linear growth area of the number of objects in the first three bins. +Fig. 3: Dependence of z00 (= − z⊙) on the model order n for the final working sample. Vertical bars show +the uncertainties of estimates. +line-of-nodes, the curve ζno(X, Y ) = z00 can be called a “curve-of-nodes”. Of course, the constructed +model ζ4(X, Y ) is real only for the area of the disk that is covered by the data used. Boundaries of areas +within which the mean error of the model is σζ4(X, Y ) ≤ 1 +6σρ = 20 pc, ≤ 1 +4σρ = 40 pc and ≤ 1 +2σρ = 59 pc +shown in Figure 4 by dotted, dashed and dashed–dotted lines, respectively, give an idea of the applicability +of the model ζ4 depending on the specified level of its uncertainty. Here σζ4(X, Y ) was calculated using +the formula (4), and σρ = 119 pc (see Table 1). + +25 +0 +pc +100Z +-25 +-50 +-75 +0 +L +N +3 +4 +5 +6 +n80 +70 +60 +Z50 +40 +30 +20 +10 +0 +.0 +.2 +.4 +.6 +.8 +Scalefactor5.0 +2.5 +0.0 +2.5 +kpc +-5.0 +7.5 +-10.0 +-12.5 +-15.0 +-5.0 +-2.5 +0.0 +2.5 +5.0 +7.5 +X, kpc10 +Nikiforov et al. +Table 1: Parameter estimates for the model ζn of the average surface of the Galactic disk of optimal order +no = 4 obtained for the final working sample of classical Cepheids. Estimates that differ from zero at a +significance level of at least 2σ are highlighted in bold. The standard deviation σρ is given in kiloparsecs, +while values of zij are in units of kpc1−i−j +Parameter +Estimation +σzi/|zi| +Parameter +Estimation +σzi/|zi| +σρ +0.1187 ± 0.0033 +0.03 +z21 +0.00289 ± 0.00094 +0.33 +z00 +−0.0243 ± 0.0079 +0.33 +z12 +0.00288 ± 0.00055 +0.19 +z10 +−0.0320 ± 0.0061 +0.19 +z03 +0.00180 ± 0.00028 +0.16 +z01 +−0.0131 ± 0.0040 +0.31 +z40 +−0.00012 ± 0.00012 +1.00 +z20 +0.0055 ± 0.0025 +0.45 +z31 +0.00013 ± 0.00014 +1.08 +z11 +0.0008 ± 0.0018 +2.25 +z22 +0.00022 ± 0.00010 +0.45 +z02 +−0.0007 ± 0.0010 +1.43 +z13 +0.000185 ± 0.000048 +0.26 +z30 +0.00203 ± 0.00087 +0.43 +z04 +0.000099 ± 0.000020 +0.20 +Table 2: Final estimates of z⊙ and σρ for the optimal models ζno(X, Y ) obtained for two samples of +Cepheids. The estimates are given in the original catalog distance scale and in the scale adjusted for cali- +bration dLMC = 18.49 ± 0.09 mag (see text) +Sample +Original distance scale +Corrected distance scale +Final working sample +z⊙ = 24.3 ± 7.9 pc +z⊙ = 27.1 ± 8.8 +�� +stat. ++1.3 +−1.2 +�� +cal. pc +(N = 615, no = 4) +σρ = 118.7 ± 3.3 pc +σρ = 132.0 ± 3.7 +�� +stat ++6.3 +−5.9 +�� +cal. pc +Final local sample +z⊙ = 25.2 ± 5.5 pc +z⊙ = 28.1 ± 6.1 +�� +stat. ± 1.3 +�� +cal. pc +(N = 153, no = 0) +σρ = 68.5 ± 3.9 pc +σρ = 76.5 ± 4.4 +�� +stat. ++3.6 +−3.4 +�� +cal. pc +An alternative representation of the resulting model is given in Figure 5, which shows the extremum +and boundary lines of the model surface ζ4(X, Y ) in projections on the planes XZ and Y Z in comparison +with the Cepheids of the final working sample. The extremum lines (there may be several of them for +each projection) are dependencies of Z-coordinates of points of local extremes of the surface ζ4(X, Y ) +with fixed X—ζmax(X, Y), ζmin(X, Y) (Figure 5, top panel) or with fixed Y —ζmax(X, Y ), ζmin(X, Y ) +(bottom panel). The boundary lines show the values of ζ4(X, Y ) at the boundaries of the final working +sample: ζ4(X, Ymin) and ζ4(X, Ymax) (top panel), and ζ4(Xmin, Y ) and ζ4(Xmax, Y ) (bottom panel), where +Xmin = −5.8 kpc, Xmax = 8.6 kpc, Ymin = −14.6 kpc, Ymax = 4.8 kpc. Figure 5 affirms that the extremum +lines mainly fall on the areas covered by the data, and the boundary lines indicate edge approximation +effects in the area X ≳ 4 kpc, Y ≳ 0 kpc. +The well-known general shape of the Galactic disk warp is clearly visible in Figure 4—in the first and +second quadrants, the model surface as a whole rises above the XY plane; in the third and fourth quadrants, +the surface decreases below this plane. In the area near X ≈ −3 kpc, Y ≈ −6 kpc, the decrease in the +average surface is observed for almost all objects. +However, in addition, we found two extrema in the first and second quadrants that do not fit into simple +warp models. The sections of the model with planes parallel to the XZ and Y Z planes and passing through + +Modeling the vertical distribution of disk objects +11 +Fig. 4: A map of the model ζ4 of the average disk surface of optimal order, constructed from the objects of +the final working sample. The color shows the value of the function ζ4(X, Y ). The objects are shown as +colored circles with color representing the Z-coordinate. The black line represents the z00-level, the red and +blue crosses depict the local extrema (maximum and minimum, respectively), the purple rectangle shows +the boundary of the local sample, the grey line denotes the solar circle R = R0 = 8 kpc. The dotted, dashed +and dashed–dotted lines are the boundaries of the applicable areas of the model σζ4, within which the mean +error of the model is σζ4(X, Y ) ≤ 1 +6σρ = 20 pc, ≤ 1 +4σρ = 40 pc and ≤ 1 +2σρ = 59 pc, respectively. +the points of extrema are shown in Figure 6. The significance S of the extrema was estimated by the formula +Sζ4(X, Y ) = ζ4(X, Y ) − z00 +σζ4(X, Y ) +. +(7) +Table 3 shows the parameters of local extrema. The given values of S show that the local minimum and +maximum are significant at the level of at least 2σ and 5σ, respectively. +In Figure 6 the slope of the model surface ζ4(X, Y ) to the nominal plane of the Galaxy XY is visible. +Having calculated the distance ρ⊙ from the Sun to the surface ζ4(X, Y ) along the normal to the latter, we + +5 +0.60 +0.30 +0 +0.15 +kpc +kpc +0.00 +N +-0.15 +-10 +-0.30 +-0.70 +-15 +-5 +X,kpc12 +Nikiforov et al. +Fig. 5: Extremum lines (thicker lines of lighter color) and boundary lines of the model surface ζ4(X, Y ) in +comparison with Cepheids of the final working sample in the projection on the plane XZ (top panel) and +Y Z (bottom panel); see text. Local maxima and lines of the largest values of ζ4 at the sample boundaries +are shown in red, while local minima and lines of the smallest boundary values are shown in blue. +Table 3: Parameters of local extrema of the optimal model ζ4(X, Y ) constructed for the final working sam- +ple: The Cartesian heliocentric coordinates X, Y of the extremum, the value of ζ4(X, Y ) at the extremum +point and the significance level Sζ4(X, Y ) of the extremum (the number of σ) +Extremum +X (kpc) +Y (kpc) +ζ4(X, Y ) (pc) +Sζ4(X, Y ) +Local maximum +−3.1 +0.6 +58.4 ± 15.0 +5.43 +Local minimum +1.4 +0.6 +−55.0 ± 11.0 +−2.81 + +(X, Ymin) +3 +((X, Ymax) +2 +min(X, ) +kpc +Smax(X, ) +1 +N +0 +O +-6 +-4 ++-2 +0 +2 +4 +6 +8 +X,kpc +(Xmin, Y) +3 +((Xmax, Y) +2 +Smin(X, 7) +kpc +Imax(X, y) +1 +N +0 +8 +-1 +C +-12 +-8 +-4 +0 +4 +Y, kpcModeling the vertical distribution of disk objects +13 +Fig. 6: Sections of the model ζ4(X, Y ) of the middle surface of the disk passing through the local maximum +(upper panels) and minimum (lower panels). Each panel shows the confidence area for the model values +ζ4(X, Y ) (±1σζ) and the area of object deviations from the model ±1σρ, as well as projections of the +position of objects located in the ±1 kpc band from the cut line. The green line indicates the position of the +plane Z = z00. +obtained an estimate of the angle of inclination of the local average surface of the Galactic disk to the plane +XY of the galactic coordinate system +γ = arccos +�ρ⊙ +z⊙ +� += 1.◦79+0.◦34 +−0.◦33. +(8) +The 1σ-uncertainty of the angle γ indicated here was found by the Monte Carlo method based on the results +of processing 50 mock data catalogs. +According to Pearson’s chi-square test, the probability density function f(ρ) (2) does not fit data well +(see Fig. 7, left panel)—the probability of accepting the null hypothesis that the observed distribution has +a probability density of the form (2) is less than 1%. This means the other functions f(ρ) or combinations +of them should be considered in the future. However, for the final local sample the chi-square test gives +a probability of acceptance of the hypothesis (2) of about 30% (see Fig. 7, right panel), i.e., the Gaussian +distribution as a model for f(ρ) should not be excluded from consideration. +Since in this paper we used a catalog (Mel’nik et al. 2015) based on observations in infrared bands (IC +and K; see Dambis et al. 2015), this allows us to expect a low selection effect due to the absorption of +light by dust in the Galactic disk. In particular, there should be no significant differential selection in the +vertical direction, i.e., statistical sample bias to the north of the disk. Indeed, due to the position of the Sun +above the average surface of the disk, a ray of light from an object located south of the average surface +of the disk passes through a larger thickness of the disk and experiences greater light absorption. Indeed, + +0.4 +O +0.0 +Q. +,kpc +N +-0.4 +{(1.383,Y) +{(1.383, Y)±0b +((1.383, Y) ±p +-0.8 +-8 +-4 +0 +4 +Y,kpc0.4 +{(X,0.555) +00 +((X, 0.555) ± 0b +((X, 0.555)±Op +o +8 +kpc +0.0 +N +O +o +o +-0.4 +-6 +-2 +0 +2 +X,kpc0.4 +8 +%0 +08 +8 +0.0 +o +o +kpc +O +0o +-0.4 +N +o +-0.8 +{(-3.107, Y) +((-3.107, Y) ± 0z +o +(-3.107, Y)±0p +-1.2 +-8 +-4 +0 +Y,kpc0.4 +7(X,0.640) +00 +((X, 0.640) ± 0z +O +((X, 0.640) ± 0p +0 +% +0.0 +880 +o +te +-0.4 +o +-6 +-4 +-2 +0 +2 +X,kpc14 +Nikiforov et al. +Fig. 7: Observed (green columns) and model (black line) distributions of object deviations ρ along the nor- +mal to the model average surface for the working sample (left panel, for no = 4 and the model parameters +given in Table 1) and local sample (right panel, for no = 1 and the parameters given in Table 2). Outliers +are shown in red. +due to the position of the Sun above the average surface of the disk, a ray of light from an object located +south of the average surface of the disk passes through a larger thickness of the disk and experiences greater +light absorption compared to a northern object at the same distance from the average surface. Therefore, in +principle, one would expect a sharper truncation of the observed distribution of deviations ρ from the model +from the negative ρ side compared to the positive ones, i.e., greater detection of northern objects compared +to southern ones. Asymmetry of this type in the distribution of deviations ρ does not really manifest itself +in a noticeable way (fig. 7). In reality, for the working sample, instead of a sharper truncation on negative ρ, +rather on the contrary, there is some deficit on ρ ∼ +0.2 kpc (fig. 7, left panel). The standard deviation +calculated for objects with ρ > 0 pc for any sample does not significantly exceed the standard deviation +found for objects with ρ < 0 kpc (see Table 4). These results support of the insignificance of north–south +selection. +Note that the increase in the selection effect with distance from the Sun in the sense of incomplete +sampling (Fig. 2, right panel) does not in itself lead to bias, i.e., to systematic errors in the position of the +average (model) surface of the disk and in estimation of the dispersion of objects relative to this surface. +Classical Cepheids belong to population I and, being quite young objects—∼107–108 years (see, e.g., +Veselova & Nikiforov 2020)— represent a thin disk of the Galaxy. Moreover, the vertical deviation of the +subsystem of classical Cepheids σρ∼130 pc (this work) is significantly smaller than the average vertical +scale hz = 300±50 pc of the thin disk as a whole (Bland-Hawthorn & Gerhard 2016). This makes classical +Cepheids a good tracer of the thin disk of the Galaxy: even in areas where there are few of them, they still + +40 +30 +20 +10 +-0.4 +0.0 +p, kpc80 +70 +60 +50 +40 +30 +20 +10 +0 +-0.8 +-0.4 +0.0 +0.4 +0.8 +P, kpcModeling the vertical distribution of disk objects +15 +Table 4: Standard deviations of Cepheids across the disk, calculated for objects of the considered samples +with only positive deviations ρ from the model, σ+ +ρ , and with only negative ones, σ− +ρ +Sample +σ+ +ρ (kpc) +σ− +ρ (kpc) +Working sample +0.1621 ± 0.0045 +0.1641 ± 0.0046 +Final working sample +0.1277 ± 0.0053 +0.1162 ± 0.0048 +Local sample +0.0765 ± 0.0043 +0.0894 ± 0.0046 +Final local sample +0.0757 ± 0.0043 +0.0792 ± 0.0043 +represent the position of the disk with a relatively small spread. In such areas the mathematical expectation +of the average Z-coordinate of the Cepheids, Z, remains equal to the true Z-coordinate of the average +surface of the disk, only the mean error of Z estimate increases, i.e., in the case of our method, the mean +error of the model σζ4 increases (isolines of σζ4(X, Y ) are shown on Figure 4). +On the other hand, Cepheids, concentrating towards spiral arms (see, e.g., Veselova & Nikiforov 2020), +like other tracers of the spiral structure of the Galaxy, often have not a uniform, but a “patchy” distribution +along the spiral arms (for example, Efremov 2011, fig. 1; Nikiforov & Veselova 2018, fig. 13; Reid et al. +2019, fig. 2; Veselova & Nikiforov 2020, fig. 5). This, as well as the growth of incomplete detection of +objects and a decrease in the density of the disk to the periphery, leads to the fact that at large distances +from the Sun, gaps appear in the distribution of Cepheids, for example at (X, Y ) ∼ (0, −7) kpc (Figure 4). +Of course, in the areas of such gaps, the constructed model should be treated only as a smooth interpolation +of the average trend between the areas represented by the data. However, when imposing a stricter restriction +on the mean error of the model, e.g., σζ4(X, Y ) ≤ 1 +6σρ = 20 pc, the internal area of applicability of the +model does not include most of these lacunae (see Figure 4). The relatively high accuracy of the model for +the area in the lower right corner of Figure 4 is, of course, only formal, due to the fact that any sufficiently +flexible model must pass through a few Cepheids in this area, located at some distance from the bulk of the +sample objects. On the other hand, these and other Cepheids at large negative Y are in the general trend +of a well-known decrease in the average surface of the disk in this area. So, in the region Y < −10 kpc, +all Cepheids of the working sample are in the range −1.2 ≤ Z ≤ −0.8 kpc (Figure 5, bottom panel), +which agrees well with the disk level in this area according to other data—e.g., −2 ≤ Z ≤ 1 kpc (Skowron +et al. 2019b, from Cepheids), Z = −1.22 kpc (Romero-G´omez et al. 2019, from red giant branch (RGB) +stars), −1 ≤ Z ≤ −0.5 kpc (Lemasle et al. 2022, from Cepheids). The vertical dispersion of objects at +large negative Y is consistent with that for the rest of the working sample (Figure 5, bottom panel). At the +same time, the use of the entire working sample, despite the gaps, does not create any fictitious ripples in +the model of the average surface of the disk in the area Y < −6 kpc (Figure 4, 6). Thus, there was no +reason to discard Cepheids at large negative Y . In addition, it was important to keep in the sample objects +representing the decline of the disk surface in the III and IV quadrants in order to test the capabilities of +the general model (1) within the framework of the proposed method to describe this well-known feature +together with other possible details of the disk surface (as it turned out, local extremes). Note that the +value of the decrease in the disk level of ∆Z∼1 kpc relative to the plane Z = z00 is much larger than the + +16 +Nikiforov et al. +Fig. 8: Distribution of optimal order values for 100 mock samples (see text). +Table 5: Same as in Table 1, but for the odd subsample of the final working sample (see text). no = 4 +Parameter +Estimation +σzi/|zi| +Parameter +Estimation +σzi/|zi| +σz +0.1218 ± 0.0050 +0.04 +z21 +0.0017 ± 0.0013 +0.76 +z00 +−0.026 ± 0.011 +0.42 +z12 +0.00214 ± 0.00080 +0.37 +z10 +−0.0321 ± 0.0093 +0.29 +z03 +0.00145 ± 0.00040 +0.28 +z01 +−0.0077 ± 0.0061 +0.79 +z40 +0.00025 ± 0.00022 +0.88 +z20 +0.0074 ± 0.0039 +0.53 +z31 +0.00026 ± 0.00021 +0.81 +z11 +−0.0002 ± 0.0026 +13.00 +z22 +−0.00008 ± 0.00018 +2.25 +z02 +−0.0013 ± 0.0014 +1.08 +z13 +0.000133 ± 0.000065 +0.49 +z30 +0.0038 ± 0.0015 +0.39 +z04 +0.000092 ± 0.000029 +0.32 +standard deviation for Cepheids (σρ ∼ 130 pc), i.e., the downward trend is detected confidently, despite the +incompleteness of the sample in this area. +We tested the algorithm used here for choosing the optimal order of the ζn model by the Monte Carlo +method. As a model of the disk surface, the constructed model ζ4 was adopted (Table 1) and 100 mock +catalogs were generated with object deviations from ζ4(X, Y ) along the normal to this surface, distributed +according to the law (2) with the value of σρ indicated in Table 1. The results are shown in Figure 8. They +show that in most cases the order of the initial model is restored exactly, and the probability that the order +of the model will be underestimated is less than 1%. These results suggest that, acting according to this +algorithm, it is possible to obtain a model that does not fully reflect some details of the real disk structure, +but it is unlikely to build an excessively complex model with fictitious details. +To check the stability of the results, we divided the final working sample (hereinafter, for short, “full +sample”) into two parts: one part included objects with odd numbers in the sample list (we will call it the +odd subsample), and the other objects with even numbers (even subsample). This separation is actually +random. On the other hand, it keeps the relative population of data of different longitude intervals approx- +imately the same for both subsamples and for the full sample, since in the catalog Mel’nik et al. (2015) +objects are ordered by their names, i.e., mainly by the names of constellations. The latter is important if we +want to check the reproducibility of the detected details on the relief of the disk surface. The calculations + +56 +50 +40 +N +30 +27 +20 +17 +10 +0 +2 +3 +4 +OrderModeling the vertical distribution of disk objects +17 +Table 6: Same as in Table 1, but for the even subsample of the final working sample (see text). no = 4 +Parameter +Estimation +σzi/|zi| +Parameter +Estimation +σzi/|zi| +σz +0.1227 ± 0.0050 +0.04 +z21 +0.0037 ± 0.0015 +0.41 +z00 +−0.028 ± 0.011 +0.39 +z12 +0.00249 ± 0.00086 +0.35 +z10 +−0.0279 ± 0.0097 +0.35 +z03 +0.00210 ± 0.00044 +0.21 +z01 +−0.0167 ± 0.0060 +0.36 +z40 +−0.00030 ± 0.00020 +0.67 +z20 +0.0046 ± 0.0036 +0.78 +z31 +0.00010 ± 0.00020 +2.00 +z11 +0.0011 ± 0.0028 +2.55 +z22 +0.00035 ± 0.00016 +0.46 +z02 +0.0011 ± 0.0016 +1.45 +z13 +0.000125 ± 0.000088 +0.70 +z30 +0.0008 ± 0.0014 +1.75 +z04 +0.000097 ± 0.000033 +0.34 +Table 7: Same as in Table 3, but for the odd subsample of the final working sample (see text) +Extremum +X (kpc) +Y (kpc) +ζ4(X, Y ) (pc) +Sζ4(X, Y ) +Local maximum +−3.0 +0.1 +53.0 ± 20 +4.01 +Local minimum +1.0 +1.0 +−50.4 ± 15 +−1.59 +were repeated for each of the two independent subsamples. The results are presented in Tables 5–7 and +in Figure 9. The optimal orders of the model ζn for both subsamples turned out to be the same and equal +to no for the full sample: no = 4. The parameters of the model ζ4 for even and odd subsamples and for the +full sample within the error limits are consistent with each other (cf. Tables 1, 5, 6). At the same time, all +significant parameters obtained from subsamples are also significant for the full sample. The characteristics +of the local extremes for the odd subsample turned out to be similar to the characteristics for the full sample +(cf. Tables 3, 7). For the even subsample, the model ζ4(X, Y ) does not formally have local extremes, but +it has areas of depression and elevation (Figure 9, right panel), in position and amplitude close to those for +models obtained from the full sample and odd subsample (Figure 4; Figure 9, left panel). The curves-of- +nodes also turned out to be similar for all three samples in the area of applicability of the model (Figures 4, +9). Thus, the topology of the resulting model as a whole is preserved even when the sample is divided. At +the same time, the drop in the significance of the details of the model surface ζ4(X, Y ) for subsamples +shows that dividing the sample into a larger number of parts is hardly meaningful. +5 DISCUSSION +Consistency of local solar offset estimate z⊙ = 28.1 ± 6.1 +�� +stat ++1.3 +−1.3 +�� +cal pc and global estimate z⊙ = +27.1 ± 8.8 +�� +stat ++1.3 +−1.2 +�� +cal pc (Table 2) also implies that the proposed algorithm is valid and the model order is +correct—when optimizing the order of the model, the value of the solar offset does not depend systemati- +cally on whether a large or small neighborhood of the Sun is considered. Our estimates also agree with the +best estimate of the solar offset z⊙ = 25 ± 5 pc (Bland-Hawthorn & Gerhard 2016) and local estimates +specifically for classical Cepheids (see below). +However, the estimate of σρ = 68.5 ± 3.9 pc obtained for the final local sample is inconsistent with +the estimate of σρ = 118.7 ± 3.3 pc for the final working sample (in original distance scale, see Table 2). + +18 +Nikiforov et al. +Fig. 9: Maps of the models ζ4 of the average disk surface of optimal order, constructed from the objects of +independent odd (left panel) and even (right panel) subsamples of the final working sample (see text). The +notation is the same as in Figure 4. +Since the estimates were obtained by optimizing the order of the model ζn, this mismatch should mainly +be a consequence of a combination of the flaring and random errors in distances. Indeed, the observed +deviation of objects relative to the model ζno is due to two factors: the natural (true, cosmic) dispersion +of objects relative to the average surface of the Galactic disk and the measuring dispersion—the deviation +of the observed positions of objects from their true positions due to random errors in distance moduli. The +latter means that more distant objects have larger distance errors. On the other hand, the natural dispersion +can grow to the periphery (the disk flaring). Indeed, at X < 0 kpc, the apparent spread of objects relative +to the model increases somewhat (Figure 6, left panels). In order to separate the contributions of these two +effects, a more complex version of the present method is required with direct consideration of random errors +in distances. This will allow you to get more reliable results about flaring itself. +Note that the contribution of the measuring dispersion to the observed dispersion σρ in this case is +very small, as shown by the following approximate estimates. For photometric distances, their standard +deviation due to measuring dispersion is σr = ln 10 +5 +σd r, where σd is the standard uncertainty of distance +moduli. In the case of local sample, the main contribution of distance errors to the observed vertical standard +deviation σZ (close to σρ) occurs due to objects at high latitudes b. Then the assumption that the standard σr +for all objects of the sample completely passes into vertical standard deviation gives an upper estimate for +the contribution of the measuring dispersion to σZ: σZ,mes < σr. For the distance r = 1σρ = 76.5 pc (see +Table 2) and σd = 0.14m (Veselova & Nikiforov 2020), this results in σZ,mes < 4.9 pc and the natural +dispersion σZ,0 = +� +σ2 +Z − σ2 +Z,mes > 76.3 pc, i.e., the correction is no more than −0.2%. +In the case of a working sample, most of the objects are located at small |b|: for characteristic distances +r = 3–6 pc (see Fig. 1) and Z = 1σρ = 132 pc (Table 2) |b| = 1.◦3–2.◦5. For sample objects, the contribution +of the measuring dispersion is σZ,mes = σr sin |b|, tan b = Z/(r cos b), then for small |b| b ≈ Z/r, σZ,mes ≈ +σr|b| = +ln 10 +5 +σd |Z|. Then for Z = 1σρ = 132 pc it turns out: σZ,mes ≈ 8.5 pc, σZ,0 ≈ 131.7 pc, i.e., + +5 +0.60 +880 +90 +0.30 +0 +00 +0.15 +kpc +kpc +-5 +0.00 +N +-0.15 +-10 +0.30 +-0.70 +-15 +5 +0 +X,kpc5 +00 +000 +0.60 +9 +0.30 +80 +a +88 +000 +0.15 +00:00 +kpc +kpc +-5 +0.00 +N +0.15 +-10 +-0.30 +-0.70 +-15 +-5 +0 +5 +X,kpcModeling the vertical distribution of disk objects +19 +correction −0.2%. Thus, in both cases, the contribution of random distance uncertainty to the observed +vertical dispersion is negligible. +From classical Cepheids on r < 2 kpc Majaess et al. (2009) obtained z⊙ = 26 ± 3 pc and the scale +height of ≤75±10pc which are consistent with our estimates (Table 2). Estimates of z⊙ = (23–24)±2 pc +and σρ = 76.4 ± 1.8 pc were found by Bobylev & Bajkova (2016a) for classical Cepheids according to +the same version of the Berdnikov et al.’s catalog as in this work. Bobylev & Bajkova (2016a) considered +the cylindrical region r ≤ 4 pc. As they used the original calibration of the catalog we can compare these +estimates with ours in the same calibration (Table 2). There is consistency with our estimates of the solar +offset for both final local and final working samples. However, the vertical scale estimates are consistent +only in the case of final local sample. Exactly the same situation is for Cepheids-based estimates in Skowron +et al. (2019a). The authors considered data on 2431 Cepheids, and the most part of the data was obtained +by the OGLE-IV project. Skowron et al. obtained an estimate of the disk scale height of 73.5 ± 3.2 pc, so +our estimate for the final local sample does not contradict this value. All this also points out the importance +of taking into account the warp of the Galactic average disk surface in order to have the ability of proper +consideration of all the data available. Otherwise, only local regions can be considered. +In addition, the fact that the estimates of z⊙ differ, as noted in Introduction, also suggests that the a +priori assumption about the flat model of the Galactic disk should be limited. Indeed, according to our +results such assumption might be made only for specific regions like the local one, i.e., close enough to +the Sun. Moreover, it can be noticed that the value of z⊙ is less dependent on the Galactic disk warping +than the value of σρ. Based on what has been said, we can conclude that any a priori assumption about the +Galactic disk warping must be carefully studied especially when the vertical scale parameter is estimated. +The detected local extrema of the average surface of the disk may be manifestations of bending waves +caused by interaction with the Sagittarius dwarf galaxy, in the form of local structures elongated in the azi- +muthal direction (G´omez et al. 2013, fig. 5; Laporte et al. 2019; Poggio et al. 2021, fig. 2), or by interaction +with the Large Magellanic Cloud (Thulasidharan et al. 2022 and references therein). +The method used after testing on classical Cepheids can now be applied to other data (in particular, to +Gaia data) and/or in other assumptions about the distribution function f(ρ). The Gaia DR2 catalog was +used in recent work by Ablimit et al. (2020) to obtain data on classical Cepheids. Despite the fact the +direct study of the Galactic disk warping was not conducted in the work of these authors, according to the +pictures plotted in the work on these data, the disk warping is clearly revealed. Unfortunately, the use of +the current version of our method with these data as is will lead to significant biases mainly because of the +dependence of distance uncertainty on distance, which will be significant due to the need to consider the +large neighborhood of the Sun. +Taking into account the uncertainty of distances may also solve the problem of establishing the form of +the vertical distribution law f(ρ). 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L., et al. 2015, ApJ, 801, 105. 3 +Yao, J., Manchester, R., & Wang, N. 2017, MNRAS, 468, 3289. 2, 3 + diff --git a/NdE2T4oBgHgl3EQfqwhx/content/tmp_files/load_file.txt b/NdE2T4oBgHgl3EQfqwhx/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..dc4459cd2c2e3b4a5cdbf9fab7ae63301e464b57 --- /dev/null +++ b/NdE2T4oBgHgl3EQfqwhx/content/tmp_files/load_file.txt @@ -0,0 +1,1134 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf,len=1133 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='04042v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='GA] 10 Jan 2023 Research in Astronomy and Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' (LATEX: ms2022-0209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' printed on January 11, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 1:30) Modeling the vertical distribution of the Milky Way’s flat subsystem objects Igor’ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Nikiforov, Vadim A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Usik and Angelina V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Veselova Saint Petersburg State University, Universitetskij Prospekt 28, Staryj Peterhof, Saint Petersburg 198504, Russia;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='nikiforov@spbu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='ru (IIN) Received 20XX Month Day;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' accepted 20XX Month Day Abstract This paper is an initial stage of consideration of the general problem of joint mod- eling of the vertical structure of a Galactic flat subsystem and the average surface of the disk of the Galaxy, taking into account the natural and measurement dispersions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' We approximate the average surface of the Galactic disk in the region covered by the data with a general (polynomial) model and determine its parameters by minimizing the squared deviations of objects along the normal to the model surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The smoothness of the model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=', its order n, is optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' An outlier elimination algorithm is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The developed method allows us to simultaneously identify significant details of the Galactic warping and estimate the offset z⊙ of the Sun relative to the average (in general, non-flat) surface of the Galactic disk and the vertical scale of the object system under consideration for an arbitrary area of the disk covered by data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The method is applied to data on classical Cepheids (Berdnikov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=', Mel’nik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Significant local extremes of the average disk surface model were found based on Cepheid data: the minimum in the first Galactic quadrant and the maximum in the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' A well- known warp (lowering of the disk surface) in the third quadrant has been confirmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The opti- mal order of the model describing all these warping details was found to be no = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The local (for a small neighborhood of the Sun, no = 0) estimate of z⊙ = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='1± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='1|stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='3|cal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' pc is close to the non-local (taking into account warping, no = 4) z⊙ = 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='1 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='8|stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='3 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='2 �� cal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' pc (statistical and calibration uncertainties are indicated), which suggests that the proposed mod- eling method eliminates the influence of warping on the z⊙ estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' However, the non-local estimate of the vertical standard deviation of Cepheids σρ = 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='7|stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' +6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='3 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='9 �� cal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' pc dif- fers significantly from the local σρ = 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='5 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='4|stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='6 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='4 �� cal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' pc, which means the need to introduce more complex models for the vertical distribution outside the Sun’s vicinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Key words: Galaxy: disk — Galaxy: structure — Galaxy: fundamental parameters — meth- ods: data analysis 2 Nikiforov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 1 INTRODUCTION The vertical distribution of objects in various Galactic subsystems contains valuable information about the origin, evolution and dynamics of our Galaxy, so determining the characteristics of this distribution is an important task of Galactic Astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The study of the vertical distribution may include consideration of many phenomena, but one of them should be taken into account necessarily—this is the offset z⊙ of the Sun relative to the plane of the Milky Way’s disk towards the North Galactic Pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Therefore, in the simplest case, modeling of the vertical distribution is reduced to determining the value of z⊙ and some dispersion parameter (standard deviation, scale height, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=') that characterizes the scattering of subsystem objects relative to the average plane of the Galactic disk (usually relative to the midplane of this subsystem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The first estimate of z⊙ = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='7 pc was obtained by van Tulder (1942) from the analysis of nearby stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Subsequently, in many papers, the solar offset was determined by different methods for various objects and Galactic subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Bland-Hawthorn & Gerhard (2016) adopted as the best (local) estimate the result of Juri´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' (2008) from the complete SDSS photometric survey, z⊙ = 25 ± 5 pc, which covers many other estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' However, the solar offset relative to the differently defined midplane of the disk does not seem to be described by a single value of z⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' For example, Bobylev & Bajkova (2016b) obtained significantly different results for reference objects (tracers) of different types: z⊙ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='5 pc for a sample of methanol masers, z⊙ = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='4 pc for data on H II regions and z⊙ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='5 pc for data on giant molecular clouds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' at the same time, Ferguson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' (2017) derived values of z⊙ = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='5 pc for a uniform selection of SDSS K and M dwarf stars and z⊙ = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='4 pc for an expanded selection, Buckner & Froebrich (2014) found an estimate z⊙ = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='2 pc for open clusters, and Majaess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' (2009) obtained values of z⊙ = 26 ± 3 pc for Cepheids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' A comparison of these and other z⊙ estimates obtained in various studies (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=', summaries in Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Skowron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2019a) shows that the differences between these estimates cannot be explained only by statistical errors, with some estimates varing significantly, even for objects of the same type (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=', for open clusters and Cepheids).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' This shows that the discrepancies reflect not only the possible objective difference in the values of z⊙ between different types of objects (subsystems of the Galaxy), but also other factors in the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' In addition to the Z-offset of the Sun, the number of already established or potential factors affecting the results of modeling the vertical distribution of objects includes: 1) the warp of the Galactic disk, 2) the dependence of the values of the characteristics of the vertical distribution on the position on the disk for the selected Galactic subsystem (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=', the flare of the Galactic disk), 3) the possible (and in the case of vertical dispersion, real) dependence of these characteristics on the type of Galactic subsystem, 4) the need to establish the functional type of the vertical distribution and its possible variations with the position on the disk and with the type of subsystem, as well as 5) taking into account the random uncertainty of helio- centric distances, systematically distorting the true vertical distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The problem in general (taking into account all these factors) has not yet been solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Meanwhile, different combinations of these factors may be responsible for discrepancy of the results (in particular, of z⊙ estimates) in different papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Subjective factors can also lead to this: the choice of the general appearance of the model of the average surface of Modeling the vertical distribution of disk objects 3 the disk, possible mismatch of the distance scales used in different works, the dependence of the results of modeling on the size and configuration of the disk area under consideration (the area covered by the data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Despite the lack of a solution to the problem in general, some of these factors and their combinations were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The most important factor is the presence of a warp of the Milky Way’s disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The warp was noticed as soon as the observation data in the 21-cm line of neutral hydrogen appeared for the southern hemi- sphere (Burke 1957;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Kerr 1957).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Subsequent studies (Oort et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 1958;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=', Binney & Merrifield 1998 and Bland-Hawthorn & Gerhard 2016 reviews and references therein;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Skowron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Chrob´akov´a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2020, among others) have shown that a significant stellar/gas warp begins outside the solar circle, and in the inner Galaxy the disk is very close to flat, including on the far side of the disk (Minniti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Various data indicate that one part of the warped disk deviates from the plane of the inner disk towards the North Galactic Pole, the other deviates in the opposite direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Not taking into account the large-scale warp (if a plane parallel to the equator of the Galactic coordinate system b = 0◦ is taken as a model of the average surface of the Galactic disk) can significantly affect the estimates of the solar offset z⊙ and the vertical scales of flat subsystems (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=', the dependence of these characteristics for planetary nebulae on the size of considered near-solar region in Bobylev & Bajkova 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' One way to avoid this is to exclude the warp zone from consideration under the assumption that in the remaining area of the disk its average surface is flat: restrictions are imposed on the selection of tracers, for example, by the heliocentric distances r (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=', r ≲ 4 kpc in Bobylev & Bajkova 2016a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' r ≲ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='5 kpc in Bobylev & Bajkova 2016b), by the predicted maximum warp offsets (<10 pc in Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2017), by the distance R to the axis of rotation of the Galaxy (R < 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0 kpc in Reid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' However, the exclusion of the warp zone requires the adoption of a specific warp model, and it is often taken simple for this and other applications: the disk in the inner Galaxy (R ≤ Rw) is considered undisturbed, and in the outer one (R > Rw) it is usually represented by a combination of a power dependence on R and a simple trigonometric function of the azimuthal coordinate (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=', Binney & Merrifield 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Pohl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Romero-G´omez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Mosenkov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' At the same time, to describe the warp in the outer Galaxy, in most of its morphological studies, simple symmetric models with a limited set of parameters are used—the radius Rw at which the disk starts bending, the phase angle of the line-of-nodes and the maximum amplitude of the warp (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=', Romero-G´omez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2019 and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' However, the warp is clearly more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Firstly, the inner part of the disk is not perfectly flat— there are corrugations on the scale of ∼30 pc (Oort et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 1958, fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Spicker & Feitzinger 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Binney & Merrifield 1998, fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' N-body simulations of the Milky Way interacting with a satellite similar to the Sagittarius dwarf galaxy show that repeated satellite passes can generate local ripples, including in the inner disk (Poggio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2020, fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' According to kinematics, the onset of the warp occurs at a guiding radius inside the Solar circle, Rg ≲ 7 kpc (Schoenrich & Dehnen 2018), or even in the center of the Galaxy (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Secondly, the outer part of the warp is also not described by a simple model—there are manifestations of lopsidedness of the warp and twisting of its line-of-nodes (Romero-G´omez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Chrob´akov´a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' (2015) detected an oscillating asymmetry in the SDSS main-sequence star counts on either side of the Galactic plane in the anticenter region, between longitudes of 110◦ < l < 229◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 4 Nikiforov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' In addition, the morphology and kinematics of the warp depend on the type/age of the tracers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=', Romero- G´omez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Chrob´akov´a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Moreover, hydrodynamic modeling of the evolution of an ensemble of stars formed in the warp shows that only younger populations trace the warp detected by HI (Khachaturyants et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2021) and that the influence of the bending waves excited by irregular gas inflow is most strongly manifested in the young populations (Khachaturyants et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' This means that the warp model, universal for all disk subsystems of the Galaxy, can hardly be accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Kinematic manifestations of the warp also indicate its asymmetry and complexity in general, as well as the dependence of its characteristics on the age of tracers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=', Romero-G´omez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2020 and references in these works).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Based on the above, the exclusion of the warp zone as a method of eliminating biases in the vertical dis- tribution parameters can only give a partial (local) solution to the problem, the accuracy of which depends on the details of the accepted warp model and on its realism in the case of the tracers under consideration and on assumptions about the boundaries of the warp-distorted area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' All these assumptions can be sources of systematic errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' That is why it seems important to us to abandon simplified warp models and consider the most general analytical warp model describing all the significant structural features of the middle sur- face of the disk identified by the tracers under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The method of excluding the warp zone is also unsuccessful due to the presence of the disk flaring, which begins at R ≳ R0, where R0 is the Galactic cen- ter distance (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=', Reid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Mosenkov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2021), since the dependence of the dispersion parameter on the accepted boundaries of the area “undisturbed” by the warp appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The warp is currently being actively explored in many ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' In particular, warp precession is actively discussed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=', Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' However, as noted by Poggio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' (2020), the precession parameters depend on our knowledge of the shape of the warp and its differences for different stellar populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' In ad- dition, Chrob´akov´a & L´opez-Corredoira (2021) even raise the question of the very existence of precession, since the application of a warp model inconsistent with the tracers used leads to a fictitious precession.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Detailed warp models are also important both for studying the dependence of z⊙ and vertical dispersion characteristics on the type/age of tracers, and for identifying the cause and dynamic nature of the warp of our Galaxy, which remain unclear (Binney & Merrifield 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Poggio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Khachaturyants et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Note also that in the framework of an alternative approach applied by Mosenkov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' (2021)— pho- tometric 3D decomposition of the Milky Way taking into account flaring and warp—the parameters of the warp disk are poorly determined, since only a 2D map is considered, whereas for creating a reliable 3D model of the warp one needs to have a 3D distribution of stars in the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Despite the fact that the best solution would be to model the Z-distribution of objects taking into account all the factors mentioned at once, due to the complexity of the overall task we focus in this paper on the task of constructing a detailed warp model with a minimum of assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' We will not consider the influence of random errors in the distance here (the selected data catalog allows this, see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 3), as well as the disk flaring, since without taking into account errors in distances, the flaring parameters may turn out to be strongly biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Modeling the vertical distribution of disk objects 5 2 METHOD We will study the spatial distribution of objects in the heliocentric Cartesian coordinate system, which does not require taking any value of R0: X-axis is directed towards the Galactic center, Y -axis is towards the rotation of the Galaxy, Z-axis is towards the North Galactic Pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' In order to free the warp model as much as possible from pre-accepted assumptions, we will consider as models the ζn(X, Y ) polynomials, each of which is a Maclaurin series expansion in the solar neighborhood up to the n-th order of the average Z-coordinate of the Galactic disk as a function of position on the XY plane: ζn(X, Y ) = n � i=0 i � j=0 zi−j, jXi−jY j = z00 + z10X + z01Y + z20X2 + z11XY + z02Y 2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' + zn0Xn + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' + z0nY n, (1) zζ = � zij �n i 0 0 , z⊙ = −z00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Here zζ is the vector of M = (n+2)(n+1)/2 parameters of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Function (1) represents the average surface of the Galactic disk, defined by the spatial distribution of objects (in the general case of matter) of the selected Galactic subsystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The distance ρ from the object to the surface ζn(R, Z) along the normal to this surface will be considered as the value of the deviation of the object from the model average surface of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' To obtain an estimate of the vector zζ, we generally use the maximum likelihood method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' In this paper, to simplify, we assume that ρ as a random variable is distributed according to a normal law with zero mean, that is, that the probability density of ρ has the form f(ρ) = 1 σρ √ 2π e − ρ2 2σ2ρ , (2) where σρ is the standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' We assume that the value of σρ is the same for all objects in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' However, in future work we propose to investigate and take into account the dependence of σρ and/or other dispersion parameters on the Galactocentric distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' With probability density (2), the likelihood function L and logarithmic likelihood function L are L = N � i=1 1 σρ √ 2π e − ρ2 i 2σ2ρ , L ≡ − ln L = N 2 ln (2π) + N ln σρ + 1 2 N � i=1 ρ2 i σ2ρ , (3) ρi = ρi(ri, li, bi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' zζ), where N is the number of objects in the sample;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' ri, li and bi are the heliocentric distance, galactic longitude and latitude of i-th object, correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Being a dispersion parameter under parametrization (2), σρ is included in the general vector of the problem parameters z = (σρ, zζ) = (z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' , zK), where K = M + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The minimum of the function L gives estimates of the parameters, including the value of the standard deviation σρ of objects across the disk, which represents the contributions of both the true (natural) vertical dispersion of objects and the random uncertainty of distance estimates (the latter contribution is negligible, see Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Thus, under assumption (2), the maximum likelihood method was reduced to 6 Nikiforov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' the nonlinear least squares method (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' (3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The resulting value of σρ was multiplied by the coefficient � N/(N − M) to obtain an unbiased estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The vector zerr of mean parameter errors and the mean model prediction error σζn(X, Y ) were calculated based on the Hessian H(L) with elements hij (Hudson 1964;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Wall & Jenkins 2012): hij = ∂L2 ∂zi∂zj , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' , K, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' , K, C = [H(L)]−1, zerr = {√cii}K i=1 , σ2 ζn(X, Y ) = {grad[ζn(X, Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' zζ)]}T C′ grad[ζn(X, Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' zζ)], (4) where cii are diagonal elements of the covariance matrix C, C′ is a submatrix of C that does not contain covariances involving σρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' here, an estimate of the vector z obtained by minimizing L is substituted for all values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' After finding the parameters, an outlier exclusion algorithm described in Nikiforov (2012) was applied to the sample of objects under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' This algorithm differs from the usual 3σ criterion in that it uses a variable exclusion limit, which increases as the number of objects increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' In order to check how well the observed distribution of sample objects by deviations ρ agrees with the model probability density function (2), we use Pearson’s chi-square test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' A priori choice of the order n of expansion (1) would lead to significant errors in all parameters, includ- ing z00 (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 4), so the value of n was also optimized using a simple algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Models of the order n = 0, 1, 2 and so on are built sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' For each model of order n, the number of parameters zij of order n (i + j = n) whose estimates differ from zero at the significance level ≥2σ (σzij/|zij| ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='5) is calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Then we find a model of the highest order such that it has at least one 2σ-significant parameter of the same order as the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' If the total number of significant parameters of the selected model is greater than the corresponding number for any lower-order model, then the selected model of order n is assumed to be optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Otherwise, a model of order n − 1 is considered as possibly optimal and is compared in the same way with models of lower orders in terms of the number of significant parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Then either the order n − 1 is assumed to be optimal, or a transition is made to the order n − 2, and so on until some order n ≥ 0 is accepted as optimal, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The importance of choosing the correct model order will be illustrated in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 3 DATA We use the catalog of classical Cepheids by Berdnikov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' (2000) in the version of Mel’nik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' (2015), which provides data for 674 Cepheids from the General Catalogue of Variable Stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=', Samus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The catalog is an updated version of the catalog of Cepheid parameters by Berdnikov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Observational data were obtained with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='4–1 m telescopes of the Maidanak Observatory (Republic of Uzbekistan), Cerro Tololo and Las Campanas observatories (Chile), Cerro Armazones Observatory of the Catholic University (Chile) and South African Astronomical Observatory (see Dambis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2015 for de- tails).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' In particular, in order to reliably determine the distances, Cepheids discovered during the CCD moni- toring of the southern sky performed as a part of the All Sky Automated Survey (ASAS) project (Pojmanski 2002) were observed, therefore, a survey of Cepheids across the entire sky was performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The peak of the distribution of Cepheids by the mean apparent magnitudes in the V band falls on the values of ⟨V ⟩ = 12– 13 mag, the limiting magnitude of the catalog is ⟨V ⟩ = 15 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The distances were obtained based on the Modeling the vertical distribution of disk objects 7 period–luminosity relation in the K infrared band and interstellar-extinction law using the period – normal color (B − V ) relation derived earlier (see Dambis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The authors of the catalog did not specify the possible value of the distance modulus error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' However, the analysis of these data in Veselova & Nikiforov (2020) showed that the nominal random errors of distance estimates given in Mel’nik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' (2015) are small (the mean error of distance moduli σd < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='14m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Using this distance catalog gives us the opportunity to apply a simpler method that does not take into account ran- dom distance errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' In the future, we intend to use newer data covering more extensive areas where taking into account the uncertainty of distances within a more complex method is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Recent versions of Berdnikov et al.’s catalog have been successfully used to study the Galactic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Based on the catalog, Mel’nik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' (2015) identified signs of ring formations in the Galaxy, Dambis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' (2015) and Veselova & Nikiforov (2020) performed spatial modeling to determine the parameters of spiral arm segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' According to the original distance scale calibration of the catalog the distance to the Large Magellanic Cloud (LMC) is d∗ LMC = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='05 mag (Berdnikov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Modern LMC calibration is dLMC = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='09 mag (de Grijs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2014), which leads to a correction factor c for the distances of the catalog used: c = r + ∆r r = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='2∆d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='117+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='053 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='050, ∆d = dLMC − d∗ LMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' (5) We analyzed the original catalog estimates of distances, and then adjusted the main results for the factor c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Spatially isolated objects were manually excluded from the initial sample, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=', those that are nominally located clearly far from the main group of catalog objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The remaining objects, the totality of which we called the working sample, were used for calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' After applying the outlier elimination algorithm (Nikiforov 2012) during the analysis of the working sample, we obtained the final working sample contain- ing 615 objects (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' We also identified a local sample, which is part of the working sample representing the largest neigh- borhood of the Sun with the property of relative completeness of the identification of objects of this type (classical Cepheids).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Assuming a homogeneous distribution of objects projected onto the XY -plane (the Galactic plane) in a vicinity of the Sun, the number of objects projected onto a section of the XY -plane should be proportional to the area of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Usually, a set of concentric rings is used to select a local, close to complete, subsample, but the working sample has an asymmetry with respect to the origin of co- ordinates on the XY plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' For this reason, it was decided to use a set of rectangular frames with borders homothetic to the rectangular border of the working sample: the lengths of each side of the frame vary according to a given scale factor, and the ratio of distances from the origin to the edges of the frames remain constant (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2, left panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' In the case of a small frame width, the area of the frame will be proportional to the outer perimeter, so for a complete sample, the number of objects in the frame should increase linearly with the size of the frame (scale factor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' On the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2 the linear growth is observed within the first three bins (frames), which are marked in red;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' objects within these frames were taken as a local sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The boundaries of this sample are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 1 (purple rectangle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The sample size is 154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' After excluding an outlier, the final local sample of 153 objects was obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 8 Nikiforov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 1: Objects of the final working sample and outliers projected onto the XY plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The purple rectangle marks the boundary of the local sample (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' X-axis is directed towards the Galactic center, Y -axis is towards the rotation of the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The Sun is placed at X = 0 kpc, Y = 0 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 4 RESULTS The influence of the accepted order of the model ζn on the results can be illustrated by the example of the dependence of z00 on n for the final working sample (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' It can be seen that for small orders the estimates of z00 strongly depend on n, while for larger orders the estimates vary insignificantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The optimal order of the model ζn for the final working sample turned out to be no = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Estimates of model parameters are presented in Table 1, significant estimates are given in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' For the local and final local samples, no is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Final results for z⊙ and σρ without and with the correction of distance scale (5) for the final working and final local samples are listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Comparison with Figure 3 shows that the choice of an underestimated order (n < no) of the model can lead to an obviously incorrect estimate of z⊙ (here at n = 0, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The model surface ζ4(X, Y ) for the final working sample is shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Here the level ζ4(X, Y ) = z00 (6) is represented by a black line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' This line is the intersection of the model of the average surface of the disk with the nominal plane of the Galaxy XY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Line (6) corresponds to the line-of-nodes in simple models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Figure 4 shows that in reality the line (6), skirting the areas of local extrema, is quite curved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' By analogy with the 2 2 kpc 6 10 Final working sample 14 Outliers 4 0 4 8 X, kpcModeling the vertical distribution of disk objects 9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2: Left: Homothetic boundaries of rectangular frames used to construct the local sample (see text), superimposed on the distribution of objects of the working sample in projection on the XY plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Right: Histogram of the distribution of objects of the working sample in rectangular frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The red color cor- responds to the frames within which the sample (local sample) can be considered complete;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' the dashed straight line shows the linear growth area of the number of objects in the first three bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 3: Dependence of z00 (= − z⊙) on the model order n for the final working sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Vertical bars show the uncertainties of estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' line-of-nodes, the curve ζno(X, Y ) = z00 can be called a “curve-of-nodes”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Of course, the constructed model ζ4(X, Y ) is real only for the area of the disk that is covered by the data used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Boundaries of areas within which the mean error of the model is σζ4(X, Y ) ≤ 1 6σρ = 20 pc, ≤ 1 4σρ = 40 pc and ≤ 1 2σρ = 59 pc shown in Figure 4 by dotted, dashed and dashed–dotted lines, respectively, give an idea of the applicability of the model ζ4 depending on the specified level of its uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Here σζ4(X, Y ) was calculated using the formula (4), and σρ = 119 pc (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 25 0 pc 100Z 25 50 75 0 L N 3 4 5 6 n80 70 60 Z50 40 30 20 10 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='8 Scalefactor5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='5 kpc 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='5 X, kpc10 Nikiforov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Table 1: Parameter estimates for the model ζn of the average surface of the Galactic disk of optimal order no = 4 obtained for the final working sample of classical Cepheids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Estimates that differ from zero at a significance level of at least 2σ are highlighted in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The standard deviation σρ is given in kiloparsecs, while values of zij are in units of kpc1−i−j Parameter Estimation σzi/|zi| Parameter Estimation σzi/|zi| σρ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='1187 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0033 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='03 z21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='00289 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='00094 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='33 z00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0243 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0079 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='33 z12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='00288 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='00055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='19 z10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0320 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0061 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='19 z03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='00180 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='00028 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='16 z01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0131 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='31 z40 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='00012 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='00012 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='00 z20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0055 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='45 z31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='00013 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='00014 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='08 z11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0008 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0018 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='25 z22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='00022 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='00010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='45 z02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0007 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0010 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='43 z13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='000185 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='000048 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='26 z30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='00203 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='00087 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='43 z04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='000099 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='000020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='20 Table 2: Final estimates of z⊙ and σρ for the optimal models ζno(X, Y ) obtained for two samples of Cepheids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The estimates are given in the original catalog distance scale and in the scale adjusted for cali- bration dLMC = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='09 mag (see text) Sample Original distance scale Corrected distance scale Final working sample z⊙ = 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='3 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='9 pc z⊙ = 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='1 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='8 �� stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='3 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='2 �� cal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' pc (N = 615, no = 4) σρ = 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='7 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='3 pc σρ = 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='7 �� stat +6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='3 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='9 �� cal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' pc Final local sample z⊙ = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='2 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='5 pc z⊙ = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='1 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='1 �� stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='3 �� cal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' pc (N = 153, no = 0) σρ = 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='5 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='9 pc σρ = 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='5 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='4 �� stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='6 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='4 �� cal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' pc An alternative representation of the resulting model is given in Figure 5, which shows the extremum and boundary lines of the model surface ζ4(X, Y ) in projections on the planes XZ and Y Z in comparison with the Cepheids of the final working sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The extremum lines (there may be several of them for each projection) are dependencies of Z-coordinates of points of local extremes of the surface ζ4(X, Y ) with fixed X—ζmax(X, Y), ζmin(X, Y) (Figure 5, top panel) or with fixed Y —ζmax(X, Y ), ζmin(X, Y ) (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The boundary lines show the values of ζ4(X, Y ) at the boundaries of the final working sample: ζ4(X, Ymin) and ζ4(X, Ymax) (top panel), and ζ4(Xmin, Y ) and ζ4(Xmax, Y ) (bottom panel), where Xmin = −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='8 kpc, Xmax = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='6 kpc, Ymin = −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='6 kpc, Ymax = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='8 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Figure 5 affirms that the extremum lines mainly fall on the areas covered by the data, and the boundary lines indicate edge approximation effects in the area X ≳ 4 kpc, Y ≳ 0 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The well-known general shape of the Galactic disk warp is clearly visible in Figure 4—in the first and second quadrants, the model surface as a whole rises above the XY plane;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' in the third and fourth quadrants, the surface decreases below this plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' In the area near X ≈ −3 kpc, Y ≈ −6 kpc, the decrease in the average surface is observed for almost all objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' However, in addition, we found two extrema in the first and second quadrants that do not fit into simple warp models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The sections of the model with planes parallel to the XZ and Y Z planes and passing through Modeling the vertical distribution of disk objects 11 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 4: A map of the model ζ4 of the average disk surface of optimal order, constructed from the objects of the final working sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The color shows the value of the function ζ4(X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The objects are shown as colored circles with color representing the Z-coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The black line represents the z00-level, the red and blue crosses depict the local extrema (maximum and minimum, respectively), the purple rectangle shows the boundary of the local sample, the grey line denotes the solar circle R = R0 = 8 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The dotted, dashed and dashed–dotted lines are the boundaries of the applicable areas of the model σζ4, within which the mean error of the model is σζ4(X, Y ) ≤ 1 6σρ = 20 pc, ≤ 1 4σρ = 40 pc and ≤ 1 2σρ = 59 pc, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' the points of extrema are shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The significance S of the extrema was estimated by the formula Sζ4(X, Y ) = ζ4(X, Y ) − z00 σζ4(X, Y ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' (7) Table 3 shows the parameters of local extrema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The given values of S show that the local minimum and maximum are significant at the level of at least 2σ and 5σ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' In Figure 6 the slope of the model surface ζ4(X, Y ) to the nominal plane of the Galaxy XY is visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Having calculated the distance ρ⊙ from the Sun to the surface ζ4(X, Y ) along the normal to the latter, we 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='30 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='15 kpc kpc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='00 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='15 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='70 15 5 X,kpc12 Nikiforov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 5: Extremum lines (thicker lines of lighter color) and boundary lines of the model surface ζ4(X, Y ) in comparison with Cepheids of the final working sample in the projection on the plane XZ (top panel) and Y Z (bottom panel);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' see text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Local maxima and lines of the largest values of ζ4 at the sample boundaries are shown in red, while local minima and lines of the smallest boundary values are shown in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Table 3: Parameters of local extrema of the optimal model ζ4(X, Y ) constructed for the final working sam- ple: The Cartesian heliocentric coordinates X, Y of the extremum, the value of ζ4(X, Y ) at the extremum point and the significance level Sζ4(X, Y ) of the extremum (the number of σ) Extremum X (kpc) Y (kpc) ζ4(X, Y ) (pc) Sζ4(X, Y ) Local maximum −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='6 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='4 ± 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='43 Local minimum 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='6 −55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0 ± 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='81 (X, Ymin) 3 ((X, Ymax) 2 min(X, ) kpc Smax(X, ) 1 N 0 O 6 4 +-2 0 2 4 6 8 X,kpc (Xmin, Y) 3 ((Xmax, Y) 2 Smin(X, 7) kpc Imax(X, y) 1 N 0 8 1 C 12 8 4 0 4 Y, kpcModeling the vertical distribution of disk objects 13 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 6: Sections of the model ζ4(X, Y ) of the middle surface of the disk passing through the local maximum (upper panels) and minimum (lower panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Each panel shows the confidence area for the model values ζ4(X, Y ) (±1σζ) and the area of object deviations from the model ±1σρ, as well as projections of the position of objects located in the ±1 kpc band from the cut line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The green line indicates the position of the plane Z = z00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' obtained an estimate of the angle of inclination of the local average surface of the Galactic disk to the plane XY of the galactic coordinate system γ = arccos �ρ⊙ z⊙ � = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='◦79+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='◦34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='◦33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' (8) The 1σ-uncertainty of the angle γ indicated here was found by the Monte Carlo method based on the results of processing 50 mock data catalogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' According to Pearson’s chi-square test, the probability density function f(ρ) (2) does not fit data well (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 7, left panel)—the probability of accepting the null hypothesis that the observed distribution has a probability density of the form (2) is less than 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' This means the other functions f(ρ) or combinations of them should be considered in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' However, for the final local sample the chi-square test gives a probability of acceptance of the hypothesis (2) of about 30% (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 7, right panel), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=', the Gaussian distribution as a model for f(ρ) should not be excluded from consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Since in this paper we used a catalog (Mel’nik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2015) based on observations in infrared bands (IC and K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' see Dambis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2015), this allows us to expect a low selection effect due to the absorption of light by dust in the Galactic disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' In particular, there should be no significant differential selection in the vertical direction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=', statistical sample bias to the north of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Indeed, due to the position of the Sun above the average surface of the disk, a ray of light from an object located south of the average surface of the disk passes through a larger thickness of the disk and experiences greater light absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Indeed, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='4 O 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' ,kpc N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='4 {(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='383,Y) {(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='383, Y)±0b ((1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='383, Y) ±p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='8 8 4 0 4 Y,kpc0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='4 {(X,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='555) 00 ((X, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='555) ± 0b ((X, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='555)±Op o 8 kpc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0 N O o o 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='4 6 2 0 2 X,kpc0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='4 8 %0 08 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0 o o kpc O 0o 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='4 N o 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='8 {(-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='107, Y) ((-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='107, Y) ± 0z o (-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='107, Y)±0p 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='2 8 4 0 Y,kpc0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='4 7(X,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='640) 00 ((X, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='640) ± 0z O ((X, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='640) ± 0p 0 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0 880 o te 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='4 o 6 4 2 0 2 X,kpc14 Nikiforov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 7: Observed (green columns) and model (black line) distributions of object deviations ρ along the nor- mal to the model average surface for the working sample (left panel, for no = 4 and the model parameters given in Table 1) and local sample (right panel, for no = 1 and the parameters given in Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Outliers are shown in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' due to the position of the Sun above the average surface of the disk, a ray of light from an object located south of the average surface of the disk passes through a larger thickness of the disk and experiences greater light absorption compared to a northern object at the same distance from the average surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Therefore, in principle, one would expect a sharper truncation of the observed distribution of deviations ρ from the model from the negative ρ side compared to the positive ones, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=', greater detection of northern objects compared to southern ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Asymmetry of this type in the distribution of deviations ρ does not really manifest itself in a noticeable way (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' In reality, for the working sample, instead of a sharper truncation on negative ρ, rather on the contrary, there is some deficit on ρ ∼ +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='2 kpc (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 7, left panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The standard deviation calculated for objects with ρ > 0 pc for any sample does not significantly exceed the standard deviation found for objects with ρ < 0 kpc (see Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' These results support of the insignificance of north–south selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Note that the increase in the selection effect with distance from the Sun in the sense of incomplete sampling (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2, right panel) does not in itself lead to bias, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=', to systematic errors in the position of the average (model) surface of the disk and in estimation of the dispersion of objects relative to this surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Classical Cepheids belong to population I and, being quite young objects—∼107–108 years (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=', Veselova & Nikiforov 2020)— represent a thin disk of the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Moreover, the vertical deviation of the subsystem of classical Cepheids σρ∼130 pc (this work) is significantly smaller than the average vertical scale hz = 300±50 pc of the thin disk as a whole (Bland-Hawthorn & Gerhard 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' This makes classical Cepheids a good tracer of the thin disk of the Galaxy: even in areas where there are few of them, they still 40 30 20 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0 p, kpc80 70 60 50 40 30 20 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='8 P, kpcModeling the vertical distribution of disk objects 15 Table 4: Standard deviations of Cepheids across the disk, calculated for objects of the considered samples with only positive deviations ρ from the model, σ+ ρ , and with only negative ones, σ− ρ Sample σ+ ρ (kpc) σ− ρ (kpc) Working sample 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='1621 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='1641 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0046 Final working sample 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='1277 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0053 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='1162 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0048 Local sample 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0765 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0894 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0046 Final local sample 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0757 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0792 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0043 represent the position of the disk with a relatively small spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' In such areas the mathematical expectation of the average Z-coordinate of the Cepheids, Z, remains equal to the true Z-coordinate of the average surface of the disk, only the mean error of Z estimate increases, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=', in the case of our method, the mean error of the model σζ4 increases (isolines of σζ4(X, Y ) are shown on Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' On the other hand, Cepheids, concentrating towards spiral arms (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=', Veselova & Nikiforov 2020), like other tracers of the spiral structure of the Galaxy, often have not a uniform, but a “patchy” distribution along the spiral arms (for example, Efremov 2011, fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Nikiforov & Veselova 2018, fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 13;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Reid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2019, fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Veselova & Nikiforov 2020, fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' This, as well as the growth of incomplete detection of objects and a decrease in the density of the disk to the periphery, leads to the fact that at large distances from the Sun, gaps appear in the distribution of Cepheids, for example at (X, Y ) ∼ (0, −7) kpc (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Of course, in the areas of such gaps, the constructed model should be treated only as a smooth interpolation of the average trend between the areas represented by the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' However, when imposing a stricter restriction on the mean error of the model, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=', σζ4(X, Y ) ≤ 1 6σρ = 20 pc, the internal area of applicability of the model does not include most of these lacunae (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The relatively high accuracy of the model for the area in the lower right corner of Figure 4 is, of course, only formal, due to the fact that any sufficiently flexible model must pass through a few Cepheids in this area, located at some distance from the bulk of the sample objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' On the other hand, these and other Cepheids at large negative Y are in the general trend of a well-known decrease in the average surface of the disk in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' So, in the region Y < −10 kpc, all Cepheids of the working sample are in the range −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='2 ≤ Z ≤ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='8 kpc (Figure 5, bottom panel), which agrees well with the disk level in this area according to other data—e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=', −2 ≤ Z ≤ 1 kpc (Skowron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2019b, from Cepheids), Z = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='22 kpc (Romero-G´omez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2019, from red giant branch (RGB) stars), −1 ≤ Z ≤ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='5 kpc (Lemasle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2022, from Cepheids).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The vertical dispersion of objects at large negative Y is consistent with that for the rest of the working sample (Figure 5, bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' At the same time, the use of the entire working sample, despite the gaps, does not create any fictitious ripples in the model of the average surface of the disk in the area Y < −6 kpc (Figure 4, 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Thus, there was no reason to discard Cepheids at large negative Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' In addition, it was important to keep in the sample objects representing the decline of the disk surface in the III and IV quadrants in order to test the capabilities of the general model (1) within the framework of the proposed method to describe this well-known feature together with other possible details of the disk surface (as it turned out, local extremes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Note that the value of the decrease in the disk level of ∆Z∼1 kpc relative to the plane Z = z00 is much larger than the 16 Nikiforov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 8: Distribution of optimal order values for 100 mock samples (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Table 5: Same as in Table 1, but for the odd subsample of the final working sample (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' no = 4 Parameter Estimation σzi/|zi| Parameter Estimation σzi/|zi| σz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='1218 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='04 z21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0017 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='76 z00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='026 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='42 z12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='00214 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='00080 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='37 z10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0321 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0093 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='29 z03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='00145 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='00040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='28 z01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0077 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0061 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='79 z40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='00025 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='00022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='88 z20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0074 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0039 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='53 z31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='00026 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='00021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='81 z11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0002 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0026 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='00 z22 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='00008 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='00018 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='25 z02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0013 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0014 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='08 z13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='000133 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='000065 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='49 z30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0038 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='39 z04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='000092 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='000029 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='32 standard deviation for Cepheids (σρ ∼ 130 pc), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=', the downward trend is detected confidently, despite the incompleteness of the sample in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' We tested the algorithm used here for choosing the optimal order of the ζn model by the Monte Carlo method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' As a model of the disk surface, the constructed model ζ4 was adopted (Table 1) and 100 mock catalogs were generated with object deviations from ζ4(X, Y ) along the normal to this surface, distributed according to the law (2) with the value of σρ indicated in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The results are shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' They show that in most cases the order of the initial model is restored exactly, and the probability that the order of the model will be underestimated is less than 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' These results suggest that, acting according to this algorithm, it is possible to obtain a model that does not fully reflect some details of the real disk structure, but it is unlikely to build an excessively complex model with fictitious details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' To check the stability of the results, we divided the final working sample (hereinafter, for short, “full sample”) into two parts: one part included objects with odd numbers in the sample list (we will call it the odd subsample), and the other objects with even numbers (even subsample).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' This separation is actually random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' On the other hand, it keeps the relative population of data of different longitude intervals approx- imately the same for both subsamples and for the full sample, since in the catalog Mel’nik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' (2015) objects are ordered by their names, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=', mainly by the names of constellations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The latter is important if we want to check the reproducibility of the detected details on the relief of the disk surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The calculations 56 50 40 N 30 27 20 17 10 0 2 3 4 OrderModeling the vertical distribution of disk objects 17 Table 6: Same as in Table 1, but for the even subsample of the final working sample (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' no = 4 Parameter Estimation σzi/|zi| Parameter Estimation σzi/|zi| σz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='1227 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='04 z21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0037 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='41 z00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='028 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='39 z12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='00249 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='00086 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='35 z10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0279 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0097 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='35 z03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='00210 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='00044 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='21 z01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0167 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0060 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='36 z40 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='00030 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='00020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='67 z20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0046 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0036 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='78 z31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='00010 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='00020 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='00 z11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0011 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0028 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='55 z22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='00035 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='00016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='46 z02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0011 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0016 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='45 z13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='000125 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='000088 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='70 z30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0008 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0014 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='75 z04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='000097 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='000033 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='34 Table 7: Same as in Table 3, but for the odd subsample of the final working sample (see text) Extremum X (kpc) Y (kpc) ζ4(X, Y ) (pc) Sζ4(X, Y ) Local maximum −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='1 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0 ± 20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='01 Local minimum 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='0 −50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='4 ± 15 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='59 were repeated for each of the two independent subsamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The results are presented in Tables 5–7 and in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The optimal orders of the model ζn for both subsamples turned out to be the same and equal to no for the full sample: no = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The parameters of the model ζ4 for even and odd subsamples and for the full sample within the error limits are consistent with each other (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Tables 1, 5, 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' At the same time, all significant parameters obtained from subsamples are also significant for the full sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The characteristics of the local extremes for the odd subsample turned out to be similar to the characteristics for the full sample (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Tables 3, 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' For the even subsample, the model ζ4(X, Y ) does not formally have local extremes, but it has areas of depression and elevation (Figure 9, right panel), in position and amplitude close to those for models obtained from the full sample and odd subsample (Figure 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Figure 9, left panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The curves-of- nodes also turned out to be similar for all three samples in the area of applicability of the model (Figures 4, 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Thus, the topology of the resulting model as a whole is preserved even when the sample is divided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' At the same time, the drop in the significance of the details of the model surface ζ4(X, Y ) for subsamples shows that dividing the sample into a larger number of parts is hardly meaningful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 5 DISCUSSION Consistency of local solar offset estimate z⊙ = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='1 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='1 �� stat +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='3 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='3 �� cal pc and global estimate z⊙ = 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='1 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='8 �� stat +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='3 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='2 �� cal pc (Table 2) also implies that the proposed algorithm is valid and the model order is correct—when optimizing the order of the model, the value of the solar offset does not depend systemati- cally on whether a large or small neighborhood of the Sun is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Our estimates also agree with the best estimate of the solar offset z⊙ = 25 ± 5 pc (Bland-Hawthorn & Gerhard 2016) and local estimates specifically for classical Cepheids (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' However, the estimate of σρ = 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='5 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='9 pc obtained for the final local sample is inconsistent with the estimate of σρ = 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='7 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='3 pc for the final working sample (in original distance scale, see Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 18 Nikiforov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 9: Maps of the models ζ4 of the average disk surface of optimal order, constructed from the objects of independent odd (left panel) and even (right panel) subsamples of the final working sample (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The notation is the same as in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Since the estimates were obtained by optimizing the order of the model ζn, this mismatch should mainly be a consequence of a combination of the flaring and random errors in distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Indeed, the observed deviation of objects relative to the model ζno is due to two factors: the natural (true, cosmic) dispersion of objects relative to the average surface of the Galactic disk and the measuring dispersion—the deviation of the observed positions of objects from their true positions due to random errors in distance moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The latter means that more distant objects have larger distance errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' On the other hand, the natural dispersion can grow to the periphery (the disk flaring).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Indeed, at X < 0 kpc, the apparent spread of objects relative to the model increases somewhat (Figure 6, left panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' In order to separate the contributions of these two effects, a more complex version of the present method is required with direct consideration of random errors in distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' This will allow you to get more reliable results about flaring itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Note that the contribution of the measuring dispersion to the observed dispersion σρ in this case is very small, as shown by the following approximate estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' For photometric distances, their standard deviation due to measuring dispersion is σr = ln 10 5 σd r, where σd is the standard uncertainty of distance moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' In the case of local sample, the main contribution of distance errors to the observed vertical standard deviation σZ (close to σρ) occurs due to objects at high latitudes b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Then the assumption that the standard σr for all objects of the sample completely passes into vertical standard deviation gives an upper estimate for the contribution of the measuring dispersion to σZ: σZ,mes < σr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' For the distance r = 1σρ = 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='5 pc (see Table 2) and σd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='14m (Veselova & Nikiforov 2020), this results in σZ,mes < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='9 pc and the natural dispersion σZ,0 = � σ2 Z − σ2 Z,mes > 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='3 pc, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=', the correction is no more than −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' In the case of a working sample, most of the objects are located at small |b|: for characteristic distances r = 3–6 pc (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 1) and Z = 1σρ = 132 pc (Table 2) |b| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='◦3–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='◦5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' For sample objects, the contribution of the measuring dispersion is σZ,mes = σr sin |b|, tan b = Z/(r cos b), then for small |b| b ≈ Z/r, σZ,mes ≈ σr|b| = ln 10 5 σd |Z|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Then for Z = 1σρ = 132 pc it turns out: σZ,mes ≈ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='5 pc, σZ,0 ≈ 131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='7 pc, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=', 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='60 880 90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='30 0 00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='15 kpc kpc 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='00 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='15 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='70 15 5 0 X,kpc5 00 000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='60 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='30 80 a 88 000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='15 00:00 kpc kpc 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='00 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='15 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='70 15 5 0 5 X,kpcModeling the vertical distribution of disk objects 19 correction −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Thus, in both cases, the contribution of random distance uncertainty to the observed vertical dispersion is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' From classical Cepheids on r < 2 kpc Majaess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' (2009) obtained z⊙ = 26 ± 3 pc and the scale height of ≤75±10pc which are consistent with our estimates (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Estimates of z⊙ = (23–24)±2 pc and σρ = 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='8 pc were found by Bobylev & Bajkova (2016a) for classical Cepheids according to the same version of the Berdnikov et al.’s catalog as in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Bobylev & Bajkova (2016a) considered the cylindrical region r ≤ 4 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' As they used the original calibration of the catalog we can compare these estimates with ours in the same calibration (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' There is consistency with our estimates of the solar offset for both final local and final working samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' However, the vertical scale estimates are consistent only in the case of final local sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Exactly the same situation is for Cepheids-based estimates in Skowron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' (2019a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The authors considered data on 2431 Cepheids, and the most part of the data was obtained by the OGLE-IV project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Skowron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' obtained an estimate of the disk scale height of 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='5 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='2 pc, so our estimate for the final local sample does not contradict this value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' All this also points out the importance of taking into account the warp of the Galactic average disk surface in order to have the ability of proper consideration of all the data available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Otherwise, only local regions can be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' In addition, the fact that the estimates of z⊙ differ, as noted in Introduction, also suggests that the a priori assumption about the flat model of the Galactic disk should be limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Indeed, according to our results such assumption might be made only for specific regions like the local one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=', close enough to the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Moreover, it can be noticed that the value of z⊙ is less dependent on the Galactic disk warping than the value of σρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Based on what has been said, we can conclude that any a priori assumption about the Galactic disk warping must be carefully studied especially when the vertical scale parameter is estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The detected local extrema of the average surface of the disk may be manifestations of bending waves caused by interaction with the Sagittarius dwarf galaxy, in the form of local structures elongated in the azi- muthal direction (G´omez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2013, fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Laporte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Poggio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2021, fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2), or by interaction with the Large Magellanic Cloud (Thulasidharan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2022 and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The method used after testing on classical Cepheids can now be applied to other data (in particular, to Gaia data) and/or in other assumptions about the distribution function f(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' The Gaia DR2 catalog was used in recent work by Ablimit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' (2020) to obtain data on classical Cepheids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Despite the fact the direct study of the Galactic disk warping was not conducted in the work of these authors, according to the pictures plotted in the work on these data, the disk warping is clearly revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Unfortunately, the use of the current version of our method with these data as is will lead to significant biases mainly because of the dependence of distance uncertainty on distance, which will be significant due to the need to consider the large neighborhood of the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Taking into account the uncertainty of distances may also solve the problem of establishing the form of the vertical distribution law f(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' Note that the analysis of the 2D distribution does not allow us to draw a definite conclusion about the functional form of this law (Mosenkov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE2T4oBgHgl3EQfqwhx/content/2301.04042v1.pdf'} +page_content=' 2021).' metadata={'source': 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mode 100644 index 0000000000000000000000000000000000000000..e0f6fd17ddc5c5907fec947818c78f6f0de455c7 --- /dev/null +++ b/O9FRT4oBgHgl3EQfITcf/content/tmp_files/2301.13491v1.pdf.txt @@ -0,0 +1,1168 @@ +MNRAS 000, 1–10 (2022) +Preprint 1 February 2023 +Compiled using MNRAS LATEX style file v3.0 +M-dwarf stars in the b294 field from the VISTA Variables in the Vía +Láctea (VVV) +Patricia Cruz1,2★, Miriam Cortés-Contreras1,2, Enrique Solano1,2, Carlos Rodrigo1,2, +Dante Minniti3,4, Javier Alonso-García5,6, Roberto K. Saito7 +1Centro de Astrobiología (CAB), CSIC-INTA, Camino Bajo del Castillo s/n, E-28692, Villanueva de la Cañada, Madrid, Spain +2Spanish Virtual Observatory +3Instituto de Astrofísica, Facultad de Ciencias Exactas, Universidad Andres Bello, Fernández Concha 700, Las Condes, Santiago, Chile +4 Vatican Observatory, Vatican City State, V-00120, Italy +5 Centro de Astronomía (CITEVA), Universidad de Antofagasta, Av. Angamos 601, Antofagasta, Chile +6 Millennium Institute of Astrophysics, Nuncio Monseñor Sotero Sanz 100, Of. 104, Providencia, Santiago, Chile +7 Departamento de Física, Universidade Federal de Santa Catarina, Trindade 88040-900, Florianópolis, SC, Brazil +Accepted 2023 January 30. Received 2023 January 27; in original form 2022 July 14 +ABSTRACT +M-dwarf stars are the dominant stellar population in the Milky Way and they are important for +a wide variety of astrophysical topics. The Gaia mission has delivered a superb collection of +data, nevertheless, ground-based photometric surveys are still needed to study faint objects. +Therefore, the present work aims to identify and characterise M-dwarf stars in the direction +of the Galactic bulge using photometric data and with the help of Virtual Observatory tools. +Using parallax measurements and proper motions from Gaia Data Release 3, in addition to +different colour-cuts based on VISTA filters, we identify and characterise 7 925 M-dwarf stars +in the b294 field from the Vista Variables in the Vía Láctea (VVV) survey. We performed +a spectral energy distribution fitting to obtain the effective temperature for all objects using +photometric information available at Virtual Observatory archives. The objects in our sample +have temperatures varying from 2800 to 3900 K. We also search for periodic signals in +VVV light curves with up to 300 epochs, approximately. As a secondary outcome, we obtain +periods for 82 M dwarfs by applying two methods: the Lomb-Scargle and Phase Dispersion +Minimization methods, independently. These objects, with periods ranging from 0.14 to 34 d, +are good candidates for future ground-based follow up. Our sample has increased significantly +the number of known M dwarfs in the direction of the Galactic bulge and within 500 pc, +showing the importance of ground-based photometric surveys in the near-infrared. +Key words: surveys – stars: fundamental parameters – stars: low-mass – astronomical data +bases: virtual observatory tools. +1 +INTRODUCTION +M dwarfs are the dominant stellar population in the Milky Way +(Kroupa 2001; Chabrier 2003). Because of their ubiquity and their +lifetimes in the main-sequence as long as the age of the Uni- +verse (Laughlin et al. 1997), these low-mass, cool dwarfs objects – +with masses from ∼0.075 to 0.6 M⊙, depending on the metallicity +(Chabrier et al. 2000), and with effective temperatures (𝑇eff) ranging +from 2 350 to 3 850 K, approximately (Pecaut & Mamajek 2013) – +are important for a wide variety of astrophysical contexts. For in- +stance, they are studied as tracers of the structure, kinematics, and +evolution of the Galaxy (e.g., Scalo 1986; Bochanski et al. 2007; +Ferguson et al. 2017) or in the discussion of formation and evolu- +★ E-mail: pcruz@cab.inta-csic.es +tion processes involving stellar objects at the end of the Herzprung- +Russell diagram (e.g., Jeffries et al. 2004; Stamatellos & Whitworth +2009; Luhman 2012). More recently, M-dwarf stars have been de- +fined as prime targets of exoplanet surveys as possible hosts of +Earth-like planets and in the search for life elsewhere in the Galaxy +(e.g., Bonfils et al. 2013; Reiners et al. 2018; Wunderlich et al. +2019). +M dwarfs emit the bulk of their energy in the near-infrared +range of the spectrum. Therefore, large area photometric surveys +operating at these wavelengths, such as for instance the Two Micron +All Sky Survey (2MASS, Skrutskie et al. 2006) and the UKIRT +Infrared Deep Sky Survey (UKIDSS, Lawrence et al. 2007), have +been valuable resources for M dwarf identification and character- +isation, in combination with broadband photometry in the visible +range. These data typically come from Pan-STARRS DR1 (Kaiser +© 2022 The Authors +arXiv:2301.13491v1 [astro-ph.SR] 31 Jan 2023 + +2 +P. Cruz et al. +et al. 2010; Chambers et al. 2016), SDSS (York et al. 2000), and +Gaia (Gaia Collaboration et al. 2016) surveys, which have been ex- +tensively used for this purpose (e.g., Cook et al. 2016; Lodieu et al. +2017; Bentley et al. 2019). +Several works have been devoted to determine the photometric +colours observed in the M dwarf regime (West et al. 2005; Cifuentes +et al. 2020). All the efforts put on the identification of M dwarfs +have generally avoided the Galactic bulge, with plenty of stars, +gas and dust. It is in this direction where the public ESO VISTA +Variables in the Via Lactea (VVV) survey operates (Minniti et al. +2010, 2018). Using colour-cuts from VVV b201 field, a field near +the bottom-right region of the Galactic bulge, Rojas-Ayala et al. +(2014) identified over 23 300 M-dwarf candidates. These authors +also estimated photometric distances, since Gaia parallaxes were +not available yet, and identified among their sample possible giant +contaminants based on a reduced proper motion diagram. +The Gaia mission has delivered a superb collection of data, +which include astrometric measurements for almost two billion +stars, a smaller set but yet an impressive amount of light curves, low- +resolution spectra and radial velocity measurements. Nevertheless, +the great majority of them are not available for targets with 𝐺-band +magnitude fainter than ∼171. Therefore, ground-based photometric +surveys still have an important role in the Gaia era, especially for +faint objects. The present work aims to identify and characterise +M-dwarf stars towards the bulge of the Galaxy using photometric +data only and with the help of Virtual Observatory tools. +In this work, we explored the VVV b294 field located in the +inner bulge region. The description of the used data is presented +in Sect. 2. The adopted sample selection methods and the list of +M-dwarf candidates are presented in Sect. 3. The stellar character- +isation based on the stellar spectral energy distribution is detailed +in Sect. 4. A comparison with M stars from Gaia is performed in +Sect. 5. As a secondary outcome, the search for periodicity in the +VVV light curves is described in 6. Finally, our conclusions are +presented in Sect. 7. +2 +VVV DATA +The VVV Survey (Minniti et al. 2010; Saito et al. 2012) is a large +ESO public near-IR surveys that mapped the bulge and inner disk +of the Milky Way in the near infrared since 2011. The observa- +tions were performed using the 4.1m-telescope at the Cerro Paranal +Observatory (Chile) and the data reduction was carried out at the +Cambridge Astronomical Survey Unit (Irwin et al. 2004; Lewis +et al. 2010). The observations cover the near-infrared 𝑍𝑌𝐽𝐻𝐾𝑠 +passbands, up to four magnitudes deeper than 2MASS. In addition, +a variability campaign was done from 2010 to 2016, where 𝐾𝑆-band +light curves were generated with up to 300 epochs (for more details, +see Saito et al. 2012; Botan et al. 2021, and references therein). +In this work we have focused on the b294 field, which spans +among the following coordinates: 3.1◦ < 𝑙 < 4.7◦ and −3.8◦ < +𝑏 < −2.5◦. For our analysis, we use the photometric catalogue for +this VVV region by Alonso-García et al. (2018), extracted using +point spread function (PSF)-fitting techniques. All the details on +image reduction, PSF photometry, and quality flags are described +in Alonso-García et al. (2018). The VVV b294 field cointains a total +of 4 592 101 detected point-source objects. +1 For details, see Gaia documentation available at https://www.cosmos. +esa.int/web/gaia/dr3. +Figure 1 shows a skymap with the location of the VVV b294 +field (red filled square), to illustrate. According to the extinction +map by Surot et al. (2020), this region presents a complex extinction +structure. However, we do not expect nearby M dwarfs to be strongly +affected, as most of the extinction is in the background. According +to the 3D maps by Schultheis et al. (2014), the expected extinction +is minimal for distances of less than 2 kpc considering the line of +sight of the b294 field. More recently, the 3D extinction maps from +Lallement et al. (2022) also show that for this region we have some +minor extinction at about 200 pc, however, it is mostly clear until +about 1 kpc, where denser clouds appear. In any case, extinction +corrections are still considered in our analysis, as described later on +section 4. +3 +SAMPLE SELECTION +3.1 +Selecting candidates using parallax +The objects from the b294 field – with ∼4.6 million objects, as +mentioned in the previous section – were cross-matched using TOP- +CAT2 (Taylor 2005, 2011) with the Gaia Data Release 3 (Gaia DR3; +Gaia Collaboration et al. 2016, 2022, epoch 2016), adopting a search +radius of 1′′. Although high proper motion objects may lie outside +the search region with such a small radius, it was chosen deliber- +ately small to minimise the number of false counterparts, since we +are dealing with a crowded field. From this search, we kept only +around ∼ 37% of the initial set (1 684 116 objects). +Only those objects with a good astrometric solution, adopted +as RUWE < 1.43, were selected, reducing the sample to 727 342 +objects. We also kept only those that presented good parallax mea- +surements (𝜋 > 0 and 𝜋err/𝜋 < 0.2; 42 198 objects). Keeping the +relative errors below 20% makes the inverse of the parallax a reli- +able distance estimator (Luri et al. 2018). We also excluded those +objects that showed Gaia magnitude errors in 𝐺 and 𝐺RP bands +greater than 10%, resulting in a sample with 38 483 objects. +The final applied criterion was to keep only those stars within a +distance of 500 pc, in order to avoid high extinction levels expected +for distant objects in the Galactic plane and, thus, effective temper- +ature - extinction degenerancies in the Spectral Energy Distribution +(SED) fitting. The obtained sample contains 2 045 M dwarf candi- +dates. +To identify the M-dwarf stars within this selection, we per- +formed a SED fitting to obtain their effective temperatures (𝑇eff). +We used VOSA4 (Virtual Observatory SED Analyzer, Bayo et al. +2008) to obtain the stellar parameters based on broad-band pho- +tometry. The details of the fitting is further described in section 4. +Those stars with estimated 𝑇eff < 4000 K, and that presented a good +fit solution (see Sect. 4), were kept as our first subset of candidates +(hereafter, sample A) which contains 1 338 M-dwarf stars. Infor- +mation on these objects can be found in the SVO archive of VVV +M-dwarfs (see appendix A). +2 TOPCAT is an interactive Tool for OPerations on Catalogues And Tables, +available at http://www.star.bris.ac.uk/~mbt/topcat/ +3 The Gaia Renormalised Unit Weight Error (RUWE) helps to identify +non-single sources or objects with problematic astrometric solution. It is +expected to be ∼ 1.0 for single stars (Arenou et al. 2018; Lindegren et al. +2018, 2021). We selected objects with RUWE < 1.4 as a compromise +between sample completeness and minimum binary contamination. +4 http://svo2.cab.inta-csic.es/theory/vosa/ +MNRAS 000, 1–10 (2022) + +M dwarfs in VVV b294 field +3 +Figure 1. Location of the VVV b294 field (red filled square) in the sky in Galactic coordinates. A Planck map in aitoff projection is displayed in the background. +3.2 +Selecting candidates using proper motion and colour-cuts +As an alternative way of selecting candidates, we applied several +different filtering criteria to derive a sample of M-dwarf candidates. +Firstly, we kept only those objects with photometric data in +all VISTA filters and with magnitude errors of less than 5%. This +resulted in a subset of around 36% of the initial sample (1 641 542 +objects). +Rojas-Ayala et al. (2014) presented the expected colours for M +dwarfs, based on VISTA filters. We applied their colour-cuts to our +sample considering only the lower limits – given for an M0 dwarf +star – as follows, which allowed us to reach later-type objects: +𝑌 − 𝐽 > 0.336; +𝑌 − 𝐻 > 0.952; +𝑌 − 𝐾s > 1.100; +𝐽 − 𝐻 > 0.432; +𝐽 − 𝐾s > 0.642; +𝐻 − 𝐾s > 0.045. +At this point, we identified 804 452 objects. +3.2.1 +Giant contaminants +We performed a cross-match with Gaia DR3 using TOPCAT, within +a search radius of 1′′, to obtain the measured proper motions (here- +after, PM), 𝜇, for our objects. Here, we kept only those with PM +errors on both directions, PMRA and PMDec, of less than 20% (a +total of 256 890 objects). Then, we calculated the reduced proper +motion in VISTA J band, 𝐻J, using the definition from the seminal +paper by Jones (1972)5. To discriminate between giant and dwarf +5 Jones (1972) defined the reduced proper motion as 𝐻 = 𝑚 + 5 · log 𝜇 + 5, +where 𝑚 is the apparent magnitude in a given photometric band. +star we adopted the following criterion described in Rojas-Ayala +et al. (2014): +𝐻𝑑 +J > 68.5 · (𝐽 − 𝐾s) − 50.7, +(1) +where 𝐻𝑑 +J indicates the value of the reduced proper motion expected +for a dwarf star as a function of (𝐽 − 𝐾s), colour that is calculated +using VISTA magnitudes. Almost half of the objects in the sample +(130 181 objects) satisfied Eq. 1, which are those expected to be +dwarf stars. +To minimise the amount of remaining giant contaminants, we +applied another criterion based on the stellar proper motion. For +that, we estimated the typical PM of M dwarfs using the SIMBAD +Astronomical Database6 (Wenger et al. 2000), starting by the dis- +tribution of giant stars – over 4 200 objects found, with a spectral +type later than M0. We searched for these objects proper motions by +cross-matching them with Gaia DR3, keeping only those with good +PM measurements (PMRA and PMDec errors of less than 20%), +resulting in 4 043. Analysing only those M giants (MIII) labelled as +stars (SIMBAD OTYPE S = ’Star’) – 2 350 of them –, we found an +average PM ∼ 11.3 mas/yr, where ∼ 64% of the giant M-dwarf sam- +ple presented PM smaller than the average (∼ 60% if we consider +𝜇 < 10.0 mas/yr). +Similarly, we searched for M dwarfs (MV) using SIMBAD, +getting a sample of 20 0007 objects. Applying the same PM con- +dition and selecting only the objects labelled as low-mass stars +(12 646; SIMBAD OTYPE S = ’low-mass’), we derived an av- +erage 𝜇 ∼ 27.1 mas/yr. Adopting the PM average for MIII of +∼ 11.3 mas/yr, we found that ∼ 76% of the tested dwarf sample +has higher PM values (∼ 81% if we consider 𝜇 > 10.0 mas/yr). +6 http://simbad.u-strasbg.fr/simbad/. +7 This is in fact a subsample of M-dwarf stars from SIMBAD, since the tool +limits the search to 20 000 results. +MNRAS 000, 1–10 (2022) + +ANCKER2LE ++60 ++45 +156 +60 +754 +P. Cruz et al. +Hence, to maximise completeness of dwarfs and minimise contam- +ination from giants, we set a cut in PM at 10.0 mas/yr. +To assess the performance of our whole approach for elimi- +nating giants, we applied the related criteria – good proper motion +values (𝑒𝑟𝑟𝜇/𝜇 < 20%), cut in 𝐻J from Eq. 1, and 𝜇 ∼ 10.0 mas/yr +– to the MIII sample from SIMBAD. From 4 252 giants (the com- +plete MIII set), only 15 giants survived the filtering, which repre- +sents 0.35% of them. If considering only those giants labeled as +stars (2 350 objects), the remaining objects represent only 0.64%, +meaning that these criteria were enough to eliminate 99.36% of +the giants. To verify the completeness of dwarf stars after applying +these criteria, we repeated the giant filtering to the sample of MV +from SIMBAD. We found that approximately half of the objects +survived all criteria, which eliminated 50.6% of the sample. Since +we are prioritizing purity over completeness, the giant filtering is +interpreted as a fair compromise between a good sample of dwarfs +and minimum giant contamination. +Therefore, we kept only those objects that have their total +proper motion higher than 10 mas/yr. This way, we identified 24 787 +candidate M-dwarf stars (or later) in our sample. +3.2.2 +Photometric distances +From the list of 24 787 candidates, we kept those objects with good +𝐺 mag photometry (with relative error of less than 10%) and with +good astrometric solutions (RUWE < 1.4), resulting in a subset of +16 652 objects. We used the Gaia and VISTA broad-band photome- +try to obtain the absolute 𝐺 magnitude, 𝑀G, as function of (𝐺 − 𝐽) +colour for our list of identified M dwarfs, considering the empirical +relation described in Cifuentes et al. (2020): +𝑀G = 16.24−13.04·(𝐺 −𝐽)+5.64·(𝐺 −𝐽)2−0.622·(𝐺 −𝐽)3. (2) +This relation was derived using 2MASS J-band photometry. How- +ever, González-Fernández et al. (2018) showed that the colour off- +sets in VISTA are modest and differences in magnitudes between +2MASS and VISTA are expected to be small, of around 5 mmag in +the J band8. It is also worth mentioning that the empirical relation +(eq. 2) is valid for stars with certain values of (𝐺 − 𝐽) colour, be- +tween 2.0 and 4.0. Applying this as condition, our sample is reduced +to 16 503 objects. +The photometric distances were then computed from the esti- +mated absolute G magnitudes and, as a final condition, only those +stars with estimated distances of less than 500 pc were selected +– 6 846 M-dwarf candidates. After performing a SED fitting with +VOSA (for details, see Sect. 4), we kept only those which had pre- +sented a good fit solution and with 𝑇eff< 4000 K as we did in +sect. 3.1. This second subset (hereafter, sample B) contains 6 747 +stars and it is also shown in the virtual observatory compliant archive +described in Appendix A. +3.3 +The final sample +We have compared the results obtained from the two methods pre- +viously described. Cross-matching both samples, we verified that +only 160 objects are common to both A and B samples. This is +8 For an A0 V star, the VISTA-2MASS colour in J band is (𝐽VISTA − +𝐽2MASS) = 0.005 ± 0.015 (González-Fernández et al. 2018, and references +therein). +Figure 2. Distribution of effective temperatures obtained from the SED +fitting with VOSA. The used grid of models has a step in temperature of +100 K, which was adopted here as the bin size. +Figure 3. Distribution of obtained distances. The distances of M-dwarf +candidates from sample A were obtained from Gaia parallaxes. For sample +B, photometric distances were adopted (see sect. 3.1 and 3.2 for details). +due to the different filtering conditions applied in each method, as +explained in detail below. +By applying to sample A the same conditions considered for the +second method – which are colour-cuts for a M0 dwarf, photometric +data in all VISTA/VVV bands with small error (less than 5%), cut +in 𝐻J (eq. 1), 𝜇 > 10 mas/yr, and photometric distances of less than +500 pc (see details in Sect. 3.2) – we recovered the same 160 objects. +Similarly, by applying to sample B the same criteria used in the first +method – which are good parallax measurements, photometric data +in Gaia DR3 𝐺 and 𝐺RP bands with small error, and distances from +parallax of less than 500 pc (see details in Sect. 3.1) –, we found 160 +objects. Therefore, our final sample contains 7 925 (1 338 + 6 747 +- 160) M-dwarf stars. Figures 2 and 3 show the distribution of the +estimated temperatures and distances of these objects. +The identified M dwarfs are placed at the colour-magnitude +diagram (CMD), as illustrated in figure 4. Grey diamonds represent a +sample of Gaia nearby stars from Torres et al. (2022), after applying +the quality metric described by Riello et al. (2021)9, illustrating +the main sequence locus. The colour (𝐽 − 𝐻) for Gaia stars was +9 For details, see Table 2, and Eqs. 6 and 18 from Riello et al. (2021). +MNRAS 000, 1–10 (2022) + +2000 +1500 +1000 +500 +0 +4000 +3800 +3600 +3400 +3200 +3000 +2800 +T eff [K]800 +700 +600 +500 +400 +300 +200 +100 +0 +100 +200 +300 +400 +500 +Distance[pc]M dwarfs in VVV b294 field +5 +Figure 4. Colour-magnitude diagram showing the M dwarf sample from +VVV b294 field. Gaia nearby stars are represented by grey diamonds and +red crosses show the M stars in our sample. +Table 1. Broad-band photometry used for the SED fitting with VOSA. The +information was compiled from the SVO Filter Profile Service (Rodrigo +et al. 2012; Rodrigo & Solano 2020). +Band +𝜆eff +𝑊eff +survey +name +(Å) +(Å) +OmegaCAM u +3607.68 +482.54 +VPHAS+ DR2 +OmegaCAM g +4679.46 +1203.25 +VPHAS+ DR2 +Gaia G +5822.39 +4052.97 +Gaia DR3 +OmegaCAM H𝛼 +6590.81 +103.76 +VPHAS+ DR2 +OmegaCAM i +7508.50 +1463.68 +VPHAS+ DR2 +VISTA Z +8789.53 +889.46 +VVV +OmegaCAM z +8884.40 +864.48 +VPHAS+ DR2 +VISTA Y +10196.43 +870.63 +VVV +VISTA J +12481.00 +1542.53 +VVV +VISTA H +16348.19 +2674.02 +VVV +VISTA Ks +21435.46 +2793.85 +VVV +IRAC I1 +35075.11 +6836.16 +GLIMPSE +IRAC I2 +44365.78 +8649.93 +GLIMPSE +IRAC I3 +56281.02 +12561.17 +GLIMPSE +IRAC I4 +75891.59 +25288.50 +GLIMPSE +obtained from a cross-correlation with 2MASS, considering a 1′′ +search radius. All objects from our sample of M stars are presented +as red crosses. +4 +STELLAR PROPERTIES WITH VOSA +As briefly mentioned in Sect. 3, we used VOSA (Bayo et al. 2008) +to obtain effective temperatures, luminosities and radii for all M- +dwarf candidates in our selected sample. VOSA is a tool developed +by the Spanish Virtual Observatory10 designed to build the SEDs of +thousands of objects at a time from a large number of photometric +catalogues, ranging from the ultraviolet to the infrared. VOSA com- +pares catalogue photometry with different collections of theoretical +models and determines which model best reproduces the observed +data, following different statistical approaches. Physical parameters +10 http://svo.cab.inta-csic.es +are then estimated for each object from the model that best fits the +data. +To construct the SED, we used the VISTA 𝑍𝑌𝐽𝐻𝐾𝑠 photome- +try obtained from the VVV survey. We also searched for additional +broad-band photometry from Gaia DR3 (only in the 𝐺 band), from +the VST Photometric H𝛼 Survey of the Southern Galactic Plane +and Bulge (VPHAS+ DR2; Drew et al. 2014), and from the Galac- +tic Legacy Infrared Mid-Plane Survey Extraordinaire (GLIMPSE +Source Catalog I + II + 3D; Spitzer Science 2009). Information on +the filters is shown in table 1. +It is worth mentioning that only a small portion of the sample +(172 objects) had VPHAS+ DR2 magnitudes available. For these +objects, the colour (𝑟 − 𝐻𝛼) from VPHAS+ DR2 was used to +search for H𝛼 emission. We found that all the objects with 𝑟 and +H𝛼 photometry available have (𝑟 − 𝐻𝛼) < 0.9, which is expected +for non-emitting M sources (Drew et al. 2014). +The BT-Settl CIFIST models – the BT-Settl theoretical spectra +by Allard et al. (2011) computed with a cloud model and using +the Caffau et al. (2011) solar abundances – were adopted for the +SED fitting, covering effective temperatures from 1200 to 7000 K +and assuming solar metallicity ([Fe/H] = 0.0). Since we expect our +candidates to be dwarf stars (see Sect. 3 for details), the surface +gravities were allowed to vary only from 4.0 to 5.5. Finally, we +defined a range of possible values for the extinction (𝐴v), going +from 𝐴v = 0.0 to 0.5, as expected according to extinction models +for the region. Extinction is left as a free parameter to be fitted +together with the stellar parameters in the SED fitting process. To +calculate the extinction in each filter, VOSA uses the extinction law +by Fitzpatrick (1999), with the improvement in the infrared region +by Indebetouw et al. (2005)11. +After the fit is performed, we opted to refine the results due to +excess present in the infrared data. By comparing the photometric +data to the best-fit model, VOSA is able to identify which data points +appear high above the model and (re)define the starting point of the +infrared excess. We then refitted those SEDs that VOSA identified to +have an excess, where all the points which were found to be affected +were not considered in the final fit. +Since the b294 tile is placed in a crowded field, some of the +detected IR excess could just be due to a wrong counterpart assign- +ment. This is illustrated in fig. 5. On the left, the example of an +object with a SED showing a strong IR excess (top left panel) but +later verified to come from a different source (bottom left panel). +For comparison, a more subtle excess in the SED of a different +object is presented on the right (top right panel), later confirmed to +come from the same emitting source (bottom right panel). All ob- +jects for which VOSA has identified an infrared excess are properly +flagged in the archive, as described in Appendix A. These detected +IR excess need to be further confirmed, as explained above. +We obtained a good fit for all objects in our sample, according +to visual goodness-of-fit value (𝑉gfb) estimated by VOSA, which +is the modified reduced 𝜒2 calculated by forcing the error in the +observed flux to be larger than 10%12. The results of the SED +fitting are also included in the archive described in Appendix A. +11 For more details on the interstellar extinction, see VOSA’s help sec- +tion at http://svo2.cab.inta-csic.es/theory/vosa/help/star/ +extinctions/. +12 The visual goodness-of-fit value, 𝑉 gfb, smaller than 12–15 is often +perceived as a good fit. For more details, see VOSA’s help page http: +//svo2.cab.inta-csic.es/theory/vosa/help/star/fit/. +MNRAS 000, 1–10 (2022) + +0 +5 +10 +15 +1.0 +0.5 +0 +0.5 +1.0 +1.5 +2.0 +J-H6 +P. Cruz et al. +Figure 5. Example of two different cases of IR excess present from GLIMPSE photometry. The top panels show the results from SED fitting using VOSA and +the identified starting point of IR excess (vertical dashed line), where blue squares represent the synthetic photometric calculated from the best-fit model, the +red filled circles are the observed photometry, and black circles indicate the detected excess. The thin grey line shows the stellar SED before correcting from +extinction. The lower panels (generated using Aladin) show the position of the same stars (marked as a pink cross at the center of each image) over a 𝐽-band +image from VISTA/VVV survey. The objects identified in the VISTA/VVV catalogue are presented as small blue circles and those in the GLIMPSE catalogue +are presented as red squares. The lower left panel illustrates an example of a star for which the GLIPMSE data came from a nearby object and, therefore, the +IR excess in the SED (left upper panel) is not real. Different from the second example (right lower panel) where the detected excess is coming from the object. +5 +COMPARISON TO GAIA M DWARFS +The sky distribution of the 7 925 M-dwarf stars identified in VVV +tile b294 is presented in figure 6, where the grey area represents the +observed field of view and the black solid line shows the Multi- +Object Coverage (MOC). The objects from samples A and B are +represented by yellow and red filled circles, respectively. +To assess the importance of our methodology and the impact of +our M stars catalogue to the identified cool stars in the studied region +– towards the Galactic bulge (see fig. 1) – we exploited the available +data from Gaia DR3 to search for known M dwarfs in the VVV tile +b294 field. From the approximately 2 million sources of the Gaia +DR3 present in tile b294 field (fig. 6, grey area), we selected those +with RUWE of less than 1.4 – thesame limit appliedfor VVV objects +and described in sect. 3 – to ensure a good astrometric solution for +a point-like source. As a matter of consistency, we selected the stars +with a relative parallax error of less than 20%, keeping those within +a distance of 500 pc. Aiming at identifying M dwarfs, we kept those +stars with 𝑇eff of less than 4000 K and log 𝑔 ≥ 4.0 – astrophysical +parameters available in Gaia DR3 catalogue, which were derived +from BP/RP spectra. The resulting sample contained only 208 stars +characterised in Gaia DR3 as M-dwarf stars. All these Gaia DR3 +M stars are shown in fig. 6 as blue crosses. Therefore, we have +increased considerably the number of M dwarfs identified in the +field, from a few hundreds to thousands of stars. +Figure 7 shows the comparison between the effective temper- +atures derived from SED fitting with VOSA and the ones given in +Gaia DR3 catalogue for the 162 objects in common. Considering +the estimated uncertainties, there is in general a good agreement +between derived values, with a mean difference of 181 K and a +standard deviation of 136 K, approximately. Among the M stars +identified in Gaia DR3, 39 of them were not included in the cross- +match with the original VVV tile b294 catalogue, considering 1′′ +radius, and 7 were lost in the applied criteria, such as colour-cuts +and proper motion limits. +Our sample has increased significantly the number of known +M dwarfs in the studied region – a crowded field towards the bulge +of the Galaxy – emphasising the importance of ground-based pho- +tometric surveys in the near-infrared. +MNRAS 000, 1–10 (2022) + +10-16 +10 +-17 +10-18 +10-19 +104 +105 +入 (A)10-16 +10-17 +10-18 +104 +入 (A)VISTA VY DR4 J +回 +白 +回 +回 +D +D +0 +回 +回 +巴 +5 +回 +回 +回 +回 +回 +回 +回 +D +8 +0 +回 +13 +. +0 +0 +0 +0 +回 +回 +回 +回 +回 +回 +0 +D +回 +D +回 +回 +回 +回 +0 +. +回 +回 +回 +N +0 +P +15. +1161'x 59.55VISTA VV DR4 J +回 +D +回 +回 +回 +回 +回 +回 +回 +6 +回 +回 +回 +回 +回 +回 +回 +回 +回 +回 +回 +回 +回 +回 +回 +回 +回 +回 +回 +回 +回 +回 +回 +回 +回 +回 +回 +回 +回 +回 +回 +回 +回 +N +回 +Lo +回 +15 +E. +1.1M dwarfs in VVV b294 field +7 +Figure 6. Sky position of the 7 925 M-dwarf stars in our sample. The VVV +tile b294 is presented as a grey area and the solid black lines represent +its MOC. Blue crosses show the M dwarfs identified in Gaia DR3 within +500 pc. +Figure 7. Comparison of 𝑇effobtained in our analysis (VOSA) with those +given in Gaia DR3, for the stars identified as M dwarfs in both sets. The +dashed line represents the identity function. +6 +SEARCHING FOR PERIODIC SIGNALS IN VVV +LIGHT CURVES +As a secondary outcome of this study, it is interesting to analyse the +multi-epoch observations performed in VVV tile b294 and search +for periodic signals. +The time series were obtained with the VVV 𝐾𝑠 filter, with over +300 observations of our field of interest. Not only this, but due to the +strategy of observation (Saito et al. 2012), every region of the field +is observed at least twice, most of the times using different chips of +the 16-chip VIRCAM instrument used in the VISTA telescope. For +our multi-epoch analysis described below, we made use of VVV +𝐾𝑠-band light curves based on PSF photometry as described in +Contreras Ramos et al. (2017). +6.1 +Light curve selection +We gathered light curves with more than 25 detections for 7 752 M +dwarfs out of the 7 925 stars in our sample. Some of them have been +detected in more than one chip due to overlap, providing a total of +8 640 different light curves in the 𝐾𝑠/VISTA band. The time-series +data covered in total 5.4 yr of observations, acquired between 12 +April 2010 and 11 September 2015 (Contreras Ramos et al. 2017). +In order to clean spurious data present in the light curves, we +carried out a two-step procedure: +(i) We ran a sigma-clipping approach aiming at identifying the +baseline emission of the star. +(ii) From the original light curve, we kept all epochs within the +first and 99th centiles and discarded those brighter than the median +value minus 3𝜎, computed from the set obtained in the previous +exercise. These detections could be ascribed as mismatches that +could contaminate the search for periodic signals. +In order to keep the sample purity, outliers were excluded from the +light-curve analysis. +All light curves that present more than three points with mag- +nitude values between the median plus 3𝜎 and the 99th percentile +were considered suitable for the analysis. A drop in brightness can +be ascribed to either a transiting object or to starspots present in the +stellar surface of active M dwarfs. We obtained 8 219 light curves +associated to 7 153 M dwarfs. An example of a raw light curve is +shown on top panel in Figure 8. +Additionally, we applied a photometric quality filter and dis- +carded all the light curves with mean magnitude errors larger than +1.5 times the Median Absolute Deviation (MAD) obtained from the +baseline emission of each star in step (i). After this, 4 952 stars with +5 817 light curves remained. They were normalised to the median +and combined in order to obtain a unique light curve per star. +6.2 +Light curve analysis +Although the light curves span over 5.4 yr, there are three main +observing blocks of nearly 6.5 months each (200 d), going from +May to November 2012, from May to December 2014, and from +March to October 2015. With the goal of providing a reliable set of +M-dwarf candidates with trustful periodic variations, we searched +for periodic signals in each of the three main blocks of observations +rather than in the whole data set. Figure 8 shows an example of a +whole light curve where the three observing blocks used for deriving +the periods are clearly differentiated. +Given the unevenly spaced observations, we obtained the pe- +riodograms from each block in the light curves using two comple- +mentary methods. Firstly, we applied the Generalized Lomb-Scargle +algorithm (GLS, Lomb 1976; Scargle 1982), which fits a sinu- +soidal model to the data at each frequency. For that, we employed +the Python astropy.timeseries package (Astropy Collaboration +et al. 2013, 2018) and obtained the best periods (i.e., the periods +with the highest peak in the periodogram). We tested them through +the false alarm probability (FAP, Scargle 1982) computed with the +Baluev approximation (Baluev 2008). We derived three periods – +one period per block – but only periods with FAP under 0.1 were +taken into consideration. Secondly, we searched for periodic sig- +nals applying the Phase Dispersion Minimization method (PDM, +Stellingwerf 1978), between 0.1 and 100 d. Unlike GLS, PDM is +unbiased towards the shape of the light curve and is able to find also +non-sinusoidal variations. +We obtained 224 periods in agreement within 20% from the +MNRAS 000, 1–10 (2022) + +4000 +K +eff,VOSA +3500 +3000 +3000 +3500 +4000 +T eff,Gaia [K]-26.0 +26.2 +26.4 +26.6 +26.8 +[deg] +27.0 +Dec +27.2 +27.4 +27.6 +-27.8 +-28.0 +28.2 +270.5 +271.0 +271.5 +272.0 +272.5 +RA [deg]8 +P. Cruz et al. +Figure 8. Top panel: Example light curve. Baseline emission data points, +detections related to variability and rejected outliers, which could also be +related to flares, are displayed in grey filled circles, blue open circles and +red crosses, respectively. Grey solid, short dashed and long dashed lines +represent the median, the median+3𝜎 and median-3𝜎 limits, respectively. +Black dashed-dotted lines represent the first and 99th quartiles. Bottom +panel: Light curve of the same object after removing outliers. Colour and +symbol code as in the figure on top. +LSG and PDM approximations. After removing near 1 d periodic +signals (from 0.9 to 1.1 d), 82 stars remain. Additionally, we use +the reduced chi-square as a variability indicator as in Botan et al. +(2021) and define two subsets of M dwarfs with periodic signals. +One is composed by 27 M dwarfs with 𝜒2 > 2, whose variability +would likely be related to the presence of a stellar companion, indi- +cating possible binary systems. The obtained periods range between +0.14 and 34 d. Among them, 2 objects were recently identified as +variables in a recent work by Molnar et al. (2022), where they were +flagged as eclipsing binary systems. The other subset is composed +by 20 M dwarfs that present 𝜒2 ∼ 1 (from 0.7 to 1.3), for which +the periodic signal could be ascribed to a planet-like transiting ob- +ject. However, the found periodicities could also be related to stellar +intrinsic variability, due to starspots for instance. Therefore, the pe- +riods need to be confirmed (or discarded) with dedicated follow-up +observations. The periods distribution is shown in Figure 9. Each +of the objects for which periodicity has been calculated is properly +flagged in the archive, as described in Appendix A. +Figure 9. Period distribution in logarithmic scale for the sample of 82 M +dwarfs. +7 +CONCLUSIONS +We searched for M-dwarf stars near the Galactic bulge in the b294 +field from the VISTA Variables in the Vía Láctea survey. We adopted +two different methodologies to identify M dwarfs. The first method +was performed using parallaxes, where we selected objects with +good astrometric solution, photometry – relative G and RP mag- +nitude errors below 10% – and parallax measurements – relative +error of less than 20% – from Gaia DR3. The second method was +based on colour-cuts and proper motions, where we kept those ob- +jects with good VISTA photometry in all 𝑌𝑍𝐽𝐻𝐾𝑠 bands, VISTA +colours among the colour-cuts defined by Rojas-Ayala et al. (2014) +for M stars, good proper motion – relative error of less than 20% – +and good astrometric solution also from Gaia DR3, and those with +a J magnitude reduced proper motion expected for dwarf stars (see +Sect. 3.2 for details). We then estimated absolute magnitudes from +the empirical relation based on colours by Cifuentes et al. (2020). +To avoid the interstellar extinction expected for the field, we +selected only the M-dwarf candidates within 500 pc, where dis- +tances where estimated from parallax (sample A) and photometric +distances based on absolute magnitudes (sample B). We then char- +acterised the remaining candidates by performing SED fittings using +VOSA, where we kept all objects with 𝑇eff < 4 000 K. Our final list +of M-dwarf candidates from the VVV b294 field has 7 925 stars, +with temperatures ranging from 2 800 to 3 900 K. +To assess the importance and impact of the identified M stars +towards the Galactic bulge, we compared our sample to all M dwarfs +characterised from BP/RP spectra available in Gaia DR3 catalogue +in the VVV tile b294 field. From nearly 2 million sources, there are +208 stars with 𝑇eff and log 𝑔 compatible with M-dwarf stars. Our +sample of 7 925 sources has significantly increased the number of +known M dwarfs within 500 pc in the studied region. +As a secondary outcome of this study, we also searched for +periodic signals in VVV light curves, with at least 25 and up to 327 +epochs. We removed outliers from the light curves and looked for +periodicities in the three main blocks of observation. We obtained +periods for 82 M dwarfs by applying two methods: the Lomb- +Scargle and Phase Dispersion Minimization, independently. These +periods range from 0.14 to 34 d. We defined two subsamples ac- +cording to the reduced chi-square (see Sect. 6.2) presenting large +and small variability (27 and 20 stars, respectively). Additional +follow-up observations and further analysis would be required for +confirming the nature of the periodic variability of these objects. +Even with the amazing collection of data delivered by Gaia +MNRAS 000, 1–10 (2022) + +13.7 +13.8 +13.9 +55500 +56000 +56500 +57000 +MJD [d]13.70 +13.75 +[mag +13.80 +55500 +56000 +56500 +57000 +MJD [d]12 +10 +8 +6 +4 +2 +0.2 +0.5 +1 +2 +5 +10 +20 +P [d]M dwarfs in VVV b294 field +9 +DR3, radial velocities and spectra are not available for all observed +objects. The methodology described in this work probed to be very +efficient on identifying and characterising M-dwarf stars in the VVV +b294 field, emphasising the importance of ground-based photomet- +ric surveys in the near-infrared. Therefore, it can be extended to +other VVV fields – and to those from the VVVX survey (Minniti +et al. 2018) – in order to increase the population of known low-mass +objects in the direction of the Galactic bulge. +ACKNOWLEDGEMENTS +We would like to thank Dr F. Jiménez-Esteban for the fruitful +discussion. P.C. acknowledges financial support from the Gov- +ernment of Comunidad Autónoma de Madrid (Spain) via post- +doctoral grant ‘Atracción de Talento Investigador’ 2019-T2/TIC- +14760. M.C.C. acknowledges financial support from the ESCAPE +project supported by the European Commission Framework Pro- +gramme Horizon 2020 Research and Innovation action under grant +agreement n. 824064. This research has made use of the Spanish +Virtual Observatory (https://svo.cab.inta-csic.es) project +funded by the Spanish Ministry of Science and Innovation/State +Agency of Research MCIN/AEI/10.13039/501100011033 through +grant PID2020-112949GB-I00 and MDM-2017-0737 at Centro +de Astrobiología (CSIC-INTA), Unidad de Excelencia María de +Maeztu. D.M. also thanks the support by the ANID BASAL +projects ACE210002 and FB210003, and Fondecyt Project No. +1220724. J.A.-G. acknowledges support from Fondecyt Regular +1201490, and ANID – Millennium Science Initiative Program – +ICN12_009 awarded to the Millennium Institute of Astrophysics +MAS. R.K.S. acknowledges support from CNPq/Brazil through +project 305902/2019-9. We gratefully acknowledge the use of data +from the ESO Public Survey program IDs 179.B-2002 and 198.B- +2004 taken with the VISTA telescope and data products from the +Cambridge Astronomical Survey Unit. We would also like to thank +R. Contreras Ramos (private communication) for the VVV light +curves used in the present work. +This publication makes use of VOSA, developed under the +Spanish Virtual Observatory project. VOSA has been partially up- +dated by using funding from the European Union’s Horizon 2020 +Research and Innovation Programme, under Grant Agreement nº +776403 (EXOPLANETS-A). This research has made use of the +SVO Filter Profile Service (http://svo2.cab.inta-csic.es/ +theory/fps/). This research has made use of "Aladin sky atlas" +developed at CDS, Strasbourg Observatory, France. This research +has made use of the SIMBAD database, operated at CDS, Stras- +bourg, France. +DATA AVAILABILITY: VIRTUAL OBSERVATORY +COMPLIANT, ONLINE CATALOGUE +In order to help the astronomical community on using our cata- +logue of VVV M dwarfs, we developed an archive system that can +be accessed from a webpage13 or through a Virtual Observatory +ConeSearch14. The content of the catalogue is presented in the +Appendix A. +13 http://svocats.cab.inta-csic.es/mdwarfs_vvv/ +14 A ConeSearch example can be seen at http://svocats.cab. +inta-csic.es/mdwarfs_vvv/cs.php?RA=271.877&DEC=-28.079& +SR=0.1&VERB=2 +The archive system implements a very simple search interface +that allows queries by coordinates and radius as well as by other +parameters of interest. The user can also select the maximum num- +ber of sources (with values from 10 to unlimited) and the number +of columns to return (minimum, default, or maximum verbosity). +The result of the query is a HTML table with all the sources found +in the archive fulfilling the search criteria. The result can also be +downloaded as a VOTable or a CSV file. Detailed information on +the output fields can be obtained placing the mouse over the ques- +tion mark located close to the name of the column. The archive also +implements the SAMP15 (Simple Application Messaging) Virtual +Observatory protocol. 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G., et al., 2000, AJ, 120, 1579 +APPENDIX A: CATALOGUE DESCRIPTION +The content of the catalogue of M-dwarf candidates from the VVV +b294 field is presented in Table A1. This catalogue can be accessed +from the dedicated webpage http://svocats.cab.inta-csic. +es/mdwarfs_vvv/ or through a Virtual Observatory ConeSearch +(e.g. http://svocats.cab.inta-csic.es/mdwarfs_vvv/cs. +php?RA=271.877&DEC=-28.079&SR=0.1&VERB=2). +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–10 (2022) + +M dwarfs in VVV b294 field +11 +Table A1. Description of the parameters contained in the VVV M-dwarf catalogue. +Parameter +Units +Description +Gaia_ID_DR3 +- +Gaia DR3 source identifier. +RAJ2000 +deg +Celestial Right Ascension (J2000). +DEJ2000 +deg +Celestial Declination (J2000). +Xmag +mag +Calibrated magnitude. X stands for 𝑍, 𝑌 , 𝐽, 𝐻 and 𝐾𝑠. +eXmag +mag +Calibrated magnitude error. X stands for 𝑍, 𝑌 , 𝐽, 𝐻 and 𝐾𝑠. +d +pc +Distance. +Ref_d +- +Reference for the distance. "Gaia" refers to parallactic distances and "This work" indicates distances are +specto-photometric derived in this work. +𝑇eff +K +Effective temperature. +Lbol +L⊙ +Bolometric luminosity. +eLbol +L⊙ +Error in the bolometric luminosity. +R +R⊙ +Stellar radius. Calculated using Lbol = 4𝜋𝑅2 𝜎 𝑇eff4. +eR +R⊙ +Error in the stellar radius. +Av +- +Visual extinction. +Method +- +Identification method of candidates. "A" stands for parallax, and "B" stands for proper motion and colour-cuts. +Flag_IR +- +Flag for stars with IR excess detected by VOSA yet to be confirmed. +Flag_lc +- +Flag for dwarfs with processed light curve. +Flag_P +- +Flag for dwarfs with periodic signal. +Flag_chi +- +Flag for M dwarfs with 𝜒2 ∼ 1 (1) and 𝜒2 > 2 (2). +MNRAS 000, 1–10 (2022) + diff --git a/O9FRT4oBgHgl3EQfITcf/content/tmp_files/load_file.txt b/O9FRT4oBgHgl3EQfITcf/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ec92973d6a3577cc0395bf9740547712c4642ad6 --- /dev/null +++ b/O9FRT4oBgHgl3EQfITcf/content/tmp_files/load_file.txt @@ -0,0 +1,953 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf,len=952 +page_content='MNRAS 000, 1–10 (2022) Preprint 1 February 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='0 M-dwarf stars in the b294 field from the VISTA Variables in the Vía Láctea (VVV) Patricia Cruz1,2★, Miriam Cortés-Contreras1,2, Enrique Solano1,2, Carlos Rodrigo1,2, Dante Minniti3,4, Javier Alonso-García5,6, Roberto K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Saito7 1Centro de Astrobiología (CAB), CSIC-INTA, Camino Bajo del Castillo s/n, E-28692, Villanueva de la Cañada, Madrid, Spain 2Spanish Virtual Observatory 3Instituto de Astrofísica, Facultad de Ciencias Exactas, Universidad Andres Bello, Fernández Concha 700, Las Condes, Santiago, Chile 4 Vatican Observatory, Vatican City State, V-00120, Italy 5 Centro de Astronomía (CITEVA), Universidad de Antofagasta, Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Angamos 601, Antofagasta, Chile 6 Millennium Institute of Astrophysics, Nuncio Monseñor Sotero Sanz 100, Of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 104, Providencia, Santiago, Chile 7 Departamento de Física, Universidade Federal de Santa Catarina, Trindade 88040-900, Florianópolis, SC, Brazil Accepted 2023 January 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Received 2023 January 27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' in original form 2022 July 14 ABSTRACT M-dwarf stars are the dominant stellar population in the Milky Way and they are important for a wide variety of astrophysical topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The Gaia mission has delivered a superb collection of data, nevertheless, ground-based photometric surveys are still needed to study faint objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Therefore, the present work aims to identify and characterise M-dwarf stars in the direction of the Galactic bulge using photometric data and with the help of Virtual Observatory tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Using parallax measurements and proper motions from Gaia Data Release 3, in addition to different colour-cuts based on VISTA filters, we identify and characterise 7 925 M-dwarf stars in the b294 field from the Vista Variables in the Vía Láctea (VVV) survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' We performed a spectral energy distribution fitting to obtain the effective temperature for all objects using photometric information available at Virtual Observatory archives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The objects in our sample have temperatures varying from 2800 to 3900 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' We also search for periodic signals in VVV light curves with up to 300 epochs, approximately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' As a secondary outcome, we obtain periods for 82 M dwarfs by applying two methods: the Lomb-Scargle and Phase Dispersion Minimization methods, independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' These objects, with periods ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='14 to 34 d, are good candidates for future ground-based follow up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Our sample has increased significantly the number of known M dwarfs in the direction of the Galactic bulge and within 500 pc, showing the importance of ground-based photometric surveys in the near-infrared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Key words: surveys – stars: fundamental parameters – stars: low-mass – astronomical data bases: virtual observatory tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 1 INTRODUCTION M dwarfs are the dominant stellar population in the Milky Way (Kroupa 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Chabrier 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Because of their ubiquity and their lifetimes in the main-sequence as long as the age of the Uni- verse (Laughlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 1997), these low-mass, cool dwarfs objects – with masses from ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='075 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='6 M⊙, depending on the metallicity (Chabrier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2000), and with effective temperatures (𝑇eff) ranging from 2 350 to 3 850 K, approximately (Pecaut & Mamajek 2013) – are important for a wide variety of astrophysical contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' For in- stance, they are studied as tracers of the structure, kinematics, and evolution of the Galaxy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=', Scalo 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Bochanski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Ferguson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2017) or in the discussion of formation and evolu- ★ E-mail: pcruz@cab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='inta-csic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='es tion processes involving stellar objects at the end of the Herzprung- Russell diagram (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=', Jeffries et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Stamatellos & Whitworth 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Luhman 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' More recently, M-dwarf stars have been de- fined as prime targets of exoplanet surveys as possible hosts of Earth-like planets and in the search for life elsewhere in the Galaxy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=', Bonfils et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Reiners et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Wunderlich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' M dwarfs emit the bulk of their energy in the near-infrared range of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Therefore, large area photometric surveys operating at these wavelengths, such as for instance the Two Micron All Sky Survey (2MASS, Skrutskie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2006) and the UKIRT Infrared Deep Sky Survey (UKIDSS, Lawrence et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2007), have been valuable resources for M dwarf identification and character- isation, in combination with broadband photometry in the visible range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' These data typically come from Pan-STARRS DR1 (Kaiser © 2022 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='13491v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='SR] 31 Jan 2023 2 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Cruz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Chambers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2016), SDSS (York et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2000), and Gaia (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2016) surveys, which have been ex- tensively used for this purpose (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=', Cook et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Lodieu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Bentley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Several works have been devoted to determine the photometric colours observed in the M dwarf regime (West et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Cifuentes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' All the efforts put on the identification of M dwarfs have generally avoided the Galactic bulge, with plenty of stars, gas and dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' It is in this direction where the public ESO VISTA Variables in the Via Lactea (VVV) survey operates (Minniti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2010, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Using colour-cuts from VVV b201 field, a field near the bottom-right region of the Galactic bulge, Rojas-Ayala et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' (2014) identified over 23 300 M-dwarf candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' These authors also estimated photometric distances, since Gaia parallaxes were not available yet, and identified among their sample possible giant contaminants based on a reduced proper motion diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The Gaia mission has delivered a superb collection of data, which include astrometric measurements for almost two billion stars, a smaller set but yet an impressive amount of light curves, low- resolution spectra and radial velocity measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Nevertheless, the great majority of them are not available for targets with 𝐺-band magnitude fainter than ∼171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Therefore, ground-based photometric surveys still have an important role in the Gaia era, especially for faint objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The present work aims to identify and characterise M-dwarf stars towards the bulge of the Galaxy using photometric data only and with the help of Virtual Observatory tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' In this work, we explored the VVV b294 field located in the inner bulge region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The description of the used data is presented in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The adopted sample selection methods and the list of M-dwarf candidates are presented in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The stellar character- isation based on the stellar spectral energy distribution is detailed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' A comparison with M stars from Gaia is performed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' As a secondary outcome, the search for periodicity in the VVV light curves is described in 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Finally, our conclusions are presented in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2 VVV DATA The VVV Survey (Minniti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Saito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2012) is a large ESO public near-IR surveys that mapped the bulge and inner disk of the Milky Way in the near infrared since 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The observa- tions were performed using the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='1m-telescope at the Cerro Paranal Observatory (Chile) and the data reduction was carried out at the Cambridge Astronomical Survey Unit (Irwin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The observations cover the near-infrared 𝑍𝑌𝐽𝐻𝐾𝑠 passbands, up to four magnitudes deeper than 2MASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' In addition, a variability campaign was done from 2010 to 2016, where 𝐾𝑆-band light curves were generated with up to 300 epochs (for more details, see Saito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Botan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2021, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' In this work we have focused on the b294 field, which spans among the following coordinates: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='1◦ < 𝑙 < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='7◦ and −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='8◦ < 𝑏 < −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='5◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' For our analysis, we use the photometric catalogue for this VVV region by Alonso-García et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' (2018), extracted using point spread function (PSF)-fitting techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' All the details on image reduction, PSF photometry, and quality flags are described in Alonso-García et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The VVV b294 field cointains a total of 4 592 101 detected point-source objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 1 For details, see Gaia documentation available at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='int/web/gaia/dr3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Figure 1 shows a skymap with the location of the VVV b294 field (red filled square), to illustrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' According to the extinction map by Surot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' (2020), this region presents a complex extinction structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' However, we do not expect nearby M dwarfs to be strongly affected, as most of the extinction is in the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' According to the 3D maps by Schultheis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' (2014), the expected extinction is minimal for distances of less than 2 kpc considering the line of sight of the b294 field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' More recently, the 3D extinction maps from Lallement et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' (2022) also show that for this region we have some minor extinction at about 200 pc, however, it is mostly clear until about 1 kpc, where denser clouds appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' In any case, extinction corrections are still considered in our analysis, as described later on section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 3 SAMPLE SELECTION 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='1 Selecting candidates using parallax The objects from the b294 field – with ∼4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='6 million objects, as mentioned in the previous section – were cross-matched using TOP- CAT2 (Taylor 2005, 2011) with the Gaia Data Release 3 (Gaia DR3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2016, 2022, epoch 2016), adopting a search radius of 1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Although high proper motion objects may lie outside the search region with such a small radius, it was chosen deliber- ately small to minimise the number of false counterparts, since we are dealing with a crowded field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' From this search, we kept only around ∼ 37% of the initial set (1 684 116 objects).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Only those objects with a good astrometric solution, adopted as RUWE < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='43, were selected, reducing the sample to 727 342 objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' We also kept only those that presented good parallax mea- surements (𝜋 > 0 and 𝜋err/𝜋 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 42 198 objects).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Keeping the relative errors below 20% makes the inverse of the parallax a reli- able distance estimator (Luri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' We also excluded those objects that showed Gaia magnitude errors in 𝐺 and 𝐺RP bands greater than 10%, resulting in a sample with 38 483 objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The final applied criterion was to keep only those stars within a distance of 500 pc, in order to avoid high extinction levels expected for distant objects in the Galactic plane and, thus, effective temper- ature - extinction degenerancies in the Spectral Energy Distribution (SED) fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The obtained sample contains 2 045 M dwarf candi- dates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' To identify the M-dwarf stars within this selection, we per- formed a SED fitting to obtain their effective temperatures (𝑇eff).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' We used VOSA4 (Virtual Observatory SED Analyzer, Bayo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2008) to obtain the stellar parameters based on broad-band pho- tometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The details of the fitting is further described in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Those stars with estimated 𝑇eff < 4000 K, and that presented a good fit solution (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 4), were kept as our first subset of candidates (hereafter, sample A) which contains 1 338 M-dwarf stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Infor- mation on these objects can be found in the SVO archive of VVV M-dwarfs (see appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2 TOPCAT is an interactive Tool for OPerations on Catalogues And Tables, available at http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='bris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='uk/~mbt/topcat/ 3 The Gaia Renormalised Unit Weight Error (RUWE) helps to identify non-single sources or objects with problematic astrometric solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' It is expected to be ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='0 for single stars (Arenou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Lindegren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2018, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' We selected objects with RUWE < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='4 as a compromise between sample completeness and minimum binary contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 4 http://svo2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='cab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='inta-csic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='es/theory/vosa/ MNRAS 000, 1–10 (2022) M dwarfs in VVV b294 field 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Location of the VVV b294 field (red filled square) in the sky in Galactic coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' A Planck map in aitoff projection is displayed in the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='2 Selecting candidates using proper motion and colour-cuts As an alternative way of selecting candidates, we applied several different filtering criteria to derive a sample of M-dwarf candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Firstly, we kept only those objects with photometric data in all VISTA filters and with magnitude errors of less than 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' This resulted in a subset of around 36% of the initial sample (1 641 542 objects).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Rojas-Ayala et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' (2014) presented the expected colours for M dwarfs, based on VISTA filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' We applied their colour-cuts to our sample considering only the lower limits – given for an M0 dwarf star – as follows, which allowed us to reach later-type objects: 𝑌 − 𝐽 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='336;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 𝑌 − 𝐻 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='952;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 𝑌 − 𝐾s > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='100;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 𝐽 − 𝐻 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='432;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 𝐽 − 𝐾s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='642;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 𝐻 − 𝐾s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='045.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' At this point, we identified 804 452 objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='1 Giant contaminants We performed a cross-match with Gaia DR3 using TOPCAT, within a search radius of 1′′, to obtain the measured proper motions (here- after, PM), 𝜇, for our objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Here, we kept only those with PM errors on both directions, PMRA and PMDec, of less than 20% (a total of 256 890 objects).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Then, we calculated the reduced proper motion in VISTA J band, 𝐻J, using the definition from the seminal paper by Jones (1972)5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' To discriminate between giant and dwarf 5 Jones (1972) defined the reduced proper motion as 𝐻 = 𝑚 + 5 · log 𝜇 + 5, where 𝑚 is the apparent magnitude in a given photometric band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' star we adopted the following criterion described in Rojas-Ayala et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' (2014): 𝐻𝑑 J > 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='5 · (𝐽 − 𝐾s) − 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='7, (1) where 𝐻𝑑 J indicates the value of the reduced proper motion expected for a dwarf star as a function of (𝐽 − 𝐾s), colour that is calculated using VISTA magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Almost half of the objects in the sample (130 181 objects) satisfied Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 1, which are those expected to be dwarf stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' To minimise the amount of remaining giant contaminants, we applied another criterion based on the stellar proper motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' For that, we estimated the typical PM of M dwarfs using the SIMBAD Astronomical Database6 (Wenger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2000), starting by the dis- tribution of giant stars – over 4 200 objects found, with a spectral type later than M0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' We searched for these objects proper motions by cross-matching them with Gaia DR3, keeping only those with good PM measurements (PMRA and PMDec errors of less than 20%), resulting in 4 043.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Analysing only those M giants (MIII) labelled as stars (SIMBAD OTYPE S = ’Star’) – 2 350 of them –, we found an average PM ∼ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='3 mas/yr, where ∼ 64% of the giant M-dwarf sam- ple presented PM smaller than the average (∼ 60% if we consider 𝜇 < 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='0 mas/yr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Similarly, we searched for M dwarfs (MV) using SIMBAD, getting a sample of 20 0007 objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Applying the same PM con- dition and selecting only the objects labelled as low-mass stars (12 646;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' SIMBAD OTYPE S = ’low-mass’), we derived an av- erage 𝜇 ∼ 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='1 mas/yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Adopting the PM average for MIII of ∼ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='3 mas/yr, we found that ∼ 76% of the tested dwarf sample has higher PM values (∼ 81% if we consider 𝜇 > 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='0 mas/yr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 6 http://simbad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='u-strasbg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='fr/simbad/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 7 This is in fact a subsample of M-dwarf stars from SIMBAD, since the tool limits the search to 20 000 results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' MNRAS 000, 1–10 (2022) ANCKER2LE +60 +45 156 60 754 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Cruz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Hence, to maximise completeness of dwarfs and minimise contam- ination from giants, we set a cut in PM at 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='0 mas/yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' To assess the performance of our whole approach for elimi- nating giants, we applied the related criteria – good proper motion values (𝑒𝑟𝑟𝜇/𝜇 < 20%), cut in 𝐻J from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 1, and 𝜇 ∼ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='0 mas/yr – to the MIII sample from SIMBAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' From 4 252 giants (the com- plete MIII set), only 15 giants survived the filtering, which repre- sents 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='35% of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' If considering only those giants labeled as stars (2 350 objects), the remaining objects represent only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='64%, meaning that these criteria were enough to eliminate 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='36% of the giants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' To verify the completeness of dwarf stars after applying these criteria, we repeated the giant filtering to the sample of MV from SIMBAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' We found that approximately half of the objects survived all criteria, which eliminated 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='6% of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Since we are prioritizing purity over completeness, the giant filtering is interpreted as a fair compromise between a good sample of dwarfs and minimum giant contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Therefore, we kept only those objects that have their total proper motion higher than 10 mas/yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' This way, we identified 24 787 candidate M-dwarf stars (or later) in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='2 Photometric distances From the list of 24 787 candidates, we kept those objects with good 𝐺 mag photometry (with relative error of less than 10%) and with good astrometric solutions (RUWE < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='4), resulting in a subset of 16 652 objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' We used the Gaia and VISTA broad-band photome- try to obtain the absolute 𝐺 magnitude, 𝑀G, as function of (𝐺 − 𝐽) colour for our list of identified M dwarfs, considering the empirical relation described in Cifuentes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' (2020): 𝑀G = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='24−13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='04·(𝐺 −𝐽)+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='64·(𝐺 −𝐽)2−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='622·(𝐺 −𝐽)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' (2) This relation was derived using 2MASS J-band photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' How- ever, González-Fernández et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' (2018) showed that the colour off- sets in VISTA are modest and differences in magnitudes between 2MASS and VISTA are expected to be small, of around 5 mmag in the J band8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' It is also worth mentioning that the empirical relation (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2) is valid for stars with certain values of (𝐺 − 𝐽) colour, be- tween 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='0 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Applying this as condition, our sample is reduced to 16 503 objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The photometric distances were then computed from the esti- mated absolute G magnitudes and, as a final condition, only those stars with estimated distances of less than 500 pc were selected – 6 846 M-dwarf candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' After performing a SED fitting with VOSA (for details, see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 4), we kept only those which had pre- sented a good fit solution and with 𝑇eff< 4000 K as we did in sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' This second subset (hereafter, sample B) contains 6 747 stars and it is also shown in the virtual observatory compliant archive described in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='3 The final sample We have compared the results obtained from the two methods pre- viously described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Cross-matching both samples, we verified that only 160 objects are common to both A and B samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' This is 8 For an A0 V star, the VISTA-2MASS colour in J band is (𝐽VISTA − 𝐽2MASS) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='005 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='015 (González-Fernández et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2018, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Distribution of effective temperatures obtained from the SED fitting with VOSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The used grid of models has a step in temperature of 100 K, which was adopted here as the bin size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Distribution of obtained distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The distances of M-dwarf candidates from sample A were obtained from Gaia parallaxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' For sample B, photometric distances were adopted (see sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='2 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' due to the different filtering conditions applied in each method, as explained in detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' By applying to sample A the same conditions considered for the second method – which are colour-cuts for a M0 dwarf, photometric data in all VISTA/VVV bands with small error (less than 5%), cut in 𝐻J (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 1), 𝜇 > 10 mas/yr, and photometric distances of less than 500 pc (see details in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='2) – we recovered the same 160 objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Similarly, by applying to sample B the same criteria used in the first method – which are good parallax measurements, photometric data in Gaia DR3 𝐺 and 𝐺RP bands with small error, and distances from parallax of less than 500 pc (see details in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='1) –, we found 160 objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Therefore, our final sample contains 7 925 (1 338 + 6 747 160) M-dwarf stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Figures 2 and 3 show the distribution of the estimated temperatures and distances of these objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The identified M dwarfs are placed at the colour-magnitude diagram (CMD), as illustrated in figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Grey diamonds represent a sample of Gaia nearby stars from Torres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' (2022), after applying the quality metric described by Riello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' (2021)9, illustrating the main sequence locus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The colour (𝐽 − 𝐻) for Gaia stars was 9 For details, see Table 2, and Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 6 and 18 from Riello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' MNRAS 000, 1–10 (2022) 2000 1500 1000 500 0 4000 3800 3600 3400 3200 3000 2800 T eff [K]800 700 600 500 400 300 200 100 0 100 200 300 400 500 Distance[pc]M dwarfs in VVV b294 field 5 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Colour-magnitude diagram showing the M dwarf sample from VVV b294 field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Gaia nearby stars are represented by grey diamonds and red crosses show the M stars in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Broad-band photometry used for the SED fitting with VOSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The information was compiled from the SVO Filter Profile Service (Rodrigo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Rodrigo & Solano 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Band 𝜆eff 𝑊eff survey name (Å) (Å) OmegaCAM u 3607.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='68 482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='54 VPHAS+ DR2 OmegaCAM g 4679.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='46 1203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='25 VPHAS+ DR2 Gaia G 5822.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='39 4052.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='97 Gaia DR3 OmegaCAM H𝛼 6590.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='81 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='76 VPHAS+ DR2 OmegaCAM i 7508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='50 1463.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='68 VPHAS+ DR2 VISTA Z 8789.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='53 889.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='46 VVV OmegaCAM z 8884.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='40 864.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='48 VPHAS+ DR2 VISTA Y 10196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='43 870.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='63 VVV VISTA J 12481.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='00 1542.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='53 VVV VISTA H 16348.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='19 2674.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='02 VVV VISTA Ks 21435.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='46 2793.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='85 VVV IRAC I1 35075.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='11 6836.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='16 GLIMPSE IRAC I2 44365.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='78 8649.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='93 GLIMPSE IRAC I3 56281.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='02 12561.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='17 GLIMPSE IRAC I4 75891.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='59 25288.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='50 GLIMPSE obtained from a cross-correlation with 2MASS, considering a 1′′ search radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' All objects from our sample of M stars are presented as red crosses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 4 STELLAR PROPERTIES WITH VOSA As briefly mentioned in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 3, we used VOSA (Bayo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2008) to obtain effective temperatures, luminosities and radii for all M- dwarf candidates in our selected sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' VOSA is a tool developed by the Spanish Virtual Observatory10 designed to build the SEDs of thousands of objects at a time from a large number of photometric catalogues, ranging from the ultraviolet to the infrared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' VOSA com- pares catalogue photometry with different collections of theoretical models and determines which model best reproduces the observed data, following different statistical approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Physical parameters 10 http://svo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='cab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='inta-csic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='es are then estimated for each object from the model that best fits the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' To construct the SED, we used the VISTA 𝑍𝑌𝐽𝐻𝐾𝑠 photome- try obtained from the VVV survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' We also searched for additional broad-band photometry from Gaia DR3 (only in the 𝐺 band), from the VST Photometric H𝛼 Survey of the Southern Galactic Plane and Bulge (VPHAS+ DR2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Drew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2014), and from the Galac- tic Legacy Infrared Mid-Plane Survey Extraordinaire (GLIMPSE Source Catalog I + II + 3D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Spitzer Science 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Information on the filters is shown in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' It is worth mentioning that only a small portion of the sample (172 objects) had VPHAS+ DR2 magnitudes available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' For these objects, the colour (𝑟 − 𝐻𝛼) from VPHAS+ DR2 was used to search for H𝛼 emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' We found that all the objects with 𝑟 and H𝛼 photometry available have (𝑟 − 𝐻𝛼) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='9, which is expected for non-emitting M sources (Drew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The BT-Settl CIFIST models – the BT-Settl theoretical spectra by Allard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' (2011) computed with a cloud model and using the Caffau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' (2011) solar abundances – were adopted for the SED fitting, covering effective temperatures from 1200 to 7000 K and assuming solar metallicity ([Fe/H] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Since we expect our candidates to be dwarf stars (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 3 for details), the surface gravities were allowed to vary only from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='0 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Finally, we defined a range of possible values for the extinction (𝐴v), going from 𝐴v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='5, as expected according to extinction models for the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Extinction is left as a free parameter to be fitted together with the stellar parameters in the SED fitting process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' To calculate the extinction in each filter, VOSA uses the extinction law by Fitzpatrick (1999), with the improvement in the infrared region by Indebetouw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' (2005)11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' After the fit is performed, we opted to refine the results due to excess present in the infrared data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' By comparing the photometric data to the best-fit model, VOSA is able to identify which data points appear high above the model and (re)define the starting point of the infrared excess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' We then refitted those SEDs that VOSA identified to have an excess, where all the points which were found to be affected were not considered in the final fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Since the b294 tile is placed in a crowded field, some of the detected IR excess could just be due to a wrong counterpart assign- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' This is illustrated in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' On the left, the example of an object with a SED showing a strong IR excess (top left panel) but later verified to come from a different source (bottom left panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' For comparison, a more subtle excess in the SED of a different object is presented on the right (top right panel), later confirmed to come from the same emitting source (bottom right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' All ob- jects for which VOSA has identified an infrared excess are properly flagged in the archive, as described in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' These detected IR excess need to be further confirmed, as explained above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' We obtained a good fit for all objects in our sample, according to visual goodness-of-fit value (𝑉gfb) estimated by VOSA, which is the modified reduced 𝜒2 calculated by forcing the error in the observed flux to be larger than 10%12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The results of the SED fitting are also included in the archive described in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 11 For more details on the interstellar extinction, see VOSA’s help sec- tion at http://svo2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='cab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='inta-csic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='es/theory/vosa/help/star/ extinctions/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 12 The visual goodness-of-fit value, 𝑉 gfb, smaller than 12–15 is often perceived as a good fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' For more details, see VOSA’s help page http: //svo2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='cab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='inta-csic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='es/theory/vosa/help/star/fit/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' MNRAS 000, 1–10 (2022) 0 5 10 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='0 J-H6 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Cruz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Example of two different cases of IR excess present from GLIMPSE photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The top panels show the results from SED fitting using VOSA and the identified starting point of IR excess (vertical dashed line), where blue squares represent the synthetic photometric calculated from the best-fit model, the red filled circles are the observed photometry, and black circles indicate the detected excess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The thin grey line shows the stellar SED before correcting from extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The lower panels (generated using Aladin) show the position of the same stars (marked as a pink cross at the center of each image) over a 𝐽-band image from VISTA/VVV survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The objects identified in the VISTA/VVV catalogue are presented as small blue circles and those in the GLIMPSE catalogue are presented as red squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The lower left panel illustrates an example of a star for which the GLIPMSE data came from a nearby object and, therefore, the IR excess in the SED (left upper panel) is not real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Different from the second example (right lower panel) where the detected excess is coming from the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 5 COMPARISON TO GAIA M DWARFS The sky distribution of the 7 925 M-dwarf stars identified in VVV tile b294 is presented in figure 6, where the grey area represents the observed field of view and the black solid line shows the Multi- Object Coverage (MOC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The objects from samples A and B are represented by yellow and red filled circles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' To assess the importance of our methodology and the impact of our M stars catalogue to the identified cool stars in the studied region – towards the Galactic bulge (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 1) – we exploited the available data from Gaia DR3 to search for known M dwarfs in the VVV tile b294 field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' From the approximately 2 million sources of the Gaia DR3 present in tile b294 field (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 6, grey area), we selected those with RUWE of less than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='4 – thesame limit appliedfor VVV objects and described in sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 3 – to ensure a good astrometric solution for a point-like source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' As a matter of consistency, we selected the stars with a relative parallax error of less than 20%, keeping those within a distance of 500 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Aiming at identifying M dwarfs, we kept those stars with 𝑇eff of less than 4000 K and log 𝑔 ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='0 – astrophysical parameters available in Gaia DR3 catalogue, which were derived from BP/RP spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The resulting sample contained only 208 stars characterised in Gaia DR3 as M-dwarf stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' All these Gaia DR3 M stars are shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 6 as blue crosses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Therefore, we have increased considerably the number of M dwarfs identified in the field, from a few hundreds to thousands of stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Figure 7 shows the comparison between the effective temper- atures derived from SED fitting with VOSA and the ones given in Gaia DR3 catalogue for the 162 objects in common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Considering the estimated uncertainties, there is in general a good agreement between derived values, with a mean difference of 181 K and a standard deviation of 136 K, approximately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Among the M stars identified in Gaia DR3, 39 of them were not included in the cross- match with the original VVV tile b294 catalogue, considering 1′′ radius, and 7 were lost in the applied criteria, such as colour-cuts and proper motion limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Our sample has increased significantly the number of known M dwarfs in the studied region – a crowded field towards the bulge of the Galaxy – emphasising the importance of ground-based pho- tometric surveys in the near-infrared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' MNRAS 000, 1–10 (2022) 10-16 10 17 10-18 10-19 104 105 入 (A)10-16 10-17 10-18 104 入 (A)VISTA VY DR4 J 回 白 回 回 D D 0 回 回 巴 5 回 回 回 回 回 回 回 D 8 0 回 13 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 0 0 0 0 回 回 回 回 回 回 0 D 回 D 回 回 回 回 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 回 回 回 N 0 P 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=" 1161'x 59." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='55VISTA VV DR4 J 回 D 回 回 回 回 回 回 回 6 回 回 回 回 回 回 回 回 回 回 回 回 回 回 回 回 回 回 回 回 回 回 回 回 回 回 回 回 回 回 回 回 回 N 回 Lo 回 15 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='1M dwarfs in VVV b294 field 7 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Sky position of the 7 925 M-dwarf stars in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The VVV tile b294 is presented as a grey area and the solid black lines represent its MOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Blue crosses show the M dwarfs identified in Gaia DR3 within 500 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Comparison of 𝑇effobtained in our analysis (VOSA) with those given in Gaia DR3, for the stars identified as M dwarfs in both sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The dashed line represents the identity function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 6 SEARCHING FOR PERIODIC SIGNALS IN VVV LIGHT CURVES As a secondary outcome of this study, it is interesting to analyse the multi-epoch observations performed in VVV tile b294 and search for periodic signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The time series were obtained with the VVV 𝐾𝑠 filter, with over 300 observations of our field of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Not only this, but due to the strategy of observation (Saito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2012), every region of the field is observed at least twice, most of the times using different chips of the 16-chip VIRCAM instrument used in the VISTA telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' For our multi-epoch analysis described below, we made use of VVV 𝐾𝑠-band light curves based on PSF photometry as described in Contreras Ramos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='1 Light curve selection We gathered light curves with more than 25 detections for 7 752 M dwarfs out of the 7 925 stars in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Some of them have been detected in more than one chip due to overlap, providing a total of 8 640 different light curves in the 𝐾𝑠/VISTA band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The time-series data covered in total 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='4 yr of observations, acquired between 12 April 2010 and 11 September 2015 (Contreras Ramos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' In order to clean spurious data present in the light curves, we carried out a two-step procedure: (i) We ran a sigma-clipping approach aiming at identifying the baseline emission of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' (ii) From the original light curve, we kept all epochs within the first and 99th centiles and discarded those brighter than the median value minus 3𝜎, computed from the set obtained in the previous exercise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' These detections could be ascribed as mismatches that could contaminate the search for periodic signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' In order to keep the sample purity, outliers were excluded from the light-curve analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' All light curves that present more than three points with mag- nitude values between the median plus 3𝜎 and the 99th percentile were considered suitable for the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' A drop in brightness can be ascribed to either a transiting object or to starspots present in the stellar surface of active M dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' We obtained 8 219 light curves associated to 7 153 M dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' An example of a raw light curve is shown on top panel in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Additionally, we applied a photometric quality filter and dis- carded all the light curves with mean magnitude errors larger than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='5 times the Median Absolute Deviation (MAD) obtained from the baseline emission of each star in step (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' After this, 4 952 stars with 5 817 light curves remained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' They were normalised to the median and combined in order to obtain a unique light curve per star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='2 Light curve analysis Although the light curves span over 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='4 yr, there are three main observing blocks of nearly 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='5 months each (200 d), going from May to November 2012, from May to December 2014, and from March to October 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' With the goal of providing a reliable set of M-dwarf candidates with trustful periodic variations, we searched for periodic signals in each of the three main blocks of observations rather than in the whole data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Figure 8 shows an example of a whole light curve where the three observing blocks used for deriving the periods are clearly differentiated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Given the unevenly spaced observations, we obtained the pe- riodograms from each block in the light curves using two comple- mentary methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Firstly, we applied the Generalized Lomb-Scargle algorithm (GLS, Lomb 1976;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Scargle 1982), which fits a sinu- soidal model to the data at each frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' For that, we employed the Python astropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='timeseries package (Astropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2013, 2018) and obtained the best periods (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=', the periods with the highest peak in the periodogram).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' We tested them through the false alarm probability (FAP, Scargle 1982) computed with the Baluev approximation (Baluev 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' We derived three periods – one period per block – but only periods with FAP under 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='1 were taken into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Secondly, we searched for periodic sig- nals applying the Phase Dispersion Minimization method (PDM, Stellingwerf 1978), between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='1 and 100 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Unlike GLS, PDM is unbiased towards the shape of the light curve and is able to find also non-sinusoidal variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' We obtained 224 periods in agreement within 20% from the MNRAS 000, 1–10 (2022) 4000 K eff,VOSA 3500 3000 3000 3500 4000 T eff,Gaia [K]-26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='0 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='2 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='4 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='6 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='8 [deg] 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='0 Dec 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='2 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='4 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='6 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='8 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='0 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='2 270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='5 271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='0 271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='5 272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='0 272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='5 RA [deg]8 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Cruz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Top panel: Example light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Baseline emission data points, detections related to variability and rejected outliers, which could also be related to flares, are displayed in grey filled circles, blue open circles and red crosses, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Grey solid, short dashed and long dashed lines represent the median, the median+3𝜎 and median-3𝜎 limits, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Black dashed-dotted lines represent the first and 99th quartiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Bottom panel: Light curve of the same object after removing outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Colour and symbol code as in the figure on top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' LSG and PDM approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' After removing near 1 d periodic signals (from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='9 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='1 d), 82 stars remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Additionally, we use the reduced chi-square as a variability indicator as in Botan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' (2021) and define two subsets of M dwarfs with periodic signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' One is composed by 27 M dwarfs with 𝜒2 > 2, whose variability would likely be related to the presence of a stellar companion, indi- cating possible binary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The obtained periods range between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='14 and 34 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Among them, 2 objects were recently identified as variables in a recent work by Molnar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' (2022), where they were flagged as eclipsing binary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The other subset is composed by 20 M dwarfs that present 𝜒2 ∼ 1 (from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='7 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='3), for which the periodic signal could be ascribed to a planet-like transiting ob- ject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' However, the found periodicities could also be related to stellar intrinsic variability, due to starspots for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Therefore, the pe- riods need to be confirmed (or discarded) with dedicated follow-up observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The periods distribution is shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Each of the objects for which periodicity has been calculated is properly flagged in the archive, as described in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Period distribution in logarithmic scale for the sample of 82 M dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 7 CONCLUSIONS We searched for M-dwarf stars near the Galactic bulge in the b294 field from the VISTA Variables in the Vía Láctea survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' We adopted two different methodologies to identify M dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The first method was performed using parallaxes, where we selected objects with good astrometric solution, photometry – relative G and RP mag- nitude errors below 10% – and parallax measurements – relative error of less than 20% – from Gaia DR3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The second method was based on colour-cuts and proper motions, where we kept those ob- jects with good VISTA photometry in all 𝑌𝑍𝐽𝐻𝐾𝑠 bands, VISTA colours among the colour-cuts defined by Rojas-Ayala et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' (2014) for M stars, good proper motion – relative error of less than 20% – and good astrometric solution also from Gaia DR3, and those with a J magnitude reduced proper motion expected for dwarf stars (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='2 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' We then estimated absolute magnitudes from the empirical relation based on colours by Cifuentes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' To avoid the interstellar extinction expected for the field, we selected only the M-dwarf candidates within 500 pc, where dis- tances where estimated from parallax (sample A) and photometric distances based on absolute magnitudes (sample B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' We then char- acterised the remaining candidates by performing SED fittings using VOSA, where we kept all objects with 𝑇eff < 4 000 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Our final list of M-dwarf candidates from the VVV b294 field has 7 925 stars, with temperatures ranging from 2 800 to 3 900 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' To assess the importance and impact of the identified M stars towards the Galactic bulge, we compared our sample to all M dwarfs characterised from BP/RP spectra available in Gaia DR3 catalogue in the VVV tile b294 field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' From nearly 2 million sources, there are 208 stars with 𝑇eff and log 𝑔 compatible with M-dwarf stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Our sample of 7 925 sources has significantly increased the number of known M dwarfs within 500 pc in the studied region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' As a secondary outcome of this study, we also searched for periodic signals in VVV light curves, with at least 25 and up to 327 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' We removed outliers from the light curves and looked for periodicities in the three main blocks of observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' We obtained periods for 82 M dwarfs by applying two methods: the Lomb- Scargle and Phase Dispersion Minimization, independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' These periods range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='14 to 34 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' We defined two subsamples ac- cording to the reduced chi-square (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='2) presenting large and small variability (27 and 20 stars, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Additional follow-up observations and further analysis would be required for confirming the nature of the periodic variability of these objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Even with the amazing collection of data delivered by Gaia MNRAS 000, 1–10 (2022) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='7 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='8 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='9 55500 56000 56500 57000 MJD [d]13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='70 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='75 [mag 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='80 55500 56000 56500 57000 MJD [d]12 10 8 6 4 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='5 1 2 5 10 20 P [d]M dwarfs in VVV b294 field 9 DR3, radial velocities and spectra are not available for all observed objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The methodology described in this work probed to be very efficient on identifying and characterising M-dwarf stars in the VVV b294 field, emphasising the importance of ground-based photomet- ric surveys in the near-infrared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Therefore, it can be extended to other VVV fields – and to those from the VVVX survey (Minniti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 2018) – in order to increase the population of known low-mass objects in the direction of the Galactic bulge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We would like to thank Dr F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Jiménez-Esteban for the fruitful discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' acknowledges financial support from the Gov- ernment of Comunidad Autónoma de Madrid (Spain) via post- doctoral grant ‘Atracción de Talento Investigador’ 2019-T2/TIC- 14760.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' acknowledges financial support from the ESCAPE project supported by the European Commission Framework Pro- gramme Horizon 2020 Research and Innovation action under grant agreement n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 824064.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' This research has made use of the Spanish Virtual Observatory (https://svo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='cab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='inta-csic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='es) project funded by the Spanish Ministry of Science and Innovation/State Agency of Research MCIN/AEI/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='13039/501100011033 through grant PID2020-112949GB-I00 and MDM-2017-0737 at Centro de Astrobiología (CSIC-INTA), Unidad de Excelencia María de Maeztu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' also thanks the support by the ANID BASAL projects ACE210002 and FB210003, and Fondecyt Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 1220724.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' acknowledges support from Fondecyt Regular 1201490, and ANID – Millennium Science Initiative Program – ICN12_009 awarded to the Millennium Institute of Astrophysics MAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' acknowledges support from CNPq/Brazil through project 305902/2019-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' We gratefully acknowledge the use of data from the ESO Public Survey program IDs 179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='B-2002 and 198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='B- 2004 taken with the VISTA telescope and data products from the Cambridge Astronomical Survey Unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' We would also like to thank R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Contreras Ramos (private communication) for the VVV light curves used in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' This publication makes use of VOSA, developed under the Spanish Virtual Observatory project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' VOSA has been partially up- dated by using funding from the European Union’s Horizon 2020 Research and Innovation Programme, under Grant Agreement nº 776403 (EXOPLANETS-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' This research has made use of the SVO Filter Profile Service (http://svo2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='cab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='inta-csic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='es/ theory/fps/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' This research has made use of "Aladin sky atlas" developed at CDS, Strasbourg Observatory, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' This research has made use of the SIMBAD database, operated at CDS, Stras- bourg, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' DATA AVAILABILITY: VIRTUAL OBSERVATORY COMPLIANT, ONLINE CATALOGUE In order to help the astronomical community on using our cata- logue of VVV M dwarfs, we developed an archive system that can be accessed from a webpage13 or through a Virtual Observatory ConeSearch14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The content of the catalogue is presented in the Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 13 http://svocats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='cab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='inta-csic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='es/mdwarfs_vvv/ 14 A ConeSearch example can be seen at http://svocats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='cab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' inta-csic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='es/mdwarfs_vvv/cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='php?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='RA=271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='877&DEC=-28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='079& SR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='1&VERB=2 The archive system implements a very simple search interface that allows queries by coordinates and radius as well as by other parameters of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The user can also select the maximum num- ber of sources (with values from 10 to unlimited) and the number of columns to return (minimum, default, or maximum verbosity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The result of the query is a HTML table with all the sources found in the archive fulfilling the search criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' The result can also be downloaded as a VOTable or a CSV file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Detailed information on the output fields can be obtained placing the mouse over the ques- tion mark located close to the name of the column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=', 2000, AJ, 120, 1579 APPENDIX A: CATALOGUE DESCRIPTION The content of the catalogue of M-dwarf candidates from the VVV b294 field is presented in Table A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' This catalogue can be accessed from the dedicated webpage http://svocats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='cab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='inta-csic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' es/mdwarfs_vvv/ or through a Virtual Observatory ConeSearch (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' http://svocats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='cab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='inta-csic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='es/mdwarfs_vvv/cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' php?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='RA=271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='877&DEC=-28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='079&SR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content='1&VERB=2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' MNRAS 000, 1–10 (2022) M dwarfs in VVV b294 field 11 Table A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Description of the parameters contained in the VVV M-dwarf catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Parameter Units Description Gaia_ID_DR3 Gaia DR3 source identifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' RAJ2000 deg Celestial Right Ascension (J2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' DEJ2000 deg Celestial Declination (J2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Xmag mag Calibrated magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' X stands for 𝑍, 𝑌 , 𝐽, 𝐻 and 𝐾𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' eXmag mag Calibrated magnitude error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' X stands for 𝑍, 𝑌 , 𝐽, 𝐻 and 𝐾𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' d pc Distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Ref_d Reference for the distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' "Gaia" refers to parallactic distances and "This work" indicates distances are specto-photometric derived in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' 𝑇eff K Effective temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Lbol L⊙ Bolometric luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' eLbol L⊙ Error in the bolometric luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' R R⊙ Stellar radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Calculated using Lbol = 4𝜋𝑅2 𝜎 𝑇eff4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' eR R⊙ Error in the stellar radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Av Visual extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Method Identification method of candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' "A" stands for parallax, and "B" stands for proper motion and colour-cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Flag_IR Flag for stars with IR excess detected by VOSA yet to be confirmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Flag_lc Flag for dwarfs with processed light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Flag_P Flag for dwarfs with periodic signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' Flag_chi Flag for M dwarfs with 𝜒2 ∼ 1 (1) and 𝜒2 > 2 (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} +page_content=' MNRAS 000, 1–10 (2022)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FRT4oBgHgl3EQfITcf/content/2301.13491v1.pdf'} diff --git a/OdE3T4oBgHgl3EQfCAmE/content/tmp_files/2301.04272v1.pdf.txt b/OdE3T4oBgHgl3EQfCAmE/content/tmp_files/2301.04272v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..79ac26954545c2c359edf0da4ea31833eaf9fd8f --- /dev/null +++ b/OdE3T4oBgHgl3EQfCAmE/content/tmp_files/2301.04272v1.pdf.txt @@ -0,0 +1,1805 @@ +Data Distillation: A Survey +Noveen Sachdeva +nosachde@ucsd.edu +Computer Science & Engineering +University of California, San Diego +Julian McAuley +jmcauley@ucsd.edu +Computer Science & Engineering +University of California, San Diego +Abstract +The popularity of deep learning has led to the curation of a vast number of massive and +multifarious datasets. +Despite having close-to-human performance on individual tasks, +training parameter-hungry models on large datasets poses multi-faceted problems such as (a) +high model-training time; (b) slow research iteration; and (c) poor eco-sustainability. As an +alternative, data distillation approaches aim to synthesize terse data summaries, which can +serve as effective drop-in replacements of the original dataset for scenarios like model training, +inference, architecture search, etc. In this survey, we present a formal framework for data +distillation, along with providing a detailed taxonomy of existing approaches. Additionally, +we cover data distillation approaches for different data modalities, namely images, graphs, +and user-item interactions (recommender systems), while also identifying current challenges +and future research directions. +1 +Introduction +(Loose) Definition 1. (Data distillation) Approaches that aim to synthesize tiny and high-fidelity data +summaries which distill the most important knowledge from a given target dataset. Such distilled summaries +are optimized to serve as effective drop-in replacements of the original dataset for efficient and accurate +data-usage applications like model training, inference, architecture search, etc. +The recent “scale-is-everything” viewpoint (Ghorbani et al., 2021; Hoffmann et al., 2022; Kaplan et al., 2020), +argues that training bigger models (i.e., consisting of a higher number of parameters) on bigger datasets, and +using larger computational resources is the sole key for advancing the frontier of artificial intelligence. On the +other hand, a well-reasoned, principled solution will arguably be more amenable to scaling, thereby leading +to faster progress (Sorscher et al., 2022). Data distillation (Definition 1), is a task rooted in the latter school +of thought. Clearly, the scale viewpoint still holds, in that if we keep increasing the amount of data (albeit +now compressed and of higher quality), we will observe an improvement in both upstream and downstream +generalization, but at a faster rate. +Motivation. A terse, high-quality data summary has use cases from a variety of standpoints. First and +foremost, it leads to a faster model-training procedure. In turn, faster model training equates to (1) compute- +cost saving and expedited research iterations, i.e., the investigative procedure of manually experimenting +different ideas; and (2) improved eco-sustainability, i.e., lowering the amount of compute time directly leads to +a lower carbon footprint from running power-hungry accelerated hardware (Gupta et al., 2022). Additionally, +a small data summary democratizes the entire pipeline, as more people can train state-of-the-art algorithms +on reasonably accessible hardware using the data summary. Finally, a high-quality data summary indirectly +also accelerates orthogonal procedures like neural architecture search (Liu et al., 2019), approximate nearest +neighbour search (Arya et al., 1998), knowledge distillation (Hinton et al., 2015), etc., where the procedure +needs to iterate over the entire dataset multiple times. +1 +arXiv:2301.04272v1 [cs.LG] 11 Jan 2023 + +Data +Distillation +Train +Train +Similar +Performance +<< 50K distilled images +50K images +Learning algorithm +[HQ Image Link] Figure 1: The premise of data distillation demonstrated using an image dataset. +Comparison with knowledge distillation & transfer learning. Despite inherently distilling some kind +of knowledge, we would like to highlight both knowledge distillation and transfer learning are orthogonal +procedures to data distillation, which can potentially work together to perform both tasks more efficiently. +Knowledge distillation (Hinton et al., 2015) entails distilling the knowledge from a trained teacher network into +a smaller student network efficiently. On the other hand, transfer learning (Pratt, 1992) is the procedure of +transferring knowledge across similar tasks, e.g., from image classification to image segmentation. Orthogonally, +data distillation aims to distill the knowledge from a given dataset into a terse data summary. Such data +summaries can be used in conjunction with knowledge distillation or transfer learning procedures for both +(1) faster learning of the teacher models; and (2) faster knowledge transfer to the student models. The +same comparison holds true for model compression techniques (LeCun et al., 1989) as well, where similar to +knowledge distillation, the goal is to reduce model storage size, rather than reducing the training time or +sample complexity. +In this survey, we intend to provide a succinct overview of various data distillation frameworks across different +data modalities. We start by presenting a formal data distillation framework in Section 2, along with a +detailed empirical comparison of existing image distillation techniques. Subsequently, in Section 3, we discuss +existing data distillation frameworks for synthesizing data of different modalities, as well as outlining the +associated challenges. In Section 4, we discuss alternative applications of synthesizing a high-fidelity data +summary rather than simply accelerating model training along with pointers to existing work. Finally, in +Section 5, we conclude by presenting common pitfalls in existing data distillation techniques, along with +proposing interesting directions for future work. +2 +The Data Distillation Framework +Before going into the specifics of data distillation, we start by outlining useful notation. Let D ≜ {(xi, yi)}|D| +i=1 +be a given dataset which needs to be distilled, where xi ∈ X are the set of input features, and yi ∈ Y is the +desired label for xi. For classification tasks, let C be the set of unique classes in Y, and Dc ≜ {(xi, yi) | yi = +c}|D| +i=1 be the subset of D with class c. We also define the matrices X ≜ [xi]|D| +i=1 and Y ≜ [yi]|D| +i=1 for convenience. +Given a data budget n ∈ Z+, data distillation techniques aim to synthesize a high-fidelity data summary +Dsyn ≜ {(˜xi, ˜yi)}n +i=1 such that n ≪ |D|. We define Dc +syn, Xsyn, and Ysyn similarly as defined for D. Let +Φθ : X �→ Y represent a learning algorithm parameterized by θ. We also assume access to a twice-differentiable +cost function l : Y × Y �→ R, and define LD(θ) ≜ E(x,y)∼D[l(Φθ(x), y)] for convenience. Notation is also +summarized in Appendix A. +For the sake of uniformity, we refer to the data synthesized by data distillation techniques as a data summary +henceforth. Inspired by the definition of coresets (Bachem et al., 2017), we formally define an ϵ−approximate +data summary, and the data distillation task as follows: +2 + +50K Real Training Images +Dataset +Distillation +10 Synthetic Training Images +Train +Train +00000 +OOO +Similar Test PerformanceData Distillation + +Trajectory +Matching +MTT, HABA, +TESLA + +Distribution +Matching + +DM, CAFE, +IT-GAN, KFS, +GCDM + +Gradient +Matching + +DC, DSA, DCC, +IDC, GCOND, +DOSCOND + +Meta-Model +Matching + +DD, KIP, RFAD, +FREPO, LINBA, +DISTILL-CF +[HQ Image Link] Figure 2: A taxonomy of existing data distillation approaches. +Definition 2. (ϵ−approximate data summary) Given a learning algorithm Φ, let θD, θDsyn represent +the optimal set of parameters for Φ estimated on D and Dsyn, and ϵ ∈ R+; we define an ϵ−approximate data +summary as one which satisfies: +sup { | l (ΦθD(x), y) − l (ΦθDsyn(x), y) | }x∼X +y∼Y +≤ ϵ +(1) +Definition 3. (Data distillation) Given a learning algorithm Φ, let θD, θDsyn represent the optimal set of +parameters for Φ estimated on D and Dsyn; we define data distillation as optimizing the following: +arg min +Dsyn,n +� +sup { | l (ΦθD(x), y) − l (ΦθDsyn(x), y) | }x∼X +y∼Y +� +(2) +From Definition 3, we highlight three cornerstones of evaluating data distillation methods: (1) Performance: +downstream evaluation of models trained on the synthesized data summary vs. the full dataset (e.g., accuracy, +FID, nDCG, etc.); (2) Efficiency: how quickly can models reach full-data performance (or even exceed it), +i.e., the scaling of n vs. downstream task-performance; and (3) Transferability: how well can data summaries +generalize to a diverse pool of learning algorithms, in terms of downstream evaluation. +No free lunch. The universal “No Free Lunch” theorem (Wolpert & Macready, 1997) applies to data +distillation as well. For example, looking at the transferability of a data summary, it is strongly dependent on +the set of encoded inductive biases, i.e., through the choice of the learning algorithm Φ used while distilling, +as well as the objective function l(·, ·). Such biases are unavoidable for any data distillation technique, in a +sense that learning algorithms closely following the set of encoded inductive biases, will be able to generalize +better on the data summary than others. +Keeping these preliminaries in mind, we now present a formal framework for data distillation, encapsulating +existing data distillation approaches. Notably, the majority of existing techniques intrinsically solve a bilevel +optimization problem, which are tractable surrogates of Equation (2). The inner-loop typically optimizes +a representative learning algorithm on the data summary, and using the optimized learning algorithm, the +outer-loop optimizes a tractable proxy of Equation (2). +Some common assumptions that existing data distillation techniques follow are: (1) static-length data +summary, i.e., n is fixed and is treated as a tunable hyper-parameter; and (2) we have on-demand access to +the target dataset D which is also assumed to be iid. Notably, the outer-loop optimization of Dsyn happens +simply through gradient descent (GD) on the analogously defined Xsyn ∈ Rn×dim(X), which is instantiated +as free parameters. Note that the labels, Ysyn ∈ Rn×dim(Y), can be similarly optimized through GD as well +(Bohdal et al., 2020). For the sake of notational clarity, we will interchangeably use optimization of Dsyn or +(Xsyn, Ysyn) henceforth. +3 + +2.1 +Data Distillation by Meta-model Matching +Meta-model matching-based data distillation approaches fundamentally optimize for the transferability of +models trained on the data summary when generalized to the original dataset: +arg min +Dsyn +LD +� +θDsyn� +s.t. +θDsyn ≜ arg min +θ +LDsyn(θ), +(3) +where intuitively, the inner-loop trains a representative learning algorithm on the data summary until +convergence, and the outer-loop subsequently optimizes the data summary for the transferability of the +optimized learning algorithm to the original dataset. Besides common assumptions mentioned earlier, the +key simplifying assumption for this family of methods is that a perfect classifier exists and can be estimated +on D, i.e., ∃ θD s.t. l(ΦθD(x), y) = 0, ∀x ∼ X, y ∼ Y. Plugging the second assumption along with the iid +assumption of D in Equation (2) directly translates to Equation (3). Despite the assumption, Equation (3) is +highly expensive both in terms of computation time and memory, due to which, methods from this family +typically resort to making further assumptions. +Wang et al. (2018) (DD) originally proposed the task of data distillation, and used the meta-model matching +framework for optimization. DD makes the expensive optimization in Equation (3) more efficient by performing +(1) local optimization à la stochastic gradient descent (SGD) in the inner-loop, and (2) outer-loop optimization +using Truncated Back-Propagation Through Time (TBPTT), i.e., unroll a limited number of inner-loop +optimization steps while optimizing the outer-loop. Formally, the modified optimization objective for DD is +as follows: +arg min +Dsyn +E +θ0∼Pθ [LD (θT )] +s.t. +θt+1 ← θt − η · ∇θLDsyn(θt), +(4) +where Pθ is a parameter initialization distribution of choice, T accounts for the truncation in TBPTT, and η +is a tunable learning rate. Notably, TBPTT has been associated with drawbacks such as (1) computationally +expensive to unroll the inner-loop at each outer-loop update (Vicol et al., 2021); (2) bias involved with +truncated unrolling (Wu et al., 2018); and (3) poorly conditioned loss landscapes, particularly with long +unrolls (Metz et al., 2019). Consequently, the TBPTT framework was empirically shown to be ineffective for +data distillation in subsequent works (Zhao et al., 2021). However, recent work (Deng & Russakovsky, 2022) +claims that using momentum-based optimizers and longer unrolling of the inner-loop can greatly improve +performance. We delay a deeper discussion of this work to Section 2.5 for clarity. +Analogously, a separate line of work focuses on using Neural Tangent Kernel (NTK) (Jacot et al., 2018) based +algorithms to solve the inner-loop in closed form. As a brief side note, the infinite-width correspondence states +that performing Kernelized Ridge Regression (KRR) using the NTK of a given neural network, is equivalent +to training the same ∞-width neural network with L2 reconstruction loss for ∞ SGD-steps. These “∞-width” +neural networks have been shown to perform reasonably compared to their finite-width counterparts, while +also being solved in closed-form (see Lee et al. (2020) for a detailed analysis on finite vs. infinite neural +networks for image classification). KIP uses the NTK of a fully-connected neural network (Nguyen et al., +2021a), or a convolutional network (Nguyen et al., 2021b) in the inner-loop of Equation (3) for efficient data +distillation. More formally, given the NTK K : X × X �→ R of a neural network architecture, KIP optimizes +the following objective: +arg min +Xsyn,Ysyn +��Y − KXXsyn · (KXsynXsyn + λI)−1 · Ysyn +��2 , +(5) +where KAB ∈ R|A|×|B| represents the gramian matrix of two sets A and B, and whose (i, j)th element +is defined by K(Ai, Bj). Although KIP doesn’t impose any additional simplifications to the meta-model +matching framework, it has an O(|D| · n · dim(X)) time and memory complexity, limiting its scalability. +Subsequently, RFAD (Loo et al., 2022) proposes using (1) the light-weight Empirical Neural Network +Gaussian Process (NNGP) kernel (Neal, 2012) instead of the NTK; and (2) a classification loss (e.g., NLL) +instead of the L2-reconstruction loss for the outer-loop to get O(n) time complexity while also having better +performance. On a similar note, FRePO (Zhou et al., 2022b) decouples the feature extractor and a linear +classifier in Φ, and alternatively optimizes (1) the data summary along with the classifier, and (2) the feature +4 + +extractor. To be precise, let fθ : X �→ X ′ be the feature extractor, gψ : X ′ �→ Y be the linear classifier, s.t. +Φ(x) ≡ gψ(fθ(x)) ∀x ∈ X; the optimization objective for FRePO can be written as: +arg min +Xsyn,Ysyn +E +θ0∼Pθ +� T +� +t=0 +���Y − Kθt +XXsyn · (Kθt +XsynXsyn + λI)−1 · Ysyn +��� +2 +� +s.t. +θt+1 ← θt − η · +E +(x,y)∼Dsyn [∇θl(gψ(fθ(x)), y)] ; Kθ +XsynXsyn ≜ fθt(Xsyn)fθt(Xsyn)T , +(6) +where T represents the number of inner-loop update steps for the feature extractor fθ. Notably, (1) a wide +architecture for fθ is crucial for distillation quality in FRePO; and (2) despite the bilevel optimization, +FRePO is shown to be more scalable compared to KIP (Equation (5)), while also being more generalizable. +2.2 +Data Distillation by Gradient Matching +Gradient matching based data distillation, at a high level, performs one-step distance matching on (1) the +network trained on the target dataset (D) vs. (2) the same network trained on the data summary (Dsyn). +In contrast to the meta-model matching framework, such an approach circumvents the unrolling of the +inner-loop, thereby making the overall optimization much more efficient. First proposed by Zhao et al. (2021) +(DC), data summaries optimized by gradient-matching significantly outperformed heuristic data samplers, +principled coreset construction techniques, as well as TBPTT-based data distillation proposed by Wang et al. +(2018). Formally, given a learning algorithm Φ, DC solves the following optimization objective: +arg min +Dsyn +E +θ0∼Pθ +c ∼ C +� T +� +t=0 +D +� +∇θLDc(θt), ∇θLDc +syn(θt) +�� +s.t. +θt+1 ← θt − η · ∇θLDsyn(θt), +(7) +where T accounts for model similarity T-steps in the future, and D : R|θ| × R|θ| �→ R is a distance metric of +choice (typically cosine distance). In addition to assumptions imposed by the meta-model matching framework +(Section 2.1), gradient-matching assumes (1) inner-loop optimization of only T steps; (2) local smoothness: +two sets of model parameters close to each other (given a distance metric) imply model similarity; and (3) +first-order approximation of θD +t : instead of exactly computing the training trajectory of optimizing θ0 on D +(say θD +t ); perform first-order approximation on the optimization trajectory of θ0 on the much smaller Dsyn (say +θDsyn +t +), i.e., approximate θD +t as a single gradient-descent update on θDsyn +t−1 using D rather than θD +t−1 (Figure 3). +Subsequently, numerous other approaches have been built atop this framework with subtle variations. DSA +(Zhao & Bilen, 2021) improves over DC by performing the same image-augmentations (e.g., crop, rotate, +jitter, etc.) on both D and Dsyn while optimizing Equation (7). Since these augmentations are universal +and are applicable across data distillation frameworks, augmentations performed by DSA have become a +common part of all methods proposed henceforth, but we omit them for notational clarity. DCC (Lee et al., +2022b) further modifies the gradient-matching objective to incorporate class contrastive signals inside each +gradient-matching step and is shown to improve stability as well as performance. With θt evolving similarly +as in Equation (7), the modified optimization objective for DCC can be written as: +arg min +Dsyn +E +θ0∼Pθ +� T +� +t=0 +D +� +E +c∈C [∇θLDc(θt)] , E +c∈C +� +∇θLDcsyn(θt) +��� +(8) +Most recently, Kim et al. (2022) (IDC) extend the gradient matching framework by: (1) multi-formation: to +synthesize a higher amount of data within the same memory budget, store the data summary (e.g., images) +in a lower resolution to remove spatial redundancies, and upsample (using e.g., bilinear, FSRCNN (Dong +et al., 2016)) to the original scale while usage; and (2) matching gradients of the network’s training trajectory +over the full dataset D rather than the data summary Dsyn. To be specific, given a k× upscaling function +f : Rd×d �→ Rkd×kd, the modified optimization objective for IDC can be formalized as: +arg min +Dsyn +E +θ0∼Pθ +c ∼ C +� T +� +t=0 +D +� +∇θLDc(θt), ∇θLf(Dcsyn)(θt) +�� +s.t. +θt+1 ← θt − η · ∇θLD(θt) +(9) +5 + +θ𝒟𝗌𝗒𝗇 +t−1 +θ𝒟𝗌𝗒𝗇 +t +θ𝒟𝗌𝗒𝗇 +t+1 +θ𝒟𝗌𝗒𝗇 +t+2 +θ𝒟𝗌𝗒𝗇 +t+N +θ𝒟 +t−1 +θ𝒟 +t +θ𝒟 +t+1 +θ𝒟 +t+2 +θ𝒟 +t+M +θ0 ∼ Pθ +˜θ𝒟 +t +ℒ𝒟 (θ𝒟𝗌𝗒𝗇 +t +) ≜ 𝔼(x,y)∼𝒟 [l (Φ +θ𝒟𝗌𝗒𝗇 +t +(x), y)] +Update + by minimizing: +𝒟𝗌𝗒𝗇 +Meta-Model Matching +Gradient Matching +Trajectory Matching +D ( +˜θ𝒟 +t+i, θ𝒟𝗌𝗒𝗇 +t+i ) +Minimize: +D (θ𝒟 +t+M, θ𝒟𝗌𝗒𝗇 +t+N ) +D (θ𝒟 +t+M, θ𝒟 +t ) +Minimize: +( +) +𝖬 ≫ 𝖭 +Update on 𝒟 +Update on 𝒟𝗌𝗒𝗇 +˜θ𝒟 +t+1 +˜θ𝒟 +t+2 +˜θ𝒟 +t+3 +˜θ𝒟 +t+N+1 +[HQ Image Link] Figure 3: The underlying optimization in various data distillation frameworks. +Kim et al. (2022) further hypothesize that training models on Dsyn instead of D in the inner-loop has two major +drawbacks: (1) strong coupling of the inner- and outer-loop resulting in a chicken-egg problem (McLachlan & +Krishnan, 2007); and (2) vanishing network gradients due to the small size of Dsyn, leading to an improper +outer-loop optimization for gradient-matching based techniques. +2.3 +Data Distillation by Trajectory Matching +Cazenavette et al. (2022) proposed MTT which aims to match the training trajectories of models trained on +D vs. Dsyn. More specifically, let {θD +t }T +t=0 represent the training trajectory of training Φθ on D; trajectory +matching algorithms aim to solve the following optimization: +arg min +Dsyn,η +E +θ0∼Pθ +� +� +T −M +� +t=0 +D +� +θD +t+M, θDsyn +t+N +� +D +� +θD +t+M, θD +t +� +� +� +s.t. +θDsyn +t+i+1 ← θDsyn +t+i − η · ∇θLDsyn(θDsyn +t+i ) +; +θDsyn +t+1 ← θD +t − η · ∇θLDsyn(θD +t ), +(10) +where D : R|θ| × R|θ| �→ R is a distance metric of choice (typically L2 distance). Such an optimization can +intuitively be seen as optimizing for similar quality models trained with N SGD steps on Dsyn, compared to +M ≫ N steps on D, thereby invoking long-horizon trajectory matching. Notably, calculating the gradient of +Equation (10) w.r.t. Dsyn encompasses gradient unrolling through N-timesteps, thereby limiting the scalability +of MTT. On the other hand, since the trajectory of training Φθ on D, i.e., {θD +t }T +t=0 is independent of the +optimization of Dsyn, it can be pre-computed for various θ0 ∼ Pθ initializations and directly substituted. +Similar to gradient matching methods (Section 2.2), the trajectory matching framework also optimizes the +first-order distance between parameters, thereby inheriting the local smoothness assumption. As a scalable +alternative, Cui et al. (2022b) proposed TESLA, which re-parameterizes the parameter-matching loss of MTT +in Equation (10) (specifically when D is set as the L2 distance), using linear algebraic manipulations to +make the bilevel optimization’s memory complexity independent of N. Furthermore, TESLA uses learnable +soft-labels (Ysyn) during the optimization for an increased compression efficiency. +2.4 +Data Distillation by Distribution Matching +Even though the aforementioned gradient-matching or trajectory-matching based data distillation techniques +have been empirically shown to synthesize high-quality data summaries, the underlying bilevel optimization, +however, is oftentimes an expensive procedure both in terms of computation time and memory. To this +6 + +end, distribution-matching techniques solve a correlated proxy task which restricts the optimization to a +single-level, leading to a much improved scalability. More specifically, instead of matching the quality of +models on D vs. Dsyn, distribution-matching techniques directly match the distribution of data in D vs. Dsyn. +The key assumption for this family of methods is that two datasets which are similar according to a particular +distribution divergence metric, also lead to similarly trained models. +First proposed by Zhao & Bilen (2023), DM uses (1) numerous parametric encoders to cast high-dimensional +data into respective low-dimensional latent spaces; and (2) an approximation of the Maximum Mean +Discrepancy to compute the distribution mismatch between D and Dsyn in each of the latent spaces. More +precisely, given a set of k encoders E ≜ {ψi : X �→ Xi}k +i=1, the optimization objective can be written as: +arg min +Dsyn +E +ψ∼E +c ∼ C +����� E +x∼Dc [ψ(x)] − +E +x∼Dcsyn +[ψ(x)] +���� +2� +(11) +DM uses a set of randomly initialized neural networks (with the same architecture) to instantiate E. They +observe similar performance when instantiated with more meaningful, task-optimized neural networks, despite +it being much less efficient. CAFE (Wang et al., 2022) further refines the distribution-matching idea by: (1) +solving a bilevel optimization problem for jointly optimizing a single encoder (Φ) and the data summary, +rather than using a pre-determined set of encoders (E); and (2) assuming a neural network encoder (Φ), +match the latent representations obtained at all intermediate layers of the encoder instead of only the last +layer. Formally, given a (L + 1)-layer neural network Φθ : X �→ Y where Φl +θ represents Φ’s output at the lth +layer, the optimization problem for CAFE can be specified as: +arg min +Dsyn +E +c ∼ C +� L +� +l=1 +���� E +x∼Dc +� +Φl +θt(x) +� +− +E +x∼Dcsyn +� +Φl +θt(x) +����� +2 +− β · +E +(x,y)∼Dc [log ˆp(y|x, θt)] +� +s.t. +θt+1 ← θt − η · ∇θLDsyn(θt) ; +ˆp(y|x, θ) ≜ softmax +y +�� +ΦL +θ (x), +E +x′∼Dy +syn +� +ΦL +θ (x′) +��� +, +(12) +where ˆp(·|·, θ) intuitively represents the nearest centroid classifier on Dsyn using the latent representations +obtained by last layer of Φθ. Analogously, IT-GAN (Zhao & Bilen, 2022) also uses the distribution-matching +framework in Equation (11) to generate data that is informative for model training, in contrast to the +traditional GAN (Goodfellow et al., 2014) which focuses on generating realistic data. +2.5 +Data Distillation by Factorization +All of the aforementioned data distillation frameworks intrinsically maintain the synthesized data summary +as a large set of free parameters, which are in turn optimized. Arguably, such a setup prohibits knowledge +sharing between synthesized data points (parameters), which might introduce data redundancy. On the other +hand, factorization-based data distillation techniques parameterize the data summary using two separate +components: (1) bases: a set of mutually independent base vectors; and (2) hallucinators: a mapping from +the bases’ vector space to the joint data- and label-space. In turn, both the bases and hallucinators are +optimized for the task of data distillation. +Formally, let B ≜ {bi ∈ B}|B| +i=1 be the set of bases, and H ≜ {hi : B �→ X × Y}|H| +i=1 be the set of hallucinators, +then the data summary is parameterized as Dsyn ≜ {h(b)}b∼B, h∼H. Even though such a two-pronged +approach seems similar to generative modeling of data, note that unlike classic generative models, (1) the +input space consists only of a fixed and optimized set of latent codes and isn’t meant to take any other inputs; +and (2) given a specific B and H, we can generate at most |B| · |H| sized data summaries. Notably, such a +hallucinator-bases data parameterization can be optimized using any of the aforementioned data optimization +frameworks (Sections 2.1 to 2.4) +This framework was concurrently proposed by Deng & Russakovsky (2022) (we take the liberty to term +their unnamed model as “Lin-ear Ba-ses”) and Liu et al. (2022c) (HaBa). LinBa modifies the general +hallucinator-bases framework by assuming (1) the bases’ vector space (B) to be the same as the task input +7 + +space (X); and (2) the hallucinator to be linear and additionally conditioned on a given predictand. More +specifically, the data parameterization can be formalized as follows: +Dsyn ≜ +� +(y HT B, y) +� +y∼C +H∼H +s.t. +B ∈ R|B|×dim(X) ≜ [bi ∈ X]|B| +i=1 +; +H ≜ +� +Hi ∈ R|B|×|C|�|H| +i=1 , +(13) +where for the sake of notational simplicity, we assume y ∈ R|C| represents the one-hot vector of the label for +which we want to generate data, and the maximum amount of data that can be synthesized n ≤ |C| · |H|. +Since the data generation (Equation (13)) is an end-to-end differentiable procedure, both B and H are jointly +optimized using the TBPTT framework discussed in Section 2.1, albeit with some crucial modifications for +vastly improved performance: (1) using momentum-based optimizers instead of vanilla SGD in the inner-loop; +and (2) longer unrolling (≥ 100 steps) of the inner-loop during TBPTT. Liu et al. (2022c) (HaBa) relax +the linear and predictand-conditional hallucinator assumption of LinBa, equating to the following data +parameterization: +Dsyn ≜ { (h(b), y) }b,y∼B +h∼H +s.t. +B ≜ { (bi ∈ X, yi ∈ Y) }|B| +i=1 +; +H ≜ {hθi : X �→ X}|H| +i=1 , +(14) +where B and H are optimized using the trajectory matching framework (Section 2.3) with an additional +contrastive constraint to promote diversity in Dsyn (cf. Liu et al. (2022c), Equation (6)). Following this setup, +HaBa can generate at most |B| · |H| sized data summaries. Furthermore, one striking difference between +HaBa (Equation (14)) and LinBa (Equation (13)) is that to generate each data point, LinBa uses a linear +combination of all the bases, whereas HaBa generates a data point using a single base vector. +Lee et al. (2022a) (KFS) further build atop this framework by maintaining a different bases’ vector space +B from the data domain X, such that dim(B) < dim(X). This parameterization allows KFS to store an +even larger number of images, with a comparable storage budget to other methods. Formally, the data +parameterization for KFS can be specified as: +Dsyn ≜ +� +c∈C +{ (h(b), c) }b∼Bc +h∼H +s.t. +B ≜ +� +c∈C +Bc +; +Bc ≜ {bc +i ∈ B}B +i=1 +; +H ≜ {hθi : B �→ X}|H| +i=1 , +(15) +where KFS stores B bases per class, equivalent to a total of n = |C| · B · |H| sized data summaries. Following +this data parameterization, B and H are optimized using the distribution matching framework for data +distillation (Equation (11)) to ensure fast, single-level optimization. +Data Distillation vs. Data Compression. We highlight that it is non-trivial to ensure a fair comparison +between data distillation techniques that (1) are “non-factorized”, i.e., maintain each synthesized data point +as a set of free-parameters (Sections 2.1 to 2.4); and (2) use factorized approaches discussed in this section to +efficiently organize the data summary. If we use the size of the data summary (n) as the efficiency metric, +factorized approaches are adversely affected as they need a much smaller storage budget to synthesize the +same-sized data summaries. On the other hand, if we use “end-to-end bytes of storage” as the efficiency +metric, non-factorized approaches are adversely affected as they perform no kind of data compression, but +focus solely on better understanding the model-to-data relationship through the lens of optimization. For a +better intuition, one can apply posthoc lossless compression (e.g., Huffman coding) on data synthesized by +non-factorized data distillation approaches to fit more images in the same storage budget (Schirrmeister et al., +2022). Such techniques unintentionally deviate from the original intent of data distillation, and progress +more toward better data compression techniques. As a potential solution, we encourage the community to +consider reporting results for both scenarios: a fixed data summary size n, as well as fixed bytes-of-storage. +Nonetheless, for the ease of empirical comparison amongst the discussed data distillation techniques, we +provide a collated set of results over four image-classification datasets in Table 1. +8 + +Table 1: Comparison of data distillation methods. Each method (1) synthesizes the data summary on +the train-set; (2) unless mentioned, trains a 128-width ConvNet (Gidaris & Komodakis, 2018) on the data +summary; and (3) evaluates it on the test-set. Confidence intervals are obtained by training at least 5 networks +on the data summary. LinBa (No Fact.) represents LinBa with the no factorization. Methods evaluated using +KRR are marked as (∞-Conv) or (∞-FC). The equivalent storage-in-bytes is used for factorization-based +techniques instead of IPC. The best method in their category is emboldened, the best-overall non-factorized +method evaluated on ConvNet is colored orange, and the best-overall factorized method is colored blue. +Dataset +MNIST +CIFAR-10 +CIFAR-100 +Tiny ImageNet +Imgs/Class (IPC) +1 +10 +50 +1 +10 +50 +1 +10 +50 +1 +10 +50 +Baselines +Random +64.9 +±3.5 +95.1 +±0.9 +97.9 +±0.2 +14.4 +±2.0 +26.0 +±1.2 +43.4 +±1.0 +4.2 +±0.3 +14.6 +±0.5 +30.0 +±0.4 +1.5 +±0.1 +6.0 +±0.8 +16.8 +±1.8 +Herding1 +89.2 +±1.6 +93.7 +±0.3 +94.9 +±0.2 +21.5 +±1.2 +31.6 +±0.7 +40.4 +±0.6 +8.4 +±0.3 +17.3 +±0.5 +33.7 +±0.5 +- +- +- +Forgetting2 +35.5 +±5.6 +68.1 +±3.3 +88.2 +±1.2 +13.5 +±1.2 +23.3 +±1.0 +23.3 +±1.1 +4.5 +±0.2 +15.1 +±0.3 +30.5 +±0.3 +- +- +- +Meta-model Matching +DD3 +- +79.5 +±8.1 +- +- +36.8 +±1.2 +- +- +- +- +- +- +- +LinBa (No Fact.)15 +95.2 +±0.3 +98.8 +±0.1 +99.2 +±0.1 +49.1 +±0.6 +62.4 +±0.4 +70.5 +±0.4 +21.3 +±0.6 +34.7 +±0.5 +- +- +- +- +KIP (ConvNet)4 +90.1 +±0.1 +97.5 +±0.0 +98.3 +±0.1 +49.9 +±0.2 +62.7 +±0.3 +68.6 +±0.2 +15.7 +±0.2 +28.3 +±0.1 +- +- +- +- +RFAD (ConvNet)5 +94.4 +±1.5 +98.5 +±0.1 +98.8 +±0.1 +53.6 +±1.2 +66.3 +±0.5 +71.1 +±0.4 +26.3 +±1.1 +33.0 +±0.3 +- +- +- +- +FRePO (ConvNet)6 +93.0 +±0.4 +98.6 +±0.1 +99.2 +±0.1 +46.8 +±0.7 +65.5 +±0.6 +71.7 +±0.2 +28.7 +±0.1 +42.5 +±0.2 +44.3 +±0.2 +15.4 +±0.3 +25.4 +±0.2 +- +KIP (∞-FC)7 +85.5 +±0.1 +97.2 +±0.2 +98.4 +±0.1 +40.5 +±0.4 +53.1 +±0.5 +58.6 +±0.4 +- +- +- +- +- +- +KIP (∞-Conv)4 +97.3 +±0.1 +99.1 +±0.1 +99.5 +±0.1 +64.7 +±0.2 +75.6 +±0.2 +80.6 +±0.1 +34.9 +±0.1 +49.5 +±0.3 +- +- +- +- +RFAD (∞-Conv)5 +97.2 +±0.2 +99.1 +±0.0 +99.1 +±0.0 +61.4 +±0.8 +73.7 +±0.2 +76.6 +±0.3 +44.1 +±0.1 +46.8 +±0.2 +- +- +- +- +FRePO (∞-Conv)6 +92.6 +±0.4 +98.6 +±0.1 +99.2 +±0.1 +47.9 +±0.6 +68.0 +±0.2 +74.4 +±0.1 +32.3 +±0.1 +44.9 +±0.2 +43.0 +±0.3 +19.1 +±0.3 +26.5 +±0.1 +- +Gradient +Matching +DC8 +91.7 +±0.5 +97.4 +±0.2 +98.2 +±0.2 +28.3 +±0.5 +44.9 +±0.5 +53.9 +±0.5 +12.8 +±0.3 +25.2 +±0.3 +30.5 +±0.3 +4.6 +±0.6 +11.2 +±1.6 +10.9 +±0.7 +DSA9 +88.7 +±0.6 +97.8 +±0.1 +99.2 +±0.1 +28.8 +±0.7 +52.1 +±0.5 +60.6 +±0.5 +13.9 +±0.3 +32.3 +±0.3 +42.8 +±0.4 +6.6 +±0.2 +14.4 +±2.0 +22.6 +±2.6 +DCC10 +- +- +- +34.0 +±0.7 +54.5 +±0.5 +64.2 +±0.4 +14.6 +±0.3 +33.5 +±0.3 +39.3 +±0.4 +- +- +- +Distr. +Matching +DM11 +89.7 +±0.6 +97.5 +±0.1 +98.6 +±0.1 +26.0 +±0.8 +48.9 +±0.6 +63.0 +±0.4 +11.4 +±0.3 +29.7 +±0.3 +43.6 +±0.4 +3.9 +±0.2 +12.9 +±0.4 +24.1 +±0.3 +CAFE12 +90.8 +±0.5 +97.5 +±0.1 +98.9 +±0.2 +31.6 +±0.8 +50.9 +±0.5 +62.3 +±0.4 +14.0 +±0.3 +31.5 +±0.2 +42.9 +±0.2 +- +- +- +Traj. +Matching +MTT13 +- +- +- +46.3 +±0.8 +65.3 +±0.7 +71.6 +±0.2 +24.3 +±0.3 +40.1 +±0.4 +47.7 +±0.2 +8.8 +±0.3 +23.2 +±0.2 +28.0 +±0.3 +TESLA14 +- +- +- +48.5 +±0.8 +66.4 +±0.8 +72.6 +±0.7 +24.8 +±0.4 +41.7 +±0.3 +47.9 +±0.3 +- +- +- +Factorization +IDC15 +- +- +- +50.0 +±0.4 +67.5 +±0.5 +74.5 +±0.1 +- +44.8 +±0.2 +- +- +- +- +LinBa16 +98.7 +±0.7 +99.3 +±0.5 +99.4 +±0.4 +66.4 +±0.4 +71.2 +±0.4 +73.6 +±0.5 +34.0 +±0.4 +42.9 +±0.7 +- +16.0 +±0.7 +- +- +HaBa17 +- +- +- +48.3 +±0.8 +69.9 +±0.4 +74.0 +±0.2 +33.4 +±0.4 +40.2 +±0.2 +47.0 +±0.2 +- +- +- +KFS18 +- +- +- +59.8 +±0.5 +72.0 +±0.3 +75.0 +±0.2 +40.0 +±0.5 +50.6 +±0.2 +- +22.7 +±0.2 +27.8 +±0.2 +- +Full Dataset +99.6 +±0.1 +84.8 +±0.1 +56.2 +±0.3 +37.6 +±0.4 +1 (Welling, 2009), 2 (Toneva et al., 2019), 3 (Wang et al., 2018), 4 (Nguyen et al., 2021b), 5 (Loo et al., 2022) +6 (Zhou et al., 2022b), 7 (Nguyen et al., 2021a), 8 (Zhao et al., 2021), 9 (Zhao & Bilen, 2021), 10 (Lee et al., 2022b) +11 (Zhao & Bilen, 2023), 12 (Wang et al., 2022), 13 (Cazenavette et al., 2022), 14 (Cui et al., 2022b) +15 (Kim et al., 2022), 16 (Deng & Russakovsky, 2022), 17 (Liu et al., 2022c), 18 (Lee et al., 2022a) +9 + +Data Distillation +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +Users +0.2 +0.1 +0.5 +0.8 +0.1 +1 +0.9 +0.8 +0.9 +0.1 +0.5 +1 +0.4 +0.4 +0.3 +0.4 +1 +0.8 +0.9 +0.2 +0.2 +Items (Movies/Ads/Songs) +Items (Movies/Ads/Songs) +Fake Users +Test accuracies +GCN: 89.4% +SGC: 89.6% +APPNP: 87.8% +GraphSAGE: 89.1% +("!,$!, %′) +Condense +(",$,%) +Test accuracies +GCN: 93.9% +SGC: 93.5% +APPNP: 94.3% +GraphSAGE: 93.0% +153,932 training nodes +154 training nodes +Test accuracies +GCN: 89.4% +SGC: 89.6% +APPNP: 87.8% +GraphSAGE: 89.1% +("!,$!, %′) +Condense +(",$,%) +Test accuracies +GCN: 93.9% +SGC: 93.5% +APPNP: 94.3% +GraphSAGE: 93.0% +153,932 training nodes +154 training nodes +Lorem ipsum dolor sit amet, consectetur +adipiscing elit, sed do eiusmod tempor +incididunt ut labore et dolore magna aliqua. +Ut enim ad minim veniam, quis nostrud +exercitation ullamco laboris nisi ut aliquip +Lorem sit adipiscing do incididunt et +aliqua. Ut minim nostrud laboris aliquip +Test accuracies +GCN: 89.4% +SGC: 89.6% +APPNP: 87.8% +GraphSAGE: 89.1% +("!,$!, %′) +Condense +(",$,%) +Test accuracies +GCN: 93.9% +SGC: 93.5% +APPNP: 94.3% +GraphSAGE: 93.0% +153,932 training nodes +154 training nodes +[HQ Image Link] Figure 4: Overview of distilling data for a few commonly observed data modalities. +3 +Data Modalities +Having learned about different kinds of optimization frameworks for data distillation, we now discuss an +orthogonal (and important) aspect of data distillation – what kinds of data can data distillation techniques +summarize? From continuous-valued images to heterogeneous, discrete, and semi-structured graphs, the +underlying data for each unique application of machine learning has its own modality, structure, and set +of assumptions. While the earliest data distillation techniques were designed to summarize images for +classification, recent steps have been taken to expand the horizon of data distillation into numerous other +scenarios. In what follows, we categorize existing data distillation techniques as per their intended data +modality, while also discussing their unique challenges. +Images. A large-portion of existing data distillation techniques are designed for image classification data +(Cazenavette et al., 2022; Deng & Russakovsky, 2022; Kim et al., 2022; Lee et al., 2022a;b; Liu et al., 2022c; +Loo et al., 2022; Nguyen et al., 2021a;b; Wang et al., 2022; 2018; Zhao & Bilen, 2021; 2022; 2023; Zhao et al., +2021; Zhou et al., 2022b) simply because images have a real-valued, continuous data-domain (X ≡ Rd×d). This +allows SGD-based optimization directly on the data, which is treated as a set of free parameters. Intuitively, +incrementally changing each pixel value can be treated as slight perturbations in the color space, and hence +given a suitable data distillation loss, can be naïvely optimized using SGD. +Text. Textual data is available in large amounts from sources like websites, news articles, academic +manuscripts, etc., and is also readily accessible with datasets like the common crawl1 which sizes up to almost +541TB. Furthermore, with the advent of large language models (LLM) (Brown et al., 2020; Devlin et al., +2019; Thoppilan et al., 2022), training such models from scratch on large datasets has become an increasingly +expensive procedure. Despite recent efforts in democratizing LLM training (Geiping & Goldstein, 2022; Scao +et al., 2022; Wolf et al., 2020), effectively distilling large-scale textual data as a solution is yet to be explored. +The key bottlenecks for distilling textual data are: (1) the inherently discrete nature of data, where a token +should belong in a limited vocabulary of words; (2) the presence of a rich underlying structure, i.e., sentences +of words (text) obey fixed patterns according to a grammar; and (3) richness of context, i.e., a given piece of +text could have wildly different semantic interpretations under different contexts. +Sucholutsky & Schonlau (2021) take a latent-embedding approach to textual data distillation. On a high +level, to circumvent the discreteness of the optimization, the authors perform distillation in a continuous +embedding space. More specifically, assuming access to a latent space specified by a fixed text-encoder, +the authors learn continuous representations of each word in the distilled text and optimize it using the +1https://commoncrawl.org/the-data/ +10 + +50K Real Training Images +Dataset +Distillation +10 Synthetic Training Images +Train +Train +00000 +OOO +Similar Test PerformanceTBPTT data-distillation framework proposed by Wang et al. (2018) (Equation (4)). Finally, the distilled +text representations are decoded by following a simple nearest-neighbor protocol. +Graphs. A wide variety of data and applications can inherently be modeled as graphs, e.g., user-item +interactions (Mittal et al., 2021; Sachdeva & McAuley, 2020; Wu et al., 2020), social networks (Fan et al., +2019), autonomous driving (Casas et al., 2020; Sachdeva et al., 2022b), etc. Taking the example of social +networks, these user-user graphs in the modern-era easily scale up to the billion-scale (Chen et al., 2021), +calling for principled scaling solutions. Graph distillation could trivially solve a majority of the scale challenges, +but synthesizing tiny, high-fidelity graphs has the following hurdles: (1) nodes in a graph can be highly +abstract, e.g., users, products, text articles, etc. some of which could be discrete, heterogeneous, or even +simply numerical IDs; (2) graphs follow a variety of intrinsic patterns (e.g., spatial (Kipf & Welling, 2017)) +which need to be retained in the distilled graphs; and (3) quadratic size of the adjacency matrix could be +computationally prohibitive even for moderate-sized graphs. +Jin et al. (2022b) propose GCond which distills graphs in the inductive node-classification setting, specified +by its node-feature matrix X, adjacency matrix A, and node-target matrix Y. GCond distills the given +graph by learning a synthetic node-feature matrix Xsyn, and using Xsyn to generate Asyn ≜ fθ(Xsyn) which +can be realized, e.g., through a parametric similarity function simθ(·, ·) between the features of two nodes, +i.e., Ai,j +syn ≜ σ(simθ(Xi +syn, Xj +syn)), where σ(·) is the sigmoid function. Finally, both Xsyn and θ are optimized +using the gradient-matching framework proposed by Zhao et al. (2021) (Equation (7)). Another work (Liu +et al., 2022a) (GCDM) shares the same framework as GCond but instead uses the distribution matching +framework proposed by Zhao & Bilen (2023) (Equation (11)) to optimize Xsyn and θ. Extending to a +graph-classification setting, Jin et al. (2022a) further propose DosCond with two major changes compared +to GCond: (1) instead of parameterizing the adjacency matrix using a similarity function on Xsyn, they +maintain a free-parameter matrix Ω with the same size as the adjacency matrix, and sample each Ai,j +syn entry +through an independent Bernoulli draw on Ωi,j as the prior using the reparameterization trick (Maddison +et al., 2017). Such a procedure ensures differentiability as well as discrete matrix synthesis; and (2) Xsyn and +Ω are still optimized using the gradient-matching framework (Equation (7)), albeit with only a single-step, +i.e., T = 1 for improved scalability and without empirically observing a loss in performance. +Recommender Systems. The amount of online user-feedback data available for training recommender +systems is rapidly increasing (Wu et al., 2022). Furthermore, typical user-facing recommender systems need to +be periodically re-trained (Naumov et al., 2019), which adds to requirements for smarter data summarization +solutions (see Sachdeva et al. (2022c) for background on sampling recommender systems data). However, +distilling recommender systems data has the following challenges: (1) the data is available in the form +of abstract and discrete (userID, itemID, relevance) tuples, which departs from the typical (features, +label) setup; (2) the distribution of both user- and item-popularity follows a strong power-law which leads +to data scarcity and unstable optimization; and (3) the data inherits a variety of inherent structures, e.g., +sequential patterns (Kang & McAuley, 2018; Sachdeva et al., 2019), user-item graph patterns (Wu et al., +2019), item-item co-occurrence patterns (Steck, 2019), missing-not-at-randomness (Sachdeva et al., 2020; +Schnabel et al., 2016), etc. +Sachdeva et al. (2022a) propose Distill-CF which distills implicit-feedback recommender systems data, i.e., +when the observed user-item relevance is binary (e.g., click or no-click). Such data can be visualized as a +binary user-item matrix R where each row represents a single user, and each column represents an item. +On a high-level, Distill-CF synthesizes fake users along with their item-consumption histories, visualized +as a synthetic user-item matrix Rsyn. Notably, to preserve semantic meaning, the item-space in Rsyn is the +same as in R. To alleviate the data discreteness problem, Distill-CF maintains a sampling-prior matrix Ω +which has the same size as Rsyn, and can in-turn be used to generate Rsyn using multi-step Gumbel sampling +with replacement (Jang et al., 2017) for each user’s prior in Ω (equivalent to each row). Such a formulation +automatically also circumvents the dynamic user- and item-popularity artifact in recommender systems data, +which can analogously be controlled by the row- and column-wise entropy of Ω. Finally, Ω is optimized using +the meta-model matching framework proposed by Nguyen et al. (2021a). Notably, Sachdeva et al. (2022a) +also propose infinite-width autoencoders which suit the task of item recommendation while also leading to +closed-form computation of the inner-loop in the meta-model matching framework (Equation (5)). +11 + +4 +Applications +While the data distillation task was originally designed to accelerate model training, there are numerous other +applications of a high-fidelity data summary. Below we briefly discuss a few such promising applications, +along with providing pointers to existing works. +Differential Privacy. Data distillation was recently shown to be a promising solution for differential privacy +as defined by Dwork (2008). Dong et al. (2022) show that data distillation techniques can perform better than +existing state-of-the-art differentially-private data generators (Cao et al., 2021; Harder et al., 2021) on both +performance and privacy grounds. Notably, the privacy benefits of data distillation techniques are virtually +free, as none of these methods were optimized for generating differentially-private data. Chen et al. (2022) +further modify the gradient matching framework (Equation (7)) by clipping and adding white noise to the +gradients obtained on the original dataset while optimization. Such a routine was shown to have better sample +utility, while also satisfying strict differential privacy guarantees. From a completely application perspective, +data distillation has been used to effectively distill sensitive medical data as well (Li et al., 2020a; 2022). +Neural Architecture Search (NAS). Automatic searching of neural-network architectures can alleviate +the manual effort, as well as lead to better models (see Elsken et al. (2019) for a detailed review). Analogous +to using model extrapolation, i.e., extrapolating the performance of an under-trained model architecture +on the full dataset; data extrapolation, on the other hand, aims to train models on a small, high-fidelity +data sample till convergence. Numerous data distillation techniques (Such et al., 2020; Zhao et al., 2021) +show promise on small NAS test-beds by employing the data extrapolation framework. However, Cui et al. +(2022a) show that data distillation does not perform well when evaluating diverse architectures on bigger +NAS test-beds, calling for better rank-preserving data distillation techniques. +Continual Learning. Never-ending learning (see Parisi et al. (2019) for a detailed review) has been +frequently associated with catastrophic forgetting (French, 1999), i.e., patterns extracted from old data/tasks +are easily forgotten when patterns from new data/tasks are learned. Data distillation has been shown as +an effective solution to alleviate catastrophic forgetting, by simply using the distilled data summary in a +replay buffer that is continually updated and used in subsequent data/task training (Rosasco et al., 2021; +Sangermano et al., 2022; Wiewel & Yang, 2021). Deng & Russakovsky (2022) show further evidence of a +simple compress-then-recall strategy outperforming existing state-of-the-art continual learning approaches. +Notably, only the data summary is stored for each task, and a new model is trained (from scratch) using all +previous data summaries, for each new incoming task. +Federated Learning. Federated or collaborative learning (see Li et al. (2020b) for a detailed survey) +involves training a learning algorithm in a decentralized fashion. A standard approach to federated learning +is to synchronize local parameter updates to a central server, instead of synchronizing the raw data itself +(Konečn`y et al., 2016). Data distillation, on the other hand, alleviates the need to synchronize large parametric +models across clients and servers, by synchronizing tiny synthesized data summaries to the central server +instead. Subsequently, the entire training happens only on the central server. Such data distillation-based +federated learning methods (Goetz & Tewari, 2020; Hu et al., 2022; Liu et al., 2022b; Song et al., 2022; Xiong +et al., 2022; Zhou et al., 2020) are shown to perform better than model-synchronization based federated +learning approaches, while also requiring multiple orders lesser client-server communication. +5 +Challenges & Future Directions +Despite achieving remarkable progress in data-efficient learning, there are numerous framework-based, theoret- +ical, and application-based directions yet to be explored in data distillation. In what follows, we highlight and +discuss such directions for the community to further explore, based either on early evidence or our intuition. +New data modalities & settings. Extending on the discussion in Section 3, existing data distillation +techniques have largely been restricted to image-classification settings, due to the easy availability of datasets, +and amenable data-optimization. However, taking a step back to the broad field of computer vision (see +12 + +Shapiro et al. (2001) for a thorough background), there are numerous equally important tasks that can benefit +from a high-quality data summary. For example, increasing the sample efficiency of training image-generation +models is both highly important due to their massive size and popularity (Ramesh et al., 2022; Rombach +et al., 2022), and is also highly non-trivial to fit into the existing data distillation framework. Similarly, a +variety of important machine learning applications don’t enjoy a continuous data domain like images, making +it hard for existing data distillation techniques to scale and work as expected. In addition to recent efforts on +distilling discrete data like graphs (Jin et al., 2022a;b) and recommender systems (Sachdeva et al., 2022a), +developing a unified, principled data distillation framework for inherently sparse and discrete data will be +useful for a variety of research communities (e.g., text, tabular-data, extreme classification, etc.). +Better scaling. Existing data distillation techniques validate their prowess only in the super low-data +regime (typically 1 − 50 data points per class). However, Cui et al. (2022a) show that as we keep scaling the +size of the data summary (larger distilled data), most distillation methods collapse to the random-sampling +baseline. While convergent behavior is expected, the distillation performance collapses much more rapidly +with larger data summaries. Analogously, for data distillation to practically replace full-data training, deeper +investigations of the causes and potential fixes of such scaling artifacts are highly necessary. +Improved optimization. A unifying thread across data distillation techniques is an underlying bilevel +optimization, which is provably NP-hard even in the linear inner-optimization case (Vicente et al., 1994). +Notably, bilevel optimization has been successfully applied in a variety of other applications like meta-learning +(Finn et al., 2017; Li et al., 2017), hyper-parameter optimization (Lorraine et al., 2020; Maclaurin et al., +2015), neural architecture search (Liu et al., 2019), coreset construction (Borsos et al., 2020; Zhou et al., +2022a), etc. Despite its success, many theoretical underpinnings are yet to be explored, e.g., the effect of +commonly-used singleton solution assumption (Franceschi et al., 2018), the effect of over-parameterization on +bilevel optimization (Vicol et al., 2022), connections to statistical influence functions (Bae et al., 2022), the +bias-variance tradeoff (Vicol et al., 2021), etc. Clearly, an overall better understanding of bilevel optimization +will directly enable the development of better data distillation techniques. +Acknowledgments +We sincerely thank Zhiwei Deng, Bo Zhao, and George Cazenavette for their feedback on early drafts of this +survey. +References +Sunil Arya, David M Mount, Nathan S Netanyahu, Ruth Silverman, and Angela Y Wu. 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In Advances +in Neural Information Processing Systems, 2022b. +A +Notation +Dataset related +D ≜ {(xi ∈ X, yi ∈ Y)}|D| +i=1 +The target dataset to be distilled +X +Data domain +Y +Predictand domain +C +Set of unique classes in Y +Dc ≜ {(xi, yi) | yi = c}|D| +i=1 +Portion of D with class c +X ≜ [xi]|D| +i=1 +Matrix of all features in D +Y ≜ [yi]|D| +i=1 +Matrix of all predictands in D +n +Size of data summary +Dsyn ≜ {(˜xi, ˜yi)}n +i=1 +Data summary +Dc +syn ≜ {(˜xi, ˜yi) | ˜yi = c}n +i=1 +Portion of Dsyn with class c +Xsyn ≜ [˜xi]n +i=1 +Matrix of all features in Dsyn +Ysyn ≜ [˜yi]n +i=1 +Matrix of all predictands in Dsyn +Learning related +Φθ : X �→ Y +Learning algorithm parameterized by θ +l : Y × Y �→ R +Twice-differentiable cost function +LD(θ) ≜ E(x,y)∼D[l(Φθ(x), y)] +Expected loss of Φ on D +LDsyn(θ) ≜ E(x,y)∼Dsyn[l(Φθ(x), y)] +Expected loss of Φ on Dsyn +General +dim(A) +Size of basis of A +|A| +Number of elements in A +sup +Supremum +arg min +θ +f(θ) +Optimum value of θ which minimizes f(θ) +E +x [f(x)] ≜ � +x p(x) · f(x) +Expected value of f(x) when domain of x is discrete +19 + diff --git a/OdE3T4oBgHgl3EQfCAmE/content/tmp_files/load_file.txt b/OdE3T4oBgHgl3EQfCAmE/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..62d2c4a68b8cf9a084ad99af17a029700a57855c --- /dev/null +++ b/OdE3T4oBgHgl3EQfCAmE/content/tmp_files/load_file.txt @@ -0,0 +1,1521 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf,len=1520 +page_content='Data Distillation: A Survey Noveen Sachdeva nosachde@ucsd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='edu Computer Science & Engineering University of California, San Diego Julian McAuley jmcauley@ucsd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='edu Computer Science & Engineering University of California, San Diego Abstract The popularity of deep learning has led to the curation of a vast number of massive and multifarious datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Despite having close-to-human performance on individual tasks, training parameter-hungry models on large datasets poses multi-faceted problems such as (a) high model-training time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' (b) slow research iteration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' and (c) poor eco-sustainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' As an alternative, data distillation approaches aim to synthesize terse data summaries, which can serve as effective drop-in replacements of the original dataset for scenarios like model training, inference, architecture search, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' In this survey, we present a formal framework for data distillation, along with providing a detailed taxonomy of existing approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Additionally, we cover data distillation approaches for different data modalities, namely images, graphs, and user-item interactions (recommender systems), while also identifying current challenges and future research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' 1 Introduction (Loose) Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' (Data distillation) Approaches that aim to synthesize tiny and high-fidelity data summaries which distill the most important knowledge from a given target dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Such distilled summaries are optimized to serve as effective drop-in replacements of the original dataset for efficient and accurate data-usage applications like model training, inference, architecture search, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' The recent “scale-is-everything” viewpoint (Ghorbani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Hoffmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Kaplan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2020), argues that training bigger models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', consisting of a higher number of parameters) on bigger datasets, and using larger computational resources is the sole key for advancing the frontier of artificial intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' On the other hand, a well-reasoned, principled solution will arguably be more amenable to scaling, thereby leading to faster progress (Sorscher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Data distillation (Definition 1), is a task rooted in the latter school of thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Clearly, the scale viewpoint still holds, in that if we keep increasing the amount of data (albeit now compressed and of higher quality), we will observe an improvement in both upstream and downstream generalization, but at a faster rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' A terse, high-quality data summary has use cases from a variety of standpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' First and foremost, it leads to a faster model-training procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' In turn, faster model training equates to (1) compute- cost saving and expedited research iterations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', the investigative procedure of manually experimenting different ideas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' and (2) improved eco-sustainability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', lowering the amount of compute time directly leads to a lower carbon footprint from running power-hungry accelerated hardware (Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Additionally, a small data summary democratizes the entire pipeline, as more people can train state-of-the-art algorithms on reasonably accessible hardware using the data summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Finally, a high-quality data summary indirectly also accelerates orthogonal procedures like neural architecture search (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2019), approximate nearest neighbour search (Arya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 1998), knowledge distillation (Hinton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2015), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', where the procedure needs to iterate over the entire dataset multiple times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='04272v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='LG] 11 Jan 2023 Data Distillation Train Train Similar Performance << 50K distilled images 50K images Learning algorithm [HQ Image Link] Figure 1: The premise of data distillation demonstrated using an image dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Comparison with knowledge distillation & transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Despite inherently distilling some kind of knowledge, we would like to highlight both knowledge distillation and transfer learning are orthogonal procedures to data distillation, which can potentially work together to perform both tasks more efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Knowledge distillation (Hinton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2015) entails distilling the knowledge from a trained teacher network into a smaller student network efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' On the other hand, transfer learning (Pratt, 1992) is the procedure of transferring knowledge across similar tasks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', from image classification to image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Orthogonally, data distillation aims to distill the knowledge from a given dataset into a terse data summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Such data summaries can be used in conjunction with knowledge distillation or transfer learning procedures for both (1) faster learning of the teacher models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' and (2) faster knowledge transfer to the student models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' The same comparison holds true for model compression techniques (LeCun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 1989) as well, where similar to knowledge distillation, the goal is to reduce model storage size, rather than reducing the training time or sample complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' In this survey, we intend to provide a succinct overview of various data distillation frameworks across different data modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' We start by presenting a formal data distillation framework in Section 2, along with a detailed empirical comparison of existing image distillation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Subsequently, in Section 3, we discuss existing data distillation frameworks for synthesizing data of different modalities, as well as outlining the associated challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' In Section 4, we discuss alternative applications of synthesizing a high-fidelity data summary rather than simply accelerating model training along with pointers to existing work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Finally, in Section 5, we conclude by presenting common pitfalls in existing data distillation techniques, along with proposing interesting directions for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' 2 The Data Distillation Framework Before going into the specifics of data distillation, we start by outlining useful notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Let D ≜ {(xi, yi)}|D| i=1 be a given dataset which needs to be distilled, where xi ∈ X are the set of input features, and yi ∈ Y is the desired label for xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' For classification tasks, let C be the set of unique classes in Y, and Dc ≜ {(xi, yi) | yi = c}|D| i=1 be the subset of D with class c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' We also define the matrices X ≜ [xi]|D| i=1 and Y ≜ [yi]|D| i=1 for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Given a data budget n ∈ Z+, data distillation techniques aim to synthesize a high-fidelity data summary Dsyn ≜ {(˜xi, ˜yi)}n i=1 such that n ≪ |D|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' We define Dc syn, Xsyn, and Ysyn similarly as defined for D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Let Φθ : X �→ Y represent a learning algorithm parameterized by θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' We also assume access to a twice-differentiable cost function l : Y × Y �→ R, and define LD(θ) ≜ E(x,y)∼D[l(Φθ(x), y)] for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Notation is also summarized in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' For the sake of uniformity, we refer to the data synthesized by data distillation techniques as a data summary henceforth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Inspired by the definition of coresets (Bachem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' 2017),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' we formally define an ϵ−approximate data summary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' and the data distillation task as follows: 2 50K Real Training Images Dataset Distillation 10 Synthetic Training Images Train Train 00000 OOO Similar Test PerformanceData Distillation Trajectory Matching MTT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' HABA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' TESLA Distribution Matching DM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' CAFE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' IT-GAN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' KFS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' GCDM Gradient Matching DC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' DSA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' DCC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' IDC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' GCOND,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' DOSCOND Meta-Model Matching DD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' KIP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' RFAD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' FREPO,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' LINBA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' DISTILL-CF [HQ Image Link] Figure 2: A taxonomy of existing data distillation approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' (ϵ−approximate data summary) Given a learning algorithm Φ, let θD, θDsyn represent the optimal set of parameters for Φ estimated on D and Dsyn, and ϵ ∈ R+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' we define an ϵ−approximate data summary as one which satisfies: sup { | l (ΦθD(x), y) − l (ΦθDsyn(x), y) | }x∼X y∼Y ≤ ϵ (1) Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' (Data distillation) Given a learning algorithm Φ, let θD, θDsyn represent the optimal set of parameters for Φ estimated on D and Dsyn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' we define data distillation as optimizing the following: arg min Dsyn,n � sup { | l (ΦθD(x), y) − l (ΦθDsyn(x), y) | }x∼X y∼Y � (2) From Definition 3, we highlight three cornerstones of evaluating data distillation methods: (1) Performance: downstream evaluation of models trained on the synthesized data summary vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' the full dataset (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', accuracy, FID, nDCG, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' (2) Efficiency: how quickly can models reach full-data performance (or even exceed it), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', the scaling of n vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' downstream task-performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' and (3) Transferability: how well can data summaries generalize to a diverse pool of learning algorithms, in terms of downstream evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' No free lunch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' The universal “No Free Lunch” theorem (Wolpert & Macready, 1997) applies to data distillation as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' For example, looking at the transferability of a data summary, it is strongly dependent on the set of encoded inductive biases, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', through the choice of the learning algorithm Φ used while distilling, as well as the objective function l(·, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Such biases are unavoidable for any data distillation technique, in a sense that learning algorithms closely following the set of encoded inductive biases, will be able to generalize better on the data summary than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Keeping these preliminaries in mind, we now present a formal framework for data distillation, encapsulating existing data distillation approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Notably, the majority of existing techniques intrinsically solve a bilevel optimization problem, which are tractable surrogates of Equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' The inner-loop typically optimizes a representative learning algorithm on the data summary, and using the optimized learning algorithm, the outer-loop optimizes a tractable proxy of Equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Some common assumptions that existing data distillation techniques follow are: (1) static-length data summary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', n is fixed and is treated as a tunable hyper-parameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' and (2) we have on-demand access to the target dataset D which is also assumed to be iid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Notably, the outer-loop optimization of Dsyn happens simply through gradient descent (GD) on the analogously defined Xsyn ∈ Rn×dim(X), which is instantiated as free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Note that the labels, Ysyn ∈ Rn×dim(Y), can be similarly optimized through GD as well (Bohdal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' For the sake of notational clarity, we will interchangeably use optimization of Dsyn or (Xsyn, Ysyn) henceforth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='1 Data Distillation by Meta-model Matching Meta-model matching-based data distillation approaches fundamentally optimize for the transferability of models trained on the data summary when generalized to the original dataset: arg min Dsyn LD � θDsyn� s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' θDsyn ≜ arg min θ LDsyn(θ), (3) where intuitively, the inner-loop trains a representative learning algorithm on the data summary until convergence, and the outer-loop subsequently optimizes the data summary for the transferability of the optimized learning algorithm to the original dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Besides common assumptions mentioned earlier, the key simplifying assumption for this family of methods is that a perfect classifier exists and can be estimated on D, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', ∃ θD s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' l(ΦθD(x), y) = 0, ∀x ∼ X, y ∼ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Plugging the second assumption along with the iid assumption of D in Equation (2) directly translates to Equation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Despite the assumption, Equation (3) is highly expensive both in terms of computation time and memory, due to which, methods from this family typically resort to making further assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' (2018) (DD) originally proposed the task of data distillation, and used the meta-model matching framework for optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' DD makes the expensive optimization in Equation (3) more efficient by performing (1) local optimization à la stochastic gradient descent (SGD) in the inner-loop, and (2) outer-loop optimization using Truncated Back-Propagation Through Time (TBPTT), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', unroll a limited number of inner-loop optimization steps while optimizing the outer-loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Formally, the modified optimization objective for DD is as follows: arg min Dsyn E θ0∼Pθ [LD (θT )] s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' θt+1 ← θt − η · ∇θLDsyn(θt), (4) where Pθ is a parameter initialization distribution of choice, T accounts for the truncation in TBPTT, and η is a tunable learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Notably, TBPTT has been associated with drawbacks such as (1) computationally expensive to unroll the inner-loop at each outer-loop update (Vicol et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' (2) bias involved with truncated unrolling (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' and (3) poorly conditioned loss landscapes, particularly with long unrolls (Metz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Consequently, the TBPTT framework was empirically shown to be ineffective for data distillation in subsequent works (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' However, recent work (Deng & Russakovsky, 2022) claims that using momentum-based optimizers and longer unrolling of the inner-loop can greatly improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' We delay a deeper discussion of this work to Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='5 for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Analogously, a separate line of work focuses on using Neural Tangent Kernel (NTK) (Jacot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2018) based algorithms to solve the inner-loop in closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' As a brief side note, the infinite-width correspondence states that performing Kernelized Ridge Regression (KRR) using the NTK of a given neural network, is equivalent to training the same ∞-width neural network with L2 reconstruction loss for ∞ SGD-steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' These “∞-width” neural networks have been shown to perform reasonably compared to their finite-width counterparts, while also being solved in closed-form (see Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' (2020) for a detailed analysis on finite vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' infinite neural networks for image classification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' KIP uses the NTK of a fully-connected neural network (Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2021a), or a convolutional network (Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2021b) in the inner-loop of Equation (3) for efficient data distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' More formally, given the NTK K : X × X �→ R of a neural network architecture, KIP optimizes the following objective: arg min Xsyn,Ysyn ��Y − KXXsyn · (KXsynXsyn + λI)−1 · Ysyn ��2 , (5) where KAB ∈ R|A|×|B| represents the gramian matrix of two sets A and B, and whose (i, j)th element is defined by K(Ai, Bj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Although KIP doesn’t impose any additional simplifications to the meta-model matching framework, it has an O(|D| · n · dim(X)) time and memory complexity, limiting its scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Subsequently, RFAD (Loo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2022) proposes using (1) the light-weight Empirical Neural Network Gaussian Process (NNGP) kernel (Neal, 2012) instead of the NTK;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' and (2) a classification loss (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', NLL) instead of the L2-reconstruction loss for the outer-loop to get O(n) time complexity while also having better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' On a similar note, FRePO (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2022b) decouples the feature extractor and a linear classifier in Φ, and alternatively optimizes (1) the data summary along with the classifier, and (2) the feature 4 extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' To be precise, let fθ : X �→ X ′ be the feature extractor, gψ : X ′ �→ Y be the linear classifier, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Φ(x) ≡ gψ(fθ(x)) ∀x ∈ X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' the optimization objective for FRePO can be written as: arg min Xsyn,Ysyn E θ0∼Pθ � T � t=0 ���Y − Kθt XXsyn · (Kθt XsynXsyn + λI)−1 · Ysyn ��� 2 � s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' θt+1 ← θt − η · E (x,y)∼Dsyn [∇θl(gψ(fθ(x)), y)] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Kθ XsynXsyn ≜ fθt(Xsyn)fθt(Xsyn)T , (6) where T represents the number of inner-loop update steps for the feature extractor fθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Notably, (1) a wide architecture for fθ is crucial for distillation quality in FRePO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' and (2) despite the bilevel optimization, FRePO is shown to be more scalable compared to KIP (Equation (5)), while also being more generalizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='2 Data Distillation by Gradient Matching Gradient matching based data distillation, at a high level, performs one-step distance matching on (1) the network trained on the target dataset (D) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' (2) the same network trained on the data summary (Dsyn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' In contrast to the meta-model matching framework, such an approach circumvents the unrolling of the inner-loop, thereby making the overall optimization much more efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' First proposed by Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' (2021) (DC), data summaries optimized by gradient-matching significantly outperformed heuristic data samplers, principled coreset construction techniques, as well as TBPTT-based data distillation proposed by Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Formally, given a learning algorithm Φ, DC solves the following optimization objective: arg min Dsyn E θ0∼Pθ c ∼ C � T � t=0 D � ∇θLDc(θt), ∇θLDc syn(θt) �� s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' θt+1 ← θt − η · ∇θLDsyn(θt), (7) where T accounts for model similarity T-steps in the future, and D : R|θ| × R|θ| �→ R is a distance metric of choice (typically cosine distance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' In addition to assumptions imposed by the meta-model matching framework (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='1), gradient-matching assumes (1) inner-loop optimization of only T steps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' (2) local smoothness: two sets of model parameters close to each other (given a distance metric) imply model similarity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' and (3) first-order approximation of θD t : instead of exactly computing the training trajectory of optimizing θ0 on D (say θD t );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' perform first-order approximation on the optimization trajectory of θ0 on the much smaller Dsyn (say θDsyn t ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', approximate θD t as a single gradient-descent update on θDsyn t−1 using D rather than θD t−1 (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Subsequently, numerous other approaches have been built atop this framework with subtle variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' DSA (Zhao & Bilen, 2021) improves over DC by performing the same image-augmentations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', crop, rotate, jitter, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=') on both D and Dsyn while optimizing Equation (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Since these augmentations are universal and are applicable across data distillation frameworks, augmentations performed by DSA have become a common part of all methods proposed henceforth, but we omit them for notational clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' DCC (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2022b) further modifies the gradient-matching objective to incorporate class contrastive signals inside each gradient-matching step and is shown to improve stability as well as performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' With θt evolving similarly as in Equation (7), the modified optimization objective for DCC can be written as: arg min Dsyn E θ0∼Pθ � T � t=0 D � E c∈C [∇θLDc(θt)] , E c∈C � ∇θLDcsyn(θt) ��� (8) Most recently, Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' (2022) (IDC) extend the gradient matching framework by: (1) multi-formation: to synthesize a higher amount of data within the same memory budget, store the data summary (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', images) in a lower resolution to remove spatial redundancies, and upsample (using e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', bilinear, FSRCNN (Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2016)) to the original scale while usage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' and (2) matching gradients of the network’s training trajectory over the full dataset D rather than the data summary Dsyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' To be specific, given a k× upscaling function f : Rd×d �→ Rkd×kd, the modified optimization objective for IDC can be formalized as: arg min Dsyn E θ0∼Pθ c ∼ C � T � t=0 D � ∇θLDc(θt), ∇θLf(Dcsyn)(θt) �� s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' θt+1 ← θt − η · ∇θLD(θt) (9) 5 θ𝒟𝗌𝗒𝗇 t−1 θ𝒟𝗌𝗒𝗇 t θ𝒟𝗌𝗒𝗇 t+1 θ𝒟𝗌𝗒𝗇 t+2 θ𝒟𝗌𝗒𝗇 t+N θ𝒟 t−1 θ𝒟 t θ𝒟 t+1 θ𝒟 t+2 θ𝒟 t+M θ0 ∼ Pθ ˜θ𝒟 t ℒ𝒟 (θ𝒟𝗌𝗒𝗇 t ) ≜ 𝔼(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='y)∼𝒟 [l (Φ θ𝒟𝗌𝗒𝗇 t (x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' y)] Update by minimizing: 𝒟𝗌𝗒𝗇 Meta-Model Matching Gradient Matching Trajectory Matching D ( ˜θ𝒟 t+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' θ𝒟𝗌𝗒𝗇 t+i ) Minimize: D (θ𝒟 t+M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' θ𝒟𝗌𝗒𝗇 t+N ) D (θ𝒟 t+M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' θ𝒟 t ) Minimize: ( ) 𝖬 ≫ 𝖭 Update on 𝒟 Update on 𝒟𝗌𝗒𝗇 ˜θ𝒟 t+1 ˜θ𝒟 t+2 ˜θ𝒟 t+3 ˜θ𝒟 t+N+1 [HQ Image Link] Figure 3: The underlying optimization in various data distillation frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' (2022) further hypothesize that training models on Dsyn instead of D in the inner-loop has two major drawbacks: (1) strong coupling of the inner- and outer-loop resulting in a chicken-egg problem (McLachlan & Krishnan, 2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' and (2) vanishing network gradients due to the small size of Dsyn, leading to an improper outer-loop optimization for gradient-matching based techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='3 Data Distillation by Trajectory Matching Cazenavette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' (2022) proposed MTT which aims to match the training trajectories of models trained on D vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Dsyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' More specifically, let {θD t }T t=0 represent the training trajectory of training Φθ on D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' trajectory matching algorithms aim to solve the following optimization: arg min Dsyn,η E θ0∼Pθ � � T −M � t=0 D � θD t+M, θDsyn t+N � D � θD t+M, θD t � � � s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' θDsyn t+i+1 ← θDsyn t+i − η · ∇θLDsyn(θDsyn t+i ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' θDsyn t+1 ← θD t − η · ∇θLDsyn(θD t ), (10) where D : R|θ| × R|θ| �→ R is a distance metric of choice (typically L2 distance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Such an optimization can intuitively be seen as optimizing for similar quality models trained with N SGD steps on Dsyn, compared to M ≫ N steps on D, thereby invoking long-horizon trajectory matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Notably, calculating the gradient of Equation (10) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Dsyn encompasses gradient unrolling through N-timesteps, thereby limiting the scalability of MTT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' On the other hand, since the trajectory of training Φθ on D, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', {θD t }T t=0 is independent of the optimization of Dsyn, it can be pre-computed for various θ0 ∼ Pθ initializations and directly substituted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Similar to gradient matching methods (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='2), the trajectory matching framework also optimizes the first-order distance between parameters, thereby inheriting the local smoothness assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' As a scalable alternative, Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' (2022b) proposed TESLA, which re-parameterizes the parameter-matching loss of MTT in Equation (10) (specifically when D is set as the L2 distance), using linear algebraic manipulations to make the bilevel optimization’s memory complexity independent of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Furthermore, TESLA uses learnable soft-labels (Ysyn) during the optimization for an increased compression efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='4 Data Distillation by Distribution Matching Even though the aforementioned gradient-matching or trajectory-matching based data distillation techniques have been empirically shown to synthesize high-quality data summaries, the underlying bilevel optimization, however, is oftentimes an expensive procedure both in terms of computation time and memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' To this 6 end, distribution-matching techniques solve a correlated proxy task which restricts the optimization to a single-level, leading to a much improved scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' More specifically, instead of matching the quality of models on D vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Dsyn, distribution-matching techniques directly match the distribution of data in D vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Dsyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' The key assumption for this family of methods is that two datasets which are similar according to a particular distribution divergence metric, also lead to similarly trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' First proposed by Zhao & Bilen (2023), DM uses (1) numerous parametric encoders to cast high-dimensional data into respective low-dimensional latent spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' and (2) an approximation of the Maximum Mean Discrepancy to compute the distribution mismatch between D and Dsyn in each of the latent spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' More precisely, given a set of k encoders E ≜ {ψi : X �→ Xi}k i=1, the optimization objective can be written as: arg min Dsyn E ψ∼E c ∼ C ����� E x∼Dc [ψ(x)] − E x∼Dcsyn [ψ(x)] ���� 2� (11) DM uses a set of randomly initialized neural networks (with the same architecture) to instantiate E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' They observe similar performance when instantiated with more meaningful, task-optimized neural networks, despite it being much less efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' CAFE (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2022) further refines the distribution-matching idea by: (1) solving a bilevel optimization problem for jointly optimizing a single encoder (Φ) and the data summary, rather than using a pre-determined set of encoders (E);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' and (2) assuming a neural network encoder (Φ), match the latent representations obtained at all intermediate layers of the encoder instead of only the last layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Formally, given a (L + 1)-layer neural network Φθ : X �→ Y where Φl θ represents Φ’s output at the lth layer, the optimization problem for CAFE can be specified as: arg min Dsyn E c ∼ C � L � l=1 ���� E x∼Dc � Φl θt(x) � − E x∼Dcsyn � Φl θt(x) ����� 2 − β · E (x,y)∼Dc [log ˆp(y|x, θt)] � s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' θt+1 ← θt − η · ∇θLDsyn(θt) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' ˆp(y|x, θ) ≜ softmax y �� ΦL θ (x), E x′∼Dy syn � ΦL θ (x′) ��� , (12) where ˆp(·|·, θ) intuitively represents the nearest centroid classifier on Dsyn using the latent representations obtained by last layer of Φθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Analogously, IT-GAN (Zhao & Bilen, 2022) also uses the distribution-matching framework in Equation (11) to generate data that is informative for model training, in contrast to the traditional GAN (Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2014) which focuses on generating realistic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='5 Data Distillation by Factorization All of the aforementioned data distillation frameworks intrinsically maintain the synthesized data summary as a large set of free parameters, which are in turn optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Arguably, such a setup prohibits knowledge sharing between synthesized data points (parameters), which might introduce data redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' On the other hand, factorization-based data distillation techniques parameterize the data summary using two separate components: (1) bases: a set of mutually independent base vectors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' and (2) hallucinators: a mapping from the bases’ vector space to the joint data- and label-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' In turn, both the bases and hallucinators are optimized for the task of data distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Formally, let B ≜ {bi ∈ B}|B| i=1 be the set of bases, and H ≜ {hi : B �→ X × Y}|H| i=1 be the set of hallucinators, then the data summary is parameterized as Dsyn ≜ {h(b)}b∼B, h∼H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Even though such a two-pronged approach seems similar to generative modeling of data, note that unlike classic generative models, (1) the input space consists only of a fixed and optimized set of latent codes and isn’t meant to take any other inputs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' and (2) given a specific B and H, we can generate at most |B| · |H| sized data summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Notably, such a hallucinator-bases data parameterization can be optimized using any of the aforementioned data optimization frameworks (Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='1 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='4) This framework was concurrently proposed by Deng & Russakovsky (2022) (we take the liberty to term their unnamed model as “Lin-ear Ba-ses”) and Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' (2022c) (HaBa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' LinBa modifies the general hallucinator-bases framework by assuming (1) the bases’ vector space (B) to be the same as the task input 7 space (X);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' and (2) the hallucinator to be linear and additionally conditioned on a given predictand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' More specifically, the data parameterization can be formalized as follows: Dsyn ≜ � (y HT B, y) � y∼C H∼H s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' B ∈ R|B|×dim(X) ≜ [bi ∈ X]|B| i=1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' H ≜ � Hi ∈ R|B|×|C|�|H| i=1 , (13) where for the sake of notational simplicity, we assume y ∈ R|C| represents the one-hot vector of the label for which we want to generate data, and the maximum amount of data that can be synthesized n ≤ |C| · |H|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Since the data generation (Equation (13)) is an end-to-end differentiable procedure, both B and H are jointly optimized using the TBPTT framework discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='1, albeit with some crucial modifications for vastly improved performance: (1) using momentum-based optimizers instead of vanilla SGD in the inner-loop;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' and (2) longer unrolling (≥ 100 steps) of the inner-loop during TBPTT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' (2022c) (HaBa) relax the linear and predictand-conditional hallucinator assumption of LinBa, equating to the following data parameterization: Dsyn ≜ { (h(b), y) }b,y∼B h∼H s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' B ≜ { (bi ∈ X, yi ∈ Y) }|B| i=1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' H ≜ {hθi : X �→ X}|H| i=1 , (14) where B and H are optimized using the trajectory matching framework (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='3) with an additional contrastive constraint to promote diversity in Dsyn (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' (2022c), Equation (6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Following this setup, HaBa can generate at most |B| · |H| sized data summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Furthermore, one striking difference between HaBa (Equation (14)) and LinBa (Equation (13)) is that to generate each data point, LinBa uses a linear combination of all the bases, whereas HaBa generates a data point using a single base vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' (2022a) (KFS) further build atop this framework by maintaining a different bases’ vector space B from the data domain X, such that dim(B) < dim(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' This parameterization allows KFS to store an even larger number of images, with a comparable storage budget to other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Formally, the data parameterization for KFS can be specified as: Dsyn ≜ � c∈C { (h(b), c) }b∼Bc h∼H s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' B ≜ � c∈C Bc ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Bc ≜ {bc i ∈ B}B i=1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' H ≜ {hθi : B �→ X}|H| i=1 , (15) where KFS stores B bases per class, equivalent to a total of n = |C| · B · |H| sized data summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Following this data parameterization, B and H are optimized using the distribution matching framework for data distillation (Equation (11)) to ensure fast, single-level optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Data Distillation vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Data Compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' We highlight that it is non-trivial to ensure a fair comparison between data distillation techniques that (1) are “non-factorized”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', maintain each synthesized data point as a set of free-parameters (Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='1 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' and (2) use factorized approaches discussed in this section to efficiently organize the data summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' If we use the size of the data summary (n) as the efficiency metric, factorized approaches are adversely affected as they need a much smaller storage budget to synthesize the same-sized data summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' On the other hand, if we use “end-to-end bytes of storage” as the efficiency metric, non-factorized approaches are adversely affected as they perform no kind of data compression, but focus solely on better understanding the model-to-data relationship through the lens of optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' For a better intuition, one can apply posthoc lossless compression (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', Huffman coding) on data synthesized by non-factorized data distillation approaches to fit more images in the same storage budget (Schirrmeister et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Such techniques unintentionally deviate from the original intent of data distillation, and progress more toward better data compression techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' As a potential solution, we encourage the community to consider reporting results for both scenarios: a fixed data summary size n, as well as fixed bytes-of-storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Nonetheless, for the ease of empirical comparison amongst the discussed data distillation techniques, we provide a collated set of results over four image-classification datasets in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' 8 Table 1: Comparison of data distillation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Each method (1) synthesizes the data summary on the train-set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' (2) unless mentioned, trains a 128-width ConvNet (Gidaris & Komodakis, 2018) on the data summary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' and (3) evaluates it on the test-set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Confidence intervals are obtained by training at least 5 networks on the data summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' LinBa (No Fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=') represents LinBa with the no factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Methods evaluated using KRR are marked as (∞-Conv) or (∞-FC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' The equivalent storage-in-bytes is used for factorization-based techniques instead of IPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' The best method in their category is emboldened, the best-overall non-factorized method evaluated on ConvNet is colored orange, and the best-overall factorized method is colored blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Dataset MNIST CIFAR-10 CIFAR-100 Tiny ImageNet Imgs/Class (IPC) 1 10 50 1 10 50 1 10 50 1 10 50 Baselines Random 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='9 ±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='5 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='1 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='9 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='9 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='2 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='4 ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='0 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='0 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='2 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='4 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='2 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='3 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='6 ±0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2022b) 15 (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2022), 16 (Deng & Russakovsky, 2022), 17 (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2022c), 18 (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2022a) 9 Data Distillation 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Users 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='1 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='2 Items (Movies/Ads/Songs) Items (Movies/Ads/Songs) Fake Users Test accuracies GCN: 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='4% SGC: 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='6% APPNP: 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='8% GraphSAGE: 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='1% ("!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=',$!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', %′) Condense (",$,%) Test accuracies GCN: 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='9% SGC: 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='5% APPNP: 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='3% GraphSAGE: 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='0% 153,932 training nodes 154 training nodes Test accuracies GCN: 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='4% SGC: 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='6% APPNP: 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='8% GraphSAGE: 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='1% ("!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=',$!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', %′) Condense (",$,%) Test accuracies GCN: 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='9% SGC: 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='5% APPNP: 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='3% GraphSAGE: 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='0% 153,932 training nodes 154 training nodes Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip Lorem sit adipiscing do incididunt et aliqua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Ut minim nostrud laboris aliquip Test accuracies GCN: 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='4% SGC: 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='6% APPNP: 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='8% GraphSAGE: 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='1% ("!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=',$!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', %′) Condense (",$,%) Test accuracies GCN: 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='9% SGC: 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='5% APPNP: 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='3% GraphSAGE: 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='0% 153,932 training nodes 154 training nodes [HQ Image Link] Figure 4: Overview of distilling data for a few commonly observed data modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' 3 Data Modalities Having learned about different kinds of optimization frameworks for data distillation, we now discuss an orthogonal (and important) aspect of data distillation – what kinds of data can data distillation techniques summarize?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' From continuous-valued images to heterogeneous, discrete, and semi-structured graphs, the underlying data for each unique application of machine learning has its own modality, structure, and set of assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' While the earliest data distillation techniques were designed to summarize images for classification, recent steps have been taken to expand the horizon of data distillation into numerous other scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' In what follows, we categorize existing data distillation techniques as per their intended data modality, while also discussing their unique challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' A large-portion of existing data distillation techniques are designed for image classification data (Cazenavette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Deng & Russakovsky, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2022c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Loo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Zhao & Bilen, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2022b) simply because images have a real-valued, continuous data-domain (X ≡ Rd×d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' This allows SGD-based optimization directly on the data, which is treated as a set of free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Intuitively, incrementally changing each pixel value can be treated as slight perturbations in the color space, and hence given a suitable data distillation loss, can be naïvely optimized using SGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Textual data is available in large amounts from sources like websites, news articles, academic manuscripts, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', and is also readily accessible with datasets like the common crawl1 which sizes up to almost 541TB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Furthermore, with the advent of large language models (LLM) (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Thoppilan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2022), training such models from scratch on large datasets has become an increasingly expensive procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Despite recent efforts in democratizing LLM training (Geiping & Goldstein, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Scao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Wolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2020), effectively distilling large-scale textual data as a solution is yet to be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' The key bottlenecks for distilling textual data are: (1) the inherently discrete nature of data, where a token should belong in a limited vocabulary of words;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' (2) the presence of a rich underlying structure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', sentences of words (text) obey fixed patterns according to a grammar;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' and (3) richness of context, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', a given piece of text could have wildly different semantic interpretations under different contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Sucholutsky & Schonlau (2021) take a latent-embedding approach to textual data distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' On a high level, to circumvent the discreteness of the optimization, the authors perform distillation in a continuous embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' More specifically, assuming access to a latent space specified by a fixed text-encoder, the authors learn continuous representations of each word in the distilled text and optimize it using the 1https://commoncrawl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='org/the-data/ 10 50K Real Training Images Dataset Distillation 10 Synthetic Training Images Train Train 00000 OOO Similar Test PerformanceTBPTT data-distillation framework proposed by Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' (2018) (Equation (4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Finally, the distilled text representations are decoded by following a simple nearest-neighbor protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' A wide variety of data and applications can inherently be modeled as graphs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', user-item interactions (Mittal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Sachdeva & McAuley, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2020), social networks (Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2019), autonomous driving (Casas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Sachdeva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2022b), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Taking the example of social networks, these user-user graphs in the modern-era easily scale up to the billion-scale (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2021), calling for principled scaling solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Graph distillation could trivially solve a majority of the scale challenges, but synthesizing tiny, high-fidelity graphs has the following hurdles: (1) nodes in a graph can be highly abstract, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', users, products, text articles, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' some of which could be discrete, heterogeneous, or even simply numerical IDs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' (2) graphs follow a variety of intrinsic patterns (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', spatial (Kipf & Welling, 2017)) which need to be retained in the distilled graphs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' and (3) quadratic size of the adjacency matrix could be computationally prohibitive even for moderate-sized graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' (2022b) propose GCond which distills graphs in the inductive node-classification setting, specified by its node-feature matrix X, adjacency matrix A, and node-target matrix Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' GCond distills the given graph by learning a synthetic node-feature matrix Xsyn, and using Xsyn to generate Asyn ≜ fθ(Xsyn) which can be realized, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', through a parametric similarity function simθ(·, ·) between the features of two nodes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', Ai,j syn ≜ σ(simθ(Xi syn, Xj syn)), where σ(·) is the sigmoid function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Finally, both Xsyn and θ are optimized using the gradient-matching framework proposed by Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' (2021) (Equation (7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Another work (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2022a) (GCDM) shares the same framework as GCond but instead uses the distribution matching framework proposed by Zhao & Bilen (2023) (Equation (11)) to optimize Xsyn and θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Extending to a graph-classification setting, Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' (2022a) further propose DosCond with two major changes compared to GCond: (1) instead of parameterizing the adjacency matrix using a similarity function on Xsyn, they maintain a free-parameter matrix Ω with the same size as the adjacency matrix, and sample each Ai,j syn entry through an independent Bernoulli draw on Ωi,j as the prior using the reparameterization trick (Maddison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Such a procedure ensures differentiability as well as discrete matrix synthesis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' and (2) Xsyn and Ω are still optimized using the gradient-matching framework (Equation (7)), albeit with only a single-step, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', T = 1 for improved scalability and without empirically observing a loss in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Recommender Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' The amount of online user-feedback data available for training recommender systems is rapidly increasing (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Furthermore, typical user-facing recommender systems need to be periodically re-trained (Naumov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2019), which adds to requirements for smarter data summarization solutions (see Sachdeva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' (2022c) for background on sampling recommender systems data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' However, distilling recommender systems data has the following challenges: (1) the data is available in the form of abstract and discrete (userID, itemID, relevance) tuples, which departs from the typical (features, label) setup;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' (2) the distribution of both user- and item-popularity follows a strong power-law which leads to data scarcity and unstable optimization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' and (3) the data inherits a variety of inherent structures, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', sequential patterns (Kang & McAuley, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Sachdeva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2019), user-item graph patterns (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2019), item-item co-occurrence patterns (Steck, 2019), missing-not-at-randomness (Sachdeva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Schnabel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2016), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Sachdeva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' (2022a) propose Distill-CF which distills implicit-feedback recommender systems data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', when the observed user-item relevance is binary (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', click or no-click).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Such data can be visualized as a binary user-item matrix R where each row represents a single user, and each column represents an item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' On a high-level, Distill-CF synthesizes fake users along with their item-consumption histories, visualized as a synthetic user-item matrix Rsyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Notably, to preserve semantic meaning, the item-space in Rsyn is the same as in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' To alleviate the data discreteness problem, Distill-CF maintains a sampling-prior matrix Ω which has the same size as Rsyn, and can in-turn be used to generate Rsyn using multi-step Gumbel sampling with replacement (Jang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2017) for each user’s prior in Ω (equivalent to each row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Such a formulation automatically also circumvents the dynamic user- and item-popularity artifact in recommender systems data, which can analogously be controlled by the row- and column-wise entropy of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Finally, Ω is optimized using the meta-model matching framework proposed by Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' (2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Notably, Sachdeva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' (2022a) also propose infinite-width autoencoders which suit the task of item recommendation while also leading to closed-form computation of the inner-loop in the meta-model matching framework (Equation (5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' 11 4 Applications While the data distillation task was originally designed to accelerate model training, there are numerous other applications of a high-fidelity data summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Below we briefly discuss a few such promising applications, along with providing pointers to existing works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Differential Privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Data distillation was recently shown to be a promising solution for differential privacy as defined by Dwork (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' (2022) show that data distillation techniques can perform better than existing state-of-the-art differentially-private data generators (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Harder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2021) on both performance and privacy grounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Notably, the privacy benefits of data distillation techniques are virtually free, as none of these methods were optimized for generating differentially-private data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' (2022) further modify the gradient matching framework (Equation (7)) by clipping and adding white noise to the gradients obtained on the original dataset while optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Such a routine was shown to have better sample utility, while also satisfying strict differential privacy guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' From a completely application perspective, data distillation has been used to effectively distill sensitive medical data as well (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Neural Architecture Search (NAS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Automatic searching of neural-network architectures can alleviate the manual effort, as well as lead to better models (see Elsken et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' (2019) for a detailed review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Analogous to using model extrapolation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', extrapolating the performance of an under-trained model architecture on the full dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' data extrapolation, on the other hand, aims to train models on a small, high-fidelity data sample till convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Numerous data distillation techniques (Such et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2021) show promise on small NAS test-beds by employing the data extrapolation framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' However, Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' (2022a) show that data distillation does not perform well when evaluating diverse architectures on bigger NAS test-beds, calling for better rank-preserving data distillation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Continual Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Never-ending learning (see Parisi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' (2019) for a detailed review) has been frequently associated with catastrophic forgetting (French, 1999), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', patterns extracted from old data/tasks are easily forgotten when patterns from new data/tasks are learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Data distillation has been shown as an effective solution to alleviate catastrophic forgetting, by simply using the distilled data summary in a replay buffer that is continually updated and used in subsequent data/task training (Rosasco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Sangermano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Wiewel & Yang, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Deng & Russakovsky (2022) show further evidence of a simple compress-then-recall strategy outperforming existing state-of-the-art continual learning approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Notably, only the data summary is stored for each task, and a new model is trained (from scratch) using all previous data summaries, for each new incoming task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Federated Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Federated or collaborative learning (see Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' (2020b) for a detailed survey) involves training a learning algorithm in a decentralized fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' A standard approach to federated learning is to synchronize local parameter updates to a central server, instead of synchronizing the raw data itself (Konečn`y et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Data distillation, on the other hand, alleviates the need to synchronize large parametric models across clients and servers, by synchronizing tiny synthesized data summaries to the central server instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Subsequently, the entire training happens only on the central server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Such data distillation-based federated learning methods (Goetz & Tewari, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Xiong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2020) are shown to perform better than model-synchronization based federated learning approaches, while also requiring multiple orders lesser client-server communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' 5 Challenges & Future Directions Despite achieving remarkable progress in data-efficient learning, there are numerous framework-based, theoret- ical, and application-based directions yet to be explored in data distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' In what follows, we highlight and discuss such directions for the community to further explore, based either on early evidence or our intuition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' New data modalities & settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Extending on the discussion in Section 3, existing data distillation techniques have largely been restricted to image-classification settings, due to the easy availability of datasets, and amenable data-optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' However, taking a step back to the broad field of computer vision (see 12 Shapiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' (2001) for a thorough background), there are numerous equally important tasks that can benefit from a high-quality data summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' For example, increasing the sample efficiency of training image-generation models is both highly important due to their massive size and popularity (Ramesh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Rombach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2022), and is also highly non-trivial to fit into the existing data distillation framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Similarly, a variety of important machine learning applications don’t enjoy a continuous data domain like images, making it hard for existing data distillation techniques to scale and work as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' In addition to recent efforts on distilling discrete data like graphs (Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='b) and recommender systems (Sachdeva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2022a), developing a unified, principled data distillation framework for inherently sparse and discrete data will be useful for a variety of research communities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', text, tabular-data, extreme classification, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Better scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Existing data distillation techniques validate their prowess only in the super low-data regime (typically 1 − 50 data points per class).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' However, Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' (2022a) show that as we keep scaling the size of the data summary (larger distilled data), most distillation methods collapse to the random-sampling baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' While convergent behavior is expected, the distillation performance collapses much more rapidly with larger data summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Analogously, for data distillation to practically replace full-data training, deeper investigations of the causes and potential fixes of such scaling artifacts are highly necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Improved optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' A unifying thread across data distillation techniques is an underlying bilevel optimization, which is provably NP-hard even in the linear inner-optimization case (Vicente et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Notably, bilevel optimization has been successfully applied in a variety of other applications like meta-learning (Finn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2017), hyper-parameter optimization (Lorraine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Maclaurin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2015), neural architecture search (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2019), coreset construction (Borsos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2022a), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Despite its success, many theoretical underpinnings are yet to be explored, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', the effect of commonly-used singleton solution assumption (Franceschi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2018), the effect of over-parameterization on bilevel optimization (Vicol et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2022), connections to statistical influence functions (Bae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2022), the bias-variance tradeoff (Vicol et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=', 2021), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Clearly, an overall better understanding of bilevel optimization will directly enable the development of better data distillation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Acknowledgments We sincerely thank Zhiwei Deng, Bo Zhao, and George Cazenavette 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' 27287–27302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' PMLR, 2022a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Yanlin Zhou, George Pu, Xiyao Ma, Xiaolin Li, and Dapeng Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Distilled one-shot federated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' arXiv preprint arXiv:2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='07999, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Yongchao Zhou, Ehsan Nezhadarya, and Jimmy Ba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' Dataset distillation using neural feature regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems, 2022b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' A Notation Dataset related D ≜ {(xi ∈ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' yi ∈ Y)}|D| i=1 The target dataset to be distilled X Data domain Y Predictand domain C Set of unique classes in Y Dc ≜ {(xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' yi) | yi = c}|D| i=1 Portion of D with class c X ≜ [xi]|D| i=1 Matrix of all features in D Y ≜ [yi]|D| i=1 Matrix of all predictands in D n Size of data summary Dsyn ≜ {(˜xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' ˜yi)}n i=1 Data summary Dc syn ≜ {(˜xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' ˜yi) | ˜yi = c}n i=1 Portion of Dsyn with class c Xsyn ≜ [˜xi]n i=1 Matrix of all features in Dsyn Ysyn ≜ [˜yi]n i=1 Matrix of all predictands in Dsyn Learning related Φθ : X �→ Y Learning algorithm parameterized by θ l : Y × Y �→ R Twice-differentiable cost function LD(θ) ≜ E(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='y)∼D[l(Φθ(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' y)] Expected loss of Φ on D LDsyn(θ) ≜ E(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content='y)∼Dsyn[l(Φθ(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} +page_content=' y)] Expected loss of Φ on Dsyn General dim(A) Size of basis of A |A| Number of elements in A sup Supremum arg min θ f(θ) Optimum value of θ which minimizes f(θ) E x [f(x)] ≜ � x p(x) · f(x) Expected value of f(x) when domain of x is discrete 19' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE3T4oBgHgl3EQfCAmE/content/2301.04272v1.pdf'} diff --git a/Q9FJT4oBgHgl3EQfKCyb/content/tmp_files/2301.11463v1.pdf.txt b/Q9FJT4oBgHgl3EQfKCyb/content/tmp_files/2301.11463v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..57e1fd795689265dad04bcb0f9f73bd6b3320b34 --- /dev/null +++ b/Q9FJT4oBgHgl3EQfKCyb/content/tmp_files/2301.11463v1.pdf.txt @@ -0,0 +1,1790 @@ +1 +Nik Defense: An Artificial Intelligence Based +Defense Mechanism against Selfish Mining in +Bitcoin +Ali Nikhalat Jahromi, Ali Mohammad Saghiri, and Mohammad Reza Meybodi +Abstract—The Bitcoin cryptocurrency has received much attention recently. In the network of Bitcoin, transactions are recorded in a +ledger. In this network, the process of recording transactions depends on some nodes called miners that execute a protocol known as +mining protocol. One of the significant aspects of mining protocol is incentive compatibility. However, literature has shown that Bitcoin +mining’s protocol is not incentive-compatible. Some nodes with high computational power can obtain more revenue than their fair share +by adopting a type of attack called the selfish mining attack. In this paper, we propose an artificial intelligence-based defense against +selfish mining attacks by applying the theory of learning automata. The proposed defense mechanism ignores private blocks by +assigning weight based on block discovery time and changes current Bitcoin’s fork resolving policy by evaluating branches’ height +difference in a self-adaptive manner utilizing learning automata. To the best of our knowledge, the proposed protocol is the literature’s +first learning-based defense mechanism. Simulation results have shown the superiority of the proposed mechanism against +tie-breaking mechanism, which is a well-known defense. The simulation results have shown that the suggested defense mechanism +increases the profit threshold up to 40% and decreases the revenue of selfish attackers. +Index Terms—Bitcoin, Selfish Mining, Defense Mechanism, Learning Automata, Distributed Systems. +! +1 +INTRODUCTION +B +ITCOIN [1], a decentralized cryptocurrency, was intro- +duced by Satoshi Nakamoto in 2009 [2], [3], [4], [5]. +It has attracted much attention because of implementing +a fully trustable decentralized financial system. In Bitcoin +network, manipulating financial transactions is done us- +ing blockchain technology. In this technology, the system +records all transactions between Bitcoin clients. The secu- +rity of the blockchain depends on a cryptographic puzzle. +Some of the participants in the blockchain’s network, called +miners, try to solve a cryptographic puzzle for putting trans- +actions in the newly discovered block. Each miner hopes to +put the newly discovered block into the main chain to ob- +tain a reward. The mining process is incentive-compatible, +meaning that a miner who solves the cryptographic puzzle +gets a reward based on resource sharing in the mining +process [6], [7], [8], [9], [10]. In the next paragraph, the +mining process and its challenges are highlighted. +In the mining process [6], [7], [8], [9], [10], miners com- +pete collectively to discover and broadcast new blocks, but +sometimes more than one block is ahead of the preceding +block. A fork has been created on top of the chain in this +situation. To resolve forks, the protocol suggests adopting +and mining on the longest chain, which has the chain with +the most work, or in the situation with the same chain +length, the miner should choose the first received block. +The following two paragraphs explain the challenges of the +mining protocol of Bitcoin, which leads to selfish mining +attacks. +• +Ali Nikhalat Jahromi, Ali Mohammad Saghiri, and Mohammad Reza +Meybodi are with the Department of Computer Engineering, Amirkabir +University of Technology, Tehran, Iran. +E-mail: {ali.nikhalat,a m saghiri,mmeybodi}@aut.ac.ir +The main drawback of Bitcoin protocols is that the +system works under some assumptions that are not true in +all situations [11], [12], [13], as explained as follows. The +Bitcoin protocols require more than half of miners to be +honest, which means that they should follow the mining +process without any changes, But Eyal and Sirer [14] have +shown that this assumption might not be correct in some +situations. In their work, they introduced a new mining +strategy, denoting the selfish mining strategy version 1, +which can be abbreviated as SM1; In the SM1’s strategy, +miners try to selfishly increase their revenue by keeping +newly discovered blocks private and creating a new fork. +Honest miners continue to mine on the public chain, while +selfish miners continue to mine on the private chain they +started. If selfish miners discover more blocks than the oth- +ers, they will try to keep newly discovered blocks private. +This effort aims to develop a more extended chain basis on +the current public chain. When the public chain approaches +the selfish miners’ private chain in length, they will reveal +blocks from their private chain to the public [14], [15]. +The selfish mining attack might threaten the decentral- +ization and fairness capabilities of the bitcoin network. By +increasing the mining power threshold, selfish miners can +obtain more revenue than their fair share. Also, honest +miners persuade to leave Bitcoin’s honest mining protocol. +Honest miners prefer to join selfish miners by following +selfish mining protocols; joining selfish miners causes them +to increase selfish mining power quickly. If selfish miners’ +computational power reaches the majority threshold, it can +lead to a new protocol that discards other miners’ discov- +ered blocks. +In this paper, we propose the Nik defense mechanism, +the first artificial intelligence-based defense against selfish +arXiv:2301.11463v1 [cs.CR] 26 Jan 2023 + +2 +mining based on one of the reinforcement learning methods +called learning automata. The proposed defense mechanism +manipulates the fork-resolving policy, leading to a novel +resolving approach. In the proposed mechanism, for the +first time in the literature, a defense algorithm based on +learning automata theory and a novel weighting mecha- +nism are suggested for managing operations on the chain +in the blockchain network in a self-organized manner. In +other words, the system can reorganize itself against selfish +mining attacks to decrease the profit of selfish miners. In +addition, another policy will be suggested to change the +fail-safe parameter by learning automaton on every miner +in the blockchain. This change in the fail-safe parameter +also makes it difficult for selfish miners to decide between +publishing or keeping newly discovered blocks. +To show the effectiveness of a new proposed defense, +we compare the proposed defense with the most famous +defense, called the tie-breaking defense [14]. The rest of the +paper is organized as follows. The preliminaries related to +the selfish mining attack are given in section 2. In section +3, related work is being discussed. In section 4, we describe +the proposed defense mechanism. In section 5, the modified +version of SM1 for evaluation is discussed. Section 6 reports +the result of the experiments, and section 7, discusses the pa- +per’s limitations and problems. Finally, section 8 concludes +the paper. +2 +PRELIMINARIES +In this section, required information about the proposed +algorithm is given. We summarize information about +blockchain, selfish mining strategies, and learning au- +tomata. In the rest of this section, at first, an overview of the +Bitcoin and its transaction is given. Then the mining process, +selfish mining attack, learning automata, and properties of +an ideal defense are explained, respectively. +Bitcoin is a distributed and decentralized cryptocur- +rency. The users of Bitcoin can transfer Bitcoins by creating +a new transaction [16], [17] and sending it to a ledger based +on the blockchain. The blockchain is an append-only ledger +protected by a group of miners in the network. Miners are +rewarded for their effort to protect the blockchain against +tampering data. A transaction in the blockchain is made of +at least one input and one output. The difference between +the total amount of inputs and outputs in a transaction is +called a transaction fee [18], [19]. The transaction fee goes to +the miner, who includes the transaction in the blockchain. +The following subsection gives more details about the min- +ing process. +2.1 +Mining Process +The state of the blockchain is changed through transactions +[20]. Transactions are grouped into blocks that are appended +to the blockchain. A typical block in blockchain consists +of two major parts: header and body. The block’s header +contains the hash of the previous block, the hash of the +current block, the Merkle root of all transactions included in +this block, and a number called the nonce. The block’s body +contains transactions that the miner decided to include in +the block [2], [3]. +A valid block contains a solution to a puzzle. To solve +the puzzle, miners try to put the correct nonce in the block’s +header so that the block’s hash is smaller than the block +difficulty target [20], [21]. The block difficulty is dynamically +adjusted such that blocks are generated at an average rate +of one every ten minutes. If a miner solves the puzzle and +puts mined block in the longest chain, it will be rewarded +with Bitcoins that did not exist before and the transaction +fees of the newly created block. +The probability of mining a new block is proportional +to the computational resources used for solving the associ- +ated puzzle. Due to the nature of the mining process, the +interval between mining events exhibits high variance from +the point of view of a single miner. Consequently, miners +typically organize themselves into mining pools. All pool +members work together to mine each block and share their +revenues when one of them successfully mines a block. +While joining a pool does not change a miner’s expected +revenue, it decreases the variance and makes the monthly +revenue more predictable [14]. +2.2 +Selfish Mining Attacks +Bitcoin’s doc illustrates the approach of releasing blocks +after mining. A miner who discovers a new valid block +should release it immediately. Eyal and Sirer [14] showed +that some miners could gain revenue more than their fair +share by deviating from Bitcoin’s main mining rules, which +they called ”selfish mining.” As noted, miners try to form +a pool to decrease revenue variance. So a selfish pool with +more than 1/3 computational power unfairly changes Bit- +coin’s rewarding system by keeping a private chain and +withholding blocks that have been mined. +Saphirshtein et al. [22] used Markov Decision Process +(MDP) to investigate the profit threshold (the minimal frac- +tion of resources required for a profitable attack). They find +the bound under which the system can be considered secure +against such attacks and modify the protocol to assess their +susceptibility to selfish mining by computing the optimal +attack under different variants. They showed circumstances +under which selfish miners can hold selfish chain even +public chain is longer. +New research areas of selfish mining in machine learning +have evolved. Most of these researches have used reinforce- +ment learning methods to improve the optimality of the +attack. [23], [24] improved MDP-based solution of [22] by +applying reinforcement learning algorithms to gain more +revenue. [25] developed a new deep reinforcement learning +framework to analyze the incentives of a rational miner in +various conditions and upper bound the security threshold +of proof-of-work based blockchain. +2.3 +Learning Automata +Learning automata are adaptive decision-making devices +that operate in unknown random environments. A Learning +automaton has a finite set of actions, and each action has a +certain probability (unknown to the automaton) of getting +rewarded by its environment. The aim is to learn to choose +the optimal action (i.e., the action with the highest prob- +ability of being rewarded) through repeated interactions +with the system. If the learning algorithm is appropriately + +3 +Fig. 1. Learning automaton (LA) +chosen, then the iterative process of interacting with the +environment can result in selecting the optimal action. The +interaction between the learning automaton and the envi- +ronment is shown in Fig 1. +Learning Automata (LAs) can be classified into two main +families, fixed and variable structure learning automata [26], +[27], [28]. Variable action set learning automata used in this +paper is a sub-set of variable structure learning automata. +Variable action set learning automata can be represented +by a sextuple < β, φ, α, P, G, T >, where β a set of inputs +actions, φ is a set of internal states, α is a set of outputs, +P denotes the state probability vector governing the choice +of the state at each stage t, G is the output mapping, and +T is the learning algorithm. The learning algorithm is a +recurrence relation used to modify the state probability +vector. +Such learning automata have a finite set of r actions +denoting < α1, α2, ..., αr >. At each stage t, the action +subset ˆα ⊆ α is available for the learning automata to +choose from. Selecting the elements of ˆα is made randomly +by an external agency. Selecting an action and updating the +action probability vector in these automata are as follows. +Let S(t) = � +αi∈ˆα(t) Pi(t) presents the sum of probabilities +of the available actions in subset ˆα. Before choosing an +action, the available actions probability vector is scaled as +Equation 1. +ˆPi(t) = pi(t) +S(t) +∀αi +(1) +The crucial factor affecting the performance of the vari- +able action set learning automata is the learning algorithm +for updating the action probabilities. Let αi be the action +chosen at time t as a sample realization from distribution +p(t). Let a and b the reward and penalty parameters, and +m denotes the number of available actions. Equations for +updating the probability vector are defined by Equation 2 +for action chosen by learning automata (i = j) and Equation +3 for other actions(i ̸= j). +Pj(n + 1) = Pj(n) + aβ(1 − Pj(n)) − b(1 − β)Pj(n) +(2) +Pj(n+1) = Pj(n)−aβPj(n)+b(1−β)[ +1 +m − 1 −Pj(n)] (3) +If a=b, the learning algorithm is called the linear reward +penalty(LRP ); if b = ϵa with ϵ < 1, then the learn- +ing algorithm is called linear reward ϵ-penalty (LRϵ−P ), +if b=0, the learning algorithm is called the linear reward +inaction(LRI), if a=0, the learning algorithm is called the +penalty inaction(LP I)and finally, if a=b=0, the learning al- +gorithm is called pure chance. +Learning automata have found applications in many +areas, such as defense mechanisms in network attacks [29], +[30], peer-to-peer networks [31], Internet of Things (IoT) +[32], and neural networks [33], to mention a few. In this +paper, for the first time, learning automata is used to create +a defense mechanism against one of the blockchain attacks. +2.4 +Properties of an Ideal Defense +By explaining of problems and weaknesses of existing de- +fenses, as [34] suggested, we can enumerate the desirable +properties of an ideal defense. +• +Decentralization: +Introducing +a +trusted +server +would open a new single point of failure. Moreover, +it violates Bitcoin’s fundamental philosophy. +• +Incentive Compatibility: The expected relative rev- +enue of a miner should be proportional to mining +power. +• +Backward compatibility: Non-miners who cannot +upgrade their clients can still participate in the net- +work. This is important for hardware products such +as Bitcoin ATMs. Specifically, the following rules +should not be changed: +– +Block validity rules: A valid block in the +current Bitcoin protocol should also be valid +within the defense. +– +Reward distribution policy: All blocks in the +main chain and no other block receive block +rewards. +– +Eventual consensus: Even when an attack +happens, old and new clients should eventu- +ally reach a consensus on the main chain. +3 +RELATED WORK +In this section, existing defenses are summarized. Summa- +rizing of these defenses is based on the classification of the +similarity of methods used in the proposed defenses. The +most popular defenses are considered. +Some of the defenses need to make fundamental changes +in blockchain structure. Generally, they require significant +updates in blockchain nodes, which are incompatible with +previous versions. First, Bahack [35] proposed a defense +with the punishment rule for all miners, including honest +miners. In this defense, all miners who fork the blockchain +will be punished. The problem with this defense is the +punishment of honest miners. Solat et al. [36] introduced +Zeroblock, where miners are forced to release their blocks +within an expected time. If miners withhold their blocks +for selfish mining and do not broadcast them within the +scheduled time, the peers in the network create their own +dummy blocks. These defenses need block validity and +reward distribution changes, so network nodes should up- +date their clients to be familiar with the new protocol. [37] + +Environment +Action +Response +Learning Automaton4 +introduced another defense that will change the structure +of transactions. In this defense, the transaction has an extra +parameter named Expected Confirmation Height. A com- +parison of the Expected Confirmation Height and expected +value for published block height will use to detect selfish +mining attacks. The following paragraph will introduce de- +fenses that operate when a new fork is seen in the network. +Defenses will decrease selfish miners’ chances when they +create a fork. The most accepted solution against selfish +mining is the tie-breaking defense. The tie-breaking defense +was proposed by Eyal and Sirer [14]. When a miner learns of +computing branches of the same length, it should propagate +all of them and choose which one to mine on uniformly +at random. As they showed in their paper, the minimum +mining power needed to start selfish mining will be about +25%. Heilman [38] proposed another backward-compatible +defense against selfish mining called Freshness Preferred +(FP). In the FP solution, Heilman suggested that each miner +use an unforgeable timestamp parameter to penalize miners +that withhold blocks by comparing the latest value of the +unforgeable timestamp. A trusted party in the network +generates an unforgeable timestamp. Heilmen claimed that +the lower bound threshold of selfish mining would increase +from 25% to 32%. The problem with this solution is using a +trusted party in the network, which conflicts with the phi- +losophy of Bitcoin’s decentralization. The following para- +graph will introduce fork-resolving policy-based defenses. +Defenses that will work based on fork-resolving policy. +In these defenses, protocols are changed so that when the +selfish chain is longer than the public chain, the defense +mechanism will work, unlike the tie-breaking defense. The +first solution in this category was published by Zhang and +Preneel [34], called Publish or Perish. In Publish or Perish, +the authors suggested neglecting blocks not published in +time and appreciating blocks that incorporate links to com- +peting blocks of their predecessors. Consequently, a block +kept secure until a competing block is published contributes +to neither or both branches; hence it confers no advantage +in winning the block race. The following paragraph will +introduce a machine learning-based algorithm to detect +selfish mining attacks. +Researches have been done to identify factors that detect +selfish mining attacks [39], [40]. These researches use exist- +ing data of selfish mining attacks to create training and test +data. This research examines factors and tries to find future +research areas in this context. +In this paper, the suggested defense algorithm utilizes +machine learning to manage parameters and decisions of +mining process in a self-adaptive manner. From this per- +spective, it cannot be compared with all defense mecha- +nisms reported in the literature because there is no machine +learning-based defense mechanism in this area. From an- +other perspective, the proposed mechanism suggests a self- +adaptive algorithm that is matched with the dynamic and +distributed nature of blockchain-based systems. +4 +PROPOSED ALGORITHM: NIK DEFENSE +In this section, the proposed algorithm is presented. At first, +the system model in which selfish mining and the proposed +Fig. 2. Structure of the miner in the system model +defense will work is described. Then, the required defini- +tions for the proposed algorithm are explained. Finally, the +proposed algorithm is demonstrated. +4.1 +System Model +In this subsection, a model of the system will be presented. +In the proposed model, we consider miner nodes in the +blockchain network, so other nodes like super nodes, light +nodes, and others will be ignored. Miner nodes in the selfish +mining attack form a mining pool to obtain revenue more +than their fair share. We will consider two groups of miners +to create the most appropriate and potent model. A group +of miners follows the selfish mining strategy, which has less +than 50% of total computing power, and a group of miners +follows Bitcoin’s mining protocol without any deviations. +Fig 2 shows a snapshot of mining pools in the proposed +defense. In this figure, selfish miners which form a mining +pool are colored red, and other miners form an honest +pool colored blue. The following paragraph will present the +distribution of computing power in the network. +To explain the distribution of computing power in the +proposed model, we assume that the selfish mining pool +has a α proportion of total computing power and other +miners have a 1 − α proportion of total computing power. +Therefore a newly discovered block with a probability of α +belongs to the selfish pool, and a probability of 1−α belongs +to other miners. The following paragraph will explain the +connectivity of nodes in the proposed model. +We will ignore block propagation delay to clarify net- +work connectivity in our proposed model. This assumption +is rational because miners in Bitcoin’s network try to send +and receive blocks as fast as possible. If they have a delay +in the process of sending and receiving, their mining pro- +cess schedule will disrupt, so they can’t find a new block +efficiently. Another critical reason to ignore propagation +delay is the effort of network researchers and developers to +decrease this delay in Bitcoin’s network. We see this result in +published papers and Bitcoin Improvement Proposals [34]. +The following paragraph will discuss the arrangement of +blocks in the network. +Blocks in every node of Bitcoin form a tree structure. +Each block refers to the previous block. For simplicity, in + +ith miner +- +I +- +4 +- +I +1 +- +- +1 +1 +1 +- +1 +-: +1 +1 +- +- +- +- +15 +our model, we don’t investigate all blocks in every branch of +the block’s tree. Instead, we can consider only two branches: +The main chain or the longest chain, which results from +consensus between nodes. Another branch is the private +branch which selfish miners have created. None of the +honest nodes can’t distinguish these two branches. The +following paragraphs will discuss the model of miners in +the network. +In networks like Bitcoin with a proof-of-work consensus +mechanism, we can assume creating a new block as an event +in the mining process. On the other hand, creating a new +block doesn’t relate to time passing, so we can consider the +mining process a discrete memoryless model. In this model, +every miner, either honest or selfish, decides on the time of +finding a new block, and their chosen action continues until +the next event finds a new one. +In our proposed model, the selfish mining attacker uses +computational power to create a private chain. In arbitrary +time t, the selfish mining attacker must choose which block +of the main chain to extend for its private chain and which +block to release to increase selfish pool revenue. +If the selfish miner realizes that the honest miner finds +a new block, it will try to substitute its private block. +Parameter γ is defined as an advertisement factor. Nodes +with γ proportion of computing power would accept selfish +miner block instead of honest miner block. In terms of +block’s height, if Bitcoin’s network is in h height, the block +of the selfish miner in h height with the probability of +γ(1 − α) would accept.Each miner node in the network +is equipped with a learning automaton like in Figure 2. +Based on ith miner in the network, the required definitions +of the proposed algorithm will be explained in the following +section. +4.2 +Required Definitions +To explain the proposed algorithm, the required definitions +of the algorithm are defined here. After related definitions, +there is an example for explaining definitions in detail. +Definition 1. Competing blocks from ith miner’s perspec- +tive are defined in this definition. Two blocks compete for +being in the main chain if both are valid in obeying protocol +rules in creating transactions and blocks and both have the +same height. Usually, these blocks form a fork in the block’s +tree structure. +Definition 2. Weight calculation from ith miner’s per- +spective is defined in this definition. Blocks in competing +forks with the same height are compared to calculate the +weight of the forks. Between blocks with the same height +but in a different fork, the block created recently or has +the most timestamp wins the competition, and the related +fork’s weight will increase one unit. If two blocks have the +same timestamp, one of them will be chosen randomly as +the competition’s winner. +Definition 3. The weight of the fork from ith miner’s +perspective is defined in this definition. The weight of a +fork will be the sum of the assigned weight after calculating +the weight of blocks of the same height in different forks. In +the proposed method, the fork’s weight is denoted by W, +and the length is denoted by L. +In Fig 3, an example is designed to illustrate definitions +1, 2, and 3. In the scenario of Figure 3, 70 blocks were mined, +Fig. 3. Structure of the block in the system model (TS denotes the +timestamp of each blocks) +and the network nodes had consent about these blocks. At +height 71, three forks were created. +At first, blocks with a height of 71 will be investigated. At +this height, fork 2 will win the race, and the weight of fork +2 will increase. Because B71 of fork 2 has the most recent +timestamp. In height 72, another time, fork 2 will win. At +height 73, the only fork with the block is fork 1. So, without +any competition, fork 1 will win. By using these definitions, +fork 1 has L = 3, which is the longest chain and fork 2 has +W = 2, which is the heaviest chain. +There should be a decision between these forks. In the +proof-of-work consensus, fork 1 will be chosen because the +height of the fork (L = 3) is longer than the others. But, +using the proposed definitions, the weight of forks should +be considered in decision making. +Definition 4. This definition defines how to choose a fork +from ith miner’s perspective. Since the miner should choose +between different forks that are created by competition of +blocks in the same height and maybe one of them is created +as a result of the selfish mining attack, need to decide +between the length of fork or weight of fork as a base of +decision for choosing a fork between different forks. A new +parameter is proposed in the proposed method, like [34]. +This parameter is called fail-safe and denoted by K. When +the length of one fork’s chain is no longer by K blocks than +other forks, the miner should select weight as a base for +choosing a chain among forks. +In the Fig 3 example, a decision should be made to +choose between the weight and height of the chain as a base. +If K = 1, the length of fork 1 is longer than the length of +fork 2, so the decision is made by Length, and fork 1 will +be chosen. On the other hand, if K > 1, the length of fork +1 is no longer than the other by K. In this situation, the +weight of the chains should be considered, and fork 2 will +be chosen. +Definition 5. Fork decision making time from ith miner’s +perspective is defined in this definition. There should be a +time for the miner to check if any forks exist and, if they +exist, decide between them. In the proposed method, this +time parameter is denoted by τ. +Definition 6. Choosing a fail-safe parameter from ith + +Fork1 +B71 +B72 +B73 +TS:709 +TS:718 +TS:731 +Fork2 +B70 +B71 +B72 +TS:700 +TS:711 +TS:720 +Fork3 +B71 +TS:7106 +Fig. 4. An example of how the miner should calculate reinforcement +signal for its learning automaton +TABLE 1 +Table of Notations +Notation +Description +α +The ratio of total computational power that selfish miner has +γ +The ratio of honest miners that prefer to mine on selfish miners block +W +Total weight of chain in the created fork +L +Length of chain in the created fork +K +Fail-safe parameter for selecting between weight or height as a base for decision +τ +Time parameter for fork checking +θ +The Time Window for changing τ +β +Reinforcement signal for miner’s learning automata +miner’s perspective is defined in this definition. To defend +against selfish mining effectively, the miner must change its +fail-safe parameter. This change is done using an installed +learning automaton on the miner. Learning automaton has +three actions (Grow-Shrink-Stop). By selecting one of these +actions, the K parameter will change. By evaluating net- +work parameters periodically, one of these actions will be +chosen. In the proposed method, this period is called Time +Window, denoted by θ. θ is a positive integer coefficient of τ. +Definition 7. Calculating reinforcement signal from ith +miner’s perspective is defined in this definition. Learning +automaton should calculate reinforcement signal, which is +denoted by β. The reinforcement signal is the feedback from +the environment, which the learning automaton applied +used for updating the learning automaton’s probability vec- +tor. In the proposed method, this signal is calculated from +the analysis of each τ decision in one θ. Equation 4 shows +how to calculate β after one θ. +β = +Number of Weight Decision +Number of {Height + Weight} Decision +(4) +Since the sum of height and weight decisions in a θ +equals the Number of τ, Equation 4 can convert to Equation +5. +β = Number of Weight Decision +Number of τ in θ +(5) +The following example is designed to illustrate defini- +tions 5, 6, and 7. In this example shown in Figure 4, the +defined θ has 5 ∗ τ. Based on past decisions, the miner +should calculate the reinforcement signal for its learning +automaton. Since in τ number 3, and 5, miners had decisions +based on weight. Therefore β equals 0.4. +To summarize all notations used in the proposed +method, Table 1 shows the definition for each notation. +4.3 +Proposed Algorithm +After modeling the system and defining the necessary +prerequisites for the proposed algorithm, this section will +present the proposed algorithm in detail. At first, the pro- +posed algorithm is explained using events that happened +while running the defense method; then sub algorithms will +be described. The proposed algorithm can respond to the +occurrence of these events: +1) +One Block Receive Event +a) +If a fork exists, the miner will check the re- +lation of a new block with existing forks by +the previous hash parameter. If the miner +needs to create a fork, it will create. +2) +Decision Making Time (τ) Event +a) +Existence of a fork will be checked. The +miner must select weather by height or +weight if a fork exists. +3) +Time Window Event +a) +Existence of a fork will be checked. The +miner must select whether by height or +weight if a fork exists. +b) +Reinforcement signal will be created for +updating learning automaton. +c) +Learning automaton will choose the next +action. This action uses for updating the K +parameter. +In Algorithm 1, the proposed algorithm is shown. The +proposed algorithm consists of five sub-algorithms. These +five sub-algorithms relate to: +• +Length Calculation: Section 4.3.1 explains this algo- +rithm. +• +Weight Calculation: Section 4.3.2 explains this algo- +rithm. +• +Chain Selection: Section 4.3.3 explains this algo- +rithm. +• +Action Selection by LA: Section 4.3.4 explains this +algorithm. +• +Updating Reinforcement Signal of LA: Section 4.3.5 +explains this algorithm. +Algorithm 1 NikDefense(event) +Notation: event denotes event enum that triggered miner +to do action {TimeWindowEvent, ForkDecisionMaking- +TimeEvent, BlockReceiveEvent}, +Kmin denotes the min value for K, +Kmax denotes the max value for K, +1: Begin +2: +switch (event) +3: +case TimeWindowEvent: +4: +ForkCreationChecking() +5: +β=CalculateUpdateSignal() +6: +LA.update(β) +7: +UpdateFailSafe(Kmin, Kmax) +8: +case ForkDecisionMakingTimeEvent: +9: +ForkCreationChecking() +10: +case BlockReceiveEvent: +11: +/*Just put the block on the correct fork’s chain, and +If needed, use ForkSelection algorithm()*/ +12: +end switch +13: End + +Time Window (0) +T Number 1 +T Number 2 +T Number3 +TNumber 4 +TNumber 5 +Height Decision +HeightDecision +Weight Decision +Height Decision +Weight Decision7 +4.3.1 +Length Calculation +To calculate the length of every chain created by the fork: 1- +Getting the height of the last block before the creation of the +fork 2-Calculating difference of the last block’s height from +the height of the last block before the creation of the fork. In +Algorithm 2, the length calculation algorithm is shown. +Algorithm 2 ForkChainsLengthCalculation(CH) +Notation: N denotes the number of chains in a fork, +Chains[i] denotes an ith element of chains array in +recent τ time i ∈ [1...N], +ChainsLength[i] denotes an ith element of chains array +length in the fork i ∈ [1...N], +CH denotes the last block’s height of the main chain +before the creation of the fork, +LastBlockHeight denotes the Height of last block in i +th fork chains +1: Begin +2: +for i ← 1 to N do +3: +ChainsLength[i] = Chains[i].LastBlockHeight-CH +4: +end for +5: +return ChainsLength +6: End +4.3.2 +Weight Calculation +To calculate the weight of every chain created by fork: 1- +Calculating max length among available chains 2-According +to max length, blocks of the different chains but of the same +height will be evaluated. Between blocks with the same +height, the chain which has the block with the most recent +timestamp will win the race 3-Calculation in part 2 will be +continued until max length. If one chain is shorter than +the others, it will not conclude in comparisons of blocks +with higher heights. In Algorithm 3, the weight calculation +algorithm is shown. +Algorithm 3 ForksWeightCalculation(LM) +Notation: N denotes the number of chains in a fork, +Blocks[j][i] denotes an ith block of jth chain in recent τ +time i ∈ [1...N], +LM denotes the max length of a fork in a chains array, +MaxTimestampIndex denotes the index of max times- +tamp in blocks with the same height in different forks +1: Begin +2: +for i ← 1 to LM do +3: +MaxTimestampIndex = 1 +4: +for j ← 1 to N do +5: +if (Blocks[j][i].Timestamp > +Blocks[MinTimestampIndex][i].Timestamp) then +6: +MaxTimestampIndex = j +7: +end if +8: +ChainsWeigth[MaxTimestampIndex] += 1 +9: +end for +10: +end for +11: +return ChainsLength +12: End +4.3.3 +Chain Selection +To choose a chain among created chains by fork condition, +miner needs to make a decision. So chain selection algorithm +of the proposed defense can be described as below: 1- +Calculating length of chains 2-Sorting chain based on length +in descending order 3-If one chain is longer than the others +by K, it will select for the next mining event 4-If no chain +longer than the others by K, the weight of all chains will +calculate by the algorithm described in section 4.3.2 5- +Sorting chain base on weight in descending order 6-Heaviest +chain will be chosen for the next mining event. In Algorithm +4, the chain selection algorithm is shown. +Algorithm 4 ChainSelection() +Notation: N denotes the number of chains in a fork, +Chains[i] denotes an ith element of chains array in +recent τ time i ∈ [1...N], +LM denotes the max length of a fork in a chains array, +CH denotes the last block’s height of the main chain +before the creation of the fork, +ChainsWeight[i] denotes an ithchain’s weight in a fork, +i ∈ [1...N], +ChainsLength[i] denotes an ith element of chains array +length in the fork, i ∈ [1...N], +LastBlockHeight denotes the Height of last block in ith +fork chains, +ChosenChain denotes chosen chain after defense +1: Begin +2: +if (N > 1) then +3: +ChainsLength = ForkChainsLengthCalculation(CH) +4: +SortDescendingly(Chains, ChainsLength) +5: +if (ChainsLength [0] - ChainsLength [1] > K) then +6: +/*Decide based on Chain’s Height*/ +7: +ChosenChain = Chains[0] +8: +CH = ChosenChain.LastBlockHeight +9: +return ChosenChain +10: +else +11: +/*Decide based on Chain’s Weight*/ +12: +LM=ChainsLength[0] +13: +ChainsWeight = ChainsWeightCalculation(LM) +14: +SortDescendingly(Chains, ChainsWeight) +15: +ChosenChain = Chains[0] +16: +CH = ChosenChain.LastBlockHeight +17: +return ChosenChain +18: +end if +19: +end if +20: End +4.3.4 +Action Selection by LA +θ consists of τ time intervals, and in any of the τ time inter- +vals particular event can occur; when the θ has finished, it is +needed to make a decision about the occurrence of events in +any τ time intervals of θ. Learning automaton should make +this decision as an AI tool. Learning automaton will choose +the next K parameter by selecting an action. Chosen action +by Learning automaton can be one of these values: 1-Grow: +Choosing this action shows that the network was under +attack. So Learning automaton increase the K value by one +unit to make chain selection harder 2-Shrink: Choosing this +action indicates that the network was not under attack or the +attack was ineffective. So learning automaton decrease the +K value by one unit 3-Stop: Choosing this action by learn- + +8 +ing automaton shows that the previous chosen action in θ +was correct, and there is no need to change the K value. The +reason for selecting variable action set learning automaton is +that when the K parameter reaches the max value (Kmax), +Grow action should omit from learning automaton’s action +selection options, and when the K parameter reaches the +min value (Kmin), Shrink action should omit from learning +automaton’s action selection options. In Algorithm 5, the +action selection algorithm has been shown. +Algorithm 5 UpdateFailSafe(LA, K, Kmin, Kmax) +Notation: N denotes the number of chains in a fork, +Chains[i] denotes an ith element of chains array in +recent τ time i ∈ [1...N], +ChainsLength[i] denotes an ith element of chains array +length in the fork i ∈ [1...N], +CH denotes the last block’s height of the main chain +before the creation of the fork, +LastBlockHeight denotes the Height of last block in i +th fork chains +1: Begin +2: +if (K = Kmax) then +3: +L = LA.choose +action([’Stop’, ’Shrink’]) +4: +else if (K = Kmin) then +5: +L = LA.choose +action([’Grow’, ’Stop’]) +6: +else +7: +L = LA.choose +action([’Grow’, ’Stop’, ’Shrink’]) +8: +end if +9: +switch (L) +10: +case ’Grow’: +11: +K = K + 1 +12: +case ’Shrink’: +13: +K = K − 1 +14: +case ’Stop’: +15: +/*Do nothing about K*/ +16: +end switch +17: +return ChainsLength +18: End +4.3.5 +Updating Reinforcement Signal of LA +By ending θ and before choosing the next action, learning +automaton needs to know how the previous action was by +receiving a reinforcement signal. If the previous action was +effective, learning automaton would be rewarded, and if +the previous action was not effective, learning automaton +would be punished. Equation 4 shows how the reinforce- +ment signal will help learning automaton to improve itself. +In this equation, every τ interval in θ is considered. At +the end of τ, If the chain is selected by height (it means +that one chain is longer than the other by K and does not +need to use the weight calculation algorithm), the counter +of height selection will increase. Otherwise, the counter of +weight calculation will increase. +5 +PROPOSED ALGORITHM: SM1 STRATEGY MOD- +IFICATION +The SM1 strategy is modified in this section to evaluate +the proposed defense in the previous section. Since the pro- +posed defense has modified the mining protocol of Bitcoin +to create the first AI defense against selfish mining using +learning automata, the effort is needed to change the SM1 +strategy. The proposed defense uses θ and τ interval pa- +rameters, so these parameters should be applied to modify +the SM1 strategy. Also, in every θ, the fail-safe parameter +will change. To make an effective attack, the attacker should +approximate the K parameter using learning automaton. +In this paper, approximated K will be shown by ˜K. The +procedure of approximating ˜K is similar to updating the +fail-safe parameter algorithm in Algorithm 5. In Algorithm +6, the modified version of SM1 is shown. +There are two scenarios in selfish mining attacks that +the selfish miner should make a decision about it. Both of +these scenarios depend on the length difference between +the selfish chain kept by the selfish miner and the honest +chain that the honest miner creates. Another factor that can +be effective in modified SM1 strategy is approximated ˜K +parameter. +The first scenario is making a decision on the selfish pool +as a decision-maker succeeds in finding a new block. If the +private chain and the public chain have one block before +finding a new block by the selfish miner, the new block will +be added to the selfish chain. In this situation, the selfish +chain is ahead, and by publishing its secret blocks, obtaining +a reward for two blocks. +The second scenario is making a decision on other min- +ers which considered as honest and succeed in finding a +new block. If no private chain is maintained by the selfish +node, by receiving an honest block, no action must be taken +because the public chain wins the race, and the attack +should reset. If the private chain was one block ahead, +receiving a new block from an honest node would lose +weight calculation due to having a timestamp less than the +new received block. So to try its chance, it won’t release the +private chain. The next block (based on mining by the selfish +miner or others) will determine the winner of the game. If +the private chain was two blocks ahead, by receiving a new +block mined by an honest node, the selfish miner would +release its private chain because the new received block +wins the first block race in weight calculation by timestamp +parameter. In this situation, there is a race between the +selfish and honest branches. If the private chain was ˜K+1 +was ahead, by receiving a new block mined by an honest +node, the selfish miner will release all of the private chain +and win ˜K+1 blocks revenue. Finally, if a private chain was +more than ˜K+1 was ahead, by receiving a new block mined +by an honest node, the selfish miner will release the first +unpublished block. +6 +EVALUATION +This section will describe an evaluation of the Nik defense +against selfish mining. To evaluate the proposed defense, +we first introduce evaluation metrics. Then, various exper- +iments have been designed to assess the proposed defense +and compare it with other defenses. +6.1 +Metric +In this subsection, metrics for the evaluation of the proposed +defense algorithm are introduced. These metrics will be + +9 +used in section 6.2 experiments. In the following items, +necessary metrics are defined. +Algorithm 6 Modified SM1() +Notation: PublicChain denotes the public chain which all +miners can see, +PrivateChain denotes the chain mined by selfish miners, +PrivateBranchLength denotes the length of the selfish +chain, +∆ denotes the difference of selfish and public chain’s +length, +˜K denotes the approximate fail-safe parameter, +Kmin denotes the min value for K, +Kmax denotes the max value for K, +WeightCalculationCounter denotes the number of weight +calculations in one θ +1: Begin +2: +On Init: +3: +PublicChain ← Publicity Known Blocks +4: +PrivateChain ← Publicity Known Blocks +5: +PrivateBranchLength = 0 +6: +Mine at the head of the private chain +7: +On ending ForkDecisionMakingTime: +8: +if (ReleaseByWeightCalculation() = true) then +9: +WeightCalculationCounter += +WeightCalculation- +Counter + 1 +10: +end if +11: +On ending TimeWindow: +12: +β = +CalculateUpdateSignal(WeightCalculationCounter) +13: +LA.update(β) +14: +ApproximateFailSafe(LA, � +K,Kmin,Kmax) +15: +WeightCalculationCounter = 0 +16: +On SelfishPool found a block: +17: +∆prev=length(PrivateChain)-length(PublicChain) +18: +Append the new block to PrivateChain +19: +PrivateBranchLength = PrivateBranchLength + 1 +20: +if (∆prev = 0 and PrivateBranchLength = 2) then +21: +Publish all of the PrivateChain //Try By Luck +22: +PrivateBranchLength = 0 +23: +end if +24: +Mine at the head of the PrivateChain +25: +On other miners found a block: +26: +∆prev=length(PrivateChain)-length(PublicChain) +27: +Append the new block to PublicChain +28: +if (∆prev = 0) then +29: +PrivateChain ← PublicChain //Others win +30: +PrivateBranchLength = 0 +31: +else if (∆prev = 1) then +32: +/*Do Nothing*/ +33: +else if (∆prev = 2) then +34: +Publish all of the PrivateChain //Try By Luck +35: +PrivateBranchLength = 0 +36: +else if (∆prev = � +K + 1) then +37: +Publish all of the PrivateChain//SelfishPool wins +38: +PrivateBranchLength = 0 +39: +else if (∆prev > � +K + 1) then +40: +Publish the first unpublished block of the private +chain //SelfishPool wins +41: +end if +42: +Mine at the head of the PrivateChain +43: End +• +Relative Revenue: This metric is used to measure a +miner’s revenue based on the revenue of the other +miners in the network. Relative revenue is obtained +by the number of total accepted mined blocks by +one miner divided by the total number of blocks in +the main chain. Equation 6 is used to show how to +calculate the relative revenue of an arbitrary miner. +Number of Mined Block by ith Miner +Total Number of Mined Blocks +(6) +Followed by Equation 6, the relative revenue of the +honest miner and the selfish miner can be shown in +Equations 7 and 8. In these equations, all selfish and +honest miners group together and assume that only +two groups of miners exist in the network. +Honest Miner W in Block +Honest Miner W in Block + Selfish Miner W in Block +(7) +Selfish Miner W in Block +Honest Miner W in Block + Selfish Miner W in Block +(8) +• +Lower Bound Threshold: This metric shows the +minimum computing power that selfish miners +should provide to start the attack. The lower bound +threshold is obtained in simulations by the inter- +section point of the selfish miner’s relative revenue +diagram and ideal defense diagram. +6.2 +Experiment Results +Inspiring by previous defenses and their simulators [14], +[22], [34] the proposed algorithm was simulated by con- +verting the mining model into a Monte Carlo simulation +process. This conversion makes it possible to distribute +newly discovered blocks among selfish and honest miners +without solving a cryptographic puzzle. For reproducibility, +the developed simulator can be found on Github 1. Ex- +periments reported in this subsection have been run using +Intel Core i7 with a 2.5 GHz frequency clock. Experiments +reported in this section are divided into four parts: +1) +Experiment 1: This experiment aims to compare the +proposed defense with the tie-breaking defense as +the previous proposed selfish mining defense using +a different kind of learning automata. Reward and +penalty parameters were changed, and then results +were reported. +2) +Experiment 2: This experiment aims to check the +effect of changing the K on the proposed defense. +By changing the K interval, results of changing this +parameter have been reported. +3) +Experiment 3: This experiment aims to check the ef- +fect of changing the τ time interval on the proposed +defense. By changing the τ time interval, the result +of changing this parameter has been reported. +4) +Experiment 4: This experiment aims to check the +effect of changing the number of τ in one θ on +the proposed defense. By changing the number of +τ, results of changing this parameter have been +reported. +For all of the experiments, 10000 blocks were generated +by the simulator. The main learning automaton used in these +experiments is LRϵ−P with a = 0.1, b = 0.01. +1. https://github.com/AliNikhalat/SelfishMining + +10 +6.2.1 +Experiment 1 +This experiment is conducted to study the impact of chang- +ing reward and penalty parameters of learning automaton +on the performance of the proposed algorithm with respect +to relative revenue. The results of proposed algorithm are +compared with the tie-breaking algorithm which is a well- +known defense mechanism against selfish mining. For this +purpose, parameter K was selected from [1,3] interval. The +value of τ time interval was about the time of mining five +blocks, and one θ has ten τ time intervals. The results of tie- +breaking are compared with those obtained for the proposed +defense in five reward and penalty rate values. The results +of this experiment are shown in Figure 5.a to Figure 5.e, +all of them perform better than the tie-breaking defense. +This means that regardless of the type of learning automata, +the defense mechanism can prevent the selfish attacker by +decreasing the relative revenue. By comparing relative rev- +enue between different learning automata, LRϵ−P performs +better than the others because the reward rate is more than +the penalty rate, and the relative revenue is near the ideal +defense. So, choosing the proper reward and penalty rate +can affect the quality of defense. +(a) Comparison with Tie breaking +defense when a=0, b=0 +(b) Comparison with Tie breaking +defense when a=0.01, b=0 +(c) Comparison with Tie breaking de- +fense when a=0, b=0.01 +(d) Comparison with Tie breaking +defense when a=0.01, b=0.01 +(e) Comparison with Tie breaking +defense when a=0.1, b=0.01 +Fig. 5. Comparison with tie-breaking defense using different penalty and +reward parameters +The lower bound threshold value for starting the selfish +mining attack is shown in Table 2. This parameter can be +TABLE 2 +The Threshold for Starting the Selfish Mining Attack +P +LR−I +LP −I +LR−P +LRε−p +a +0 +0.01 +0 +0.01 +0.1 +b +0 +0 +0.01 +0.01 +0.01 +Nik +0.43 +0.44 +0.42 +0.44 +0.45 +Tie-Breaking +0.25 +0.25 +0.25 +0.25 +0.25 +used to compare starting point of the selfish mining attack’s +profitability. The lower threshold value is, the less selfish +miner attacker is successful. This can be used to show the +effectiveness of the defense. According to the results of this +experiment, the proposed defense has a higher threshold +value than the tie-breaking defense. This threshold increases +from 0.25 to 0.4. By increasing the threshold, the selfish +miner needs to increase its pool size to start an effective +selfish mining attack. +6.2.2 +Experiment 2 +This experiment is conducted to study the impact of the +parameter K on the performance of the proposed algorithm +with respect to relative revenue. For this purpose, the pa- +rameter K was tested for three intervals [1,3], [2,4] and [1,5]. +The value of parameter τ time interval is about the time +of mining five blocks, and one θ has ten τ time intervals. +The results obtained from different values of K are shown +in Figure 6. By comparing K intervals, the advantage of +choosing bigger values for K is obvious. Figure 6 shows [1, +5] interval for K value has the most powerful defense. This +can lead to two important factors: 1-learning automaton can +choose value 5 for K 2-learning automaton has the freedom +to a choose value for K in a bigger interval(choosing four +values for K). It shows that the interval with bigger values +of K and more options for K can reduce the relative revenue +of the attacker. +Fig. 6. The impact of K intervals on the proposed defense +On the other hand, the effect of changing the value of +parameter K’s on the attack threshold can be seen in Figure + +100 +K=[1,3] +K=[2,4] +K=[1,5] +80 - +.IdealDefense +UpperBound +Relative Revenue +60 +40 +20 +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +Poolsize100 +. +NikDefense +Tie Breaking +No Defense +80 +IdealDefense +Upper Bound +Relative Revenue +60 +40 +20 +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +Pool size100 +.. +Nik Defense +Tie Breaking +NoDefense +IdealDefense +80 +Upper Bound +Relative Revenue +60 +40 +20 +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +Pool size100 +NikDefense +Tie Breaking +NoDefense +IdealDefense +80 +Upper Bound +Relative Revenue +60 +40 +20 - +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +Pool size100 +NikDefense +Tie Breaking +No Defense +80 - +IdealDefense +UpperBound +Relative Revenue +60 +40 +20- +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +Pool size100 +.... +Nik Defense +Tie Breaking +NoDefense +80 - +IdealDefense +Upper Bound +Relative Revenue +60 +40 +20 - +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +Pool size11 +6. [1, 5] has the most threshold for starting an attack. The +increasing threshold of starting attack for bigger values of +K in the chosen interval can be the result of selecting a +bigger K value. Overall, the result shows the bigger values +of K leads to the less powerful attack in the network. +6.2.3 +Experiment 3 +This experiment is conducted to study the impact of the τ +time interval parameter on the performance of the proposed +algorithm with respect to relative revenue. For this purpose, +the parameter K was selected from [1,3] interval. One θ has +ten τ time intervals. τ time interval was tested for 5, 9, and +15 mining blocks time. For example, if τ = 6, the τ time +interval is about the time of mining six blocks. The results +from different values of τ time intervals are shown in Figure +7. This figure doesn’t show noticeable changes in relative +revenue metric after selecting different values for the τ time +interval. This means that small changes in the value of τ +doesn’t have a meaningful impact on the performance of +the proposed algorithm. +Another important factor shown in Figure 7 is the thresh- +old for starting the attack. It shows that a lower value +τ time interval causes an increment in the threshold for +starting the attack. As expected, this happened because +the lower τ value in a constant number of τ (ten time +intervals in this experiment) in one θ can increase the +number of θ in simulation for the specified number of +blocks. Increasing the number of θ can force the learning +automaton to examinate more decisions. By increasing the +number of decisions, the learning automaton can tune the +β parameter effectively, and as a result, the threshold for +starting defense will increase. Overall, the result shows a +lower value for τ improves the performance of the proposed +defense mechanism. +Fig. 7. The impact of τ time interval on the proposed defense +6.2.4 +Experiment 4 +This experiment is conducted to study the impact of the +number of τ time intervals parameter in one θ on the +performance of the proposed algorithm with respect to +relative revenue. For this purpose, the parameter K was +in the [1,3] interval. τ time interval was about the time +of mining five blocks, and one θ has 6, 12, and 18 τ time +intervals. The results from the different numbers for τ in +one θ are shown in Figure 8. By comparing the results of this +experiment in Figure 8, the lower number of τ has a better +impact on decreasing the relative revenue of selfish miner. +As predicted from the behavior of the learning automaton, +the less value for τ in one θ, leads to more numbers of θ +in simulation for the specified number of blocks. Increasing +the number of θ can force the learning automaton to make +more decisions. By increasing the number of decisions, the +learning automaton can tune the β parameter effectively, +and as a result, the threshold for starting defense will +increase. It means that the lower number of τ time intervals +parameter in one θ will decrease the effect of the selfish +mining attacker in the network. +In addition, Figure 8 shows that the number of τ time +intervals creates a little difference in the threshold of starting +a selfish mining attack. τ = 6 performs better than the +others. This happened because of the effective adjusting +of the β parameter. Eventually, the result shows a lower +number of τ time intervals in one θ prevents selfish mining +efficiently. +Fig. 8. The impact of a number of τ time intervals in one θ on the +proposed defense +7 +DISCUSSION +Selfish +mining +attack +shows +how +decentralization +of +blockchain and Bitcoin as one of the most important im- +plementations of blockchain can be threatened. Consid- +ering Bitcoin’s popularity, capabilities, and the existence +of this threat to Bitcoin, a solution should find to solve +this problem. Various solutions have been studied earlier. +Existing solutions suffer a lack of self-adaptability. Using +learning automata as one of the reinforcement learning tools +for designing self-adaptive systems in developing the first +AI-based defense, the profitability of selfish mining can +decrease. We know the problems and limitations of our +proposed defense. Our work is designed and implemented +in perfect conditions and may not cover all of Bitcoin’s +protocols, while Bitcoin’s network and its protocols are so +complicated. In the following, some critical questions have +been answered for better discussion. + +100 +Blocks in t = 5 +Blocks in t = 9 +Blocks in T = 15 +.Ideal Defense +80 +Upper Bound +Relative Revenue +60 +40 +20 +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +Pool size100- +.... T=6 + t = 12 + t= 18 +. Ideal Defense +80 - +Upper Bound +Revenue +60 - +Relative +40 - +20 - +0.25 +0.30 +0.35 +0.40 +0.45 +0.50 +Pool size12 +• +Q1: What are the assumptions used in the proposed +defense mechanism? +In the proposed defense mechanism, we assume that +every new block has propagated to all nodes before +discovering a new block. There are communication +delays in sending and receiving messages between +nodes. These delays are ignored. This assumption +leads to the highest propagation speed to synchro- +nize all nodes as soon as possible. +• +Q2: How to set the interval for τ time interval? +As examined by experiments performed while writ- +ing this paper, time interval has no significant im- +pact on Nik defense before setting τ time interval. +It doesn’t mean that time interval is unimportant, +but choosing a reasonable interval can help defense +mechanism. +• +Q3: How to set the number of τ time intervals in +one θ? +Before setting the number of τ time intervals, some +experiments should be done based on the type of +learning automata as we did in this paper. On the +other hand, by answering Q1 in this section, commu- +nication delays in real situations should be consid- +ered in choosing the number of τ time intervals. +• +Q4: Can different fail-safe parameters collapse the +Bitcoin network? +Choosing a fail-safe parameter can be hard as a syn- +chronization mechanism. Miners may have different +values for fail-safe parameters. Our model assumes +that all nodes in the network are synced. Different +values of the fail-safe parameter in the synced net- +work may lead to a large-scale voting system for +choosing the winning branch. At last, miners can +choose the winning branch, decreasing the selfish +attack’s profitability. +• +Q5: Why do we use AI for the Nik defense as a +self-organized mechanism? +These days, AI has been increasingly used for de- +tecting and preventing attacks. In blockchain based +networks, The events are executed very fast and +proposing self-organized and reactive mechanisms +that are able to execute appropriate operation after +execution of critical events is vital. Therefore, we +have decided to use learning automata based self- +adaptive algorithm as an AI tool to prevent selfish +mining attacks considering this potential. Blockchain +is an unknown space, and since we don’t have +any information about forks, reinforcement learning +of learning automata is a good choice for defense. +Hence learning automata were used. +• +Q6: What development hurdles can we face while +adopting the proposed mechanism? +The main development hurdle that affect the pro- +posed mechanism is the synchronization algorithms +of blocks in the network. The assumption of the +upper limit for propagation time of Nik defense can +be challenging. It should be noted that this is not +limitation of our model and other defense models +reported in [34], [35], [36] face with this problem. +8 +CONCLUSION AND FUTURE RESEARCH DIREC- +TIONS +The selfish mining attack hurt Bitcoin’s incentive com- +patibility by disrupting the rewarding system’s fairness.In +addition, the decentralization capability of Bitcoin can be +threatened by it. To solve this problem, a new defense based +on learning automata as an AI tool was proposed. The pro- +posed method suggested a set of policies to reinforce chain +selection algorithm in a self-organized manner. Experiments +have shown the superiority of the proposed defense in +Bitcoin’s network in comparison with existing mechanisms. +Compared to existing defenses, the proposed defense can +noticeably decrease the profitability of the selfish mining +attack. On the other hand, it can increase the threshold for +starting the attack’s profitability from 25% to 40% of the +network’s computational power. +Since the proposed study is the first use of an AI-based +algorithm for organizing a self-organized defense mecha- +nism, several works in the area of AI to detect and prevent +selfish mining attacks can be considered as future directions. +In this paper, learning automata theory was used as a self- +adaptive decision-maker, but many other models such as +Q-learning can be used. This means that other AI tools can +modify the proposed algorithm to have a better solution +against selfish mining attacks. +In addition, many open problems in the area of proof-of- +work consensus need to be studied and explored to develop +a more usable and efficient industrial consensus without +the selfish mining attack. Alternative consensus algorithms +and integration with other systems and architectures are +examples of several open issues that can be considered +as future directions. This study opens a new horizon for +designing defenses mechanism based on AI. +REFERENCES +[1] Nakamoto, S. Bitcoin: A peer-to-peer electronic cash system. (2008) +[2] Narayanan, A., Bonneau, J., Felten, E., Miller, A. & Goldfeder, S. +Bitcoin and cryptocurrency technologies: a comprehensive intro- +duction. (Princeton University Press,2016) +[3] Antonopoulos, A. Mastering Bitcoin: unlocking digital cryptocur- +rencies. 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International Conference On Cyber +Warfare And Security. 17 pp. 237-243 (2022) +Ali Nikhalat Jahromi He received the B.Sc de- +gree in Electrical Engineering in 2018 and then +the M.Sc degree in Computer Engineering in +2021, both from Amirkabir University of Technol- +ogy, Tehran, Iran. His research interests include +Software Systems, Parallel and Distributed Sys- +tems, and Programming Languages. +Ali Mohammad Saghiri He received the B.Sc. +degree from University of Science and Culture, +in 2007 and the M.Sc. and Ph.D. degrees from +AmirKabir University of Technology, Tehran, Iran, +in 2010 and 2017, respectively, all in computer +engineering. He published more than 40 sci- +entific papers on international conferences and +journals among which Journal of Network and +Computer Applications(JNCA), Applied Intelli- +gence, and international journal of communica- +tion systems. His research interests include the +Internet of Things, Blockchain, and Artificial Intelligence. +Mohammad Reza Meybodi He received the +B.Sc. and M.Sc. degrees in economics from +Shahid Beheshti University, Tehran, Iran, in 1973 +and 1977, respectively, the M.Sc. and Ph.D. de- +grees in computer science from Oklahoma Uni- +versity, Norman, OK, USA, in 1980 and 1983, +respectively.,He was an Assistant Professor with +Western Michigan University, Kalamazoo, MI, +USA, from 1983 to 1985, and an Associate Pro- +fessor with Ohio University, Athens, OH, USA, +from 1985 to 1991. He is currently a Full Profes- +sor with the Computer Engineering Department, Amirkabir University of +Technology, Tehran. His research interests include wireless networks, +fault tolerant systems, learning systems, parallel algorithms, soft com- +puting, and software development. + diff --git a/Q9FJT4oBgHgl3EQfKCyb/content/tmp_files/load_file.txt b/Q9FJT4oBgHgl3EQfKCyb/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e6e031408bf095b1811073d339fd4f744375ef76 --- /dev/null +++ b/Q9FJT4oBgHgl3EQfKCyb/content/tmp_files/load_file.txt @@ -0,0 +1,1024 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf,len=1023 +page_content='1 Nik Defense: An Artificial Intelligence Based Defense Mechanism against Selfish Mining in Bitcoin Ali Nikhalat Jahromi, Ali Mohammad Saghiri, and Mohammad Reza Meybodi Abstract—The Bitcoin cryptocurrency has received much attention recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In the network of Bitcoin, transactions are recorded in a ledger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In this network, the process of recording transactions depends on some nodes called miners that execute a protocol known as mining protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' One of the significant aspects of mining protocol is incentive compatibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' However, literature has shown that Bitcoin mining’s protocol is not incentive-compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Some nodes with high computational power can obtain more revenue than their fair share by adopting a type of attack called the selfish mining attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In this paper, we propose an artificial intelligence-based defense against selfish mining attacks by applying the theory of learning automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The proposed defense mechanism ignores private blocks by assigning weight based on block discovery time and changes current Bitcoin’s fork resolving policy by evaluating branches’ height difference in a self-adaptive manner utilizing learning automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' To the best of our knowledge, the proposed protocol is the literature’s first learning-based defense mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Simulation results have shown the superiority of the proposed mechanism against tie-breaking mechanism, which is a well-known defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The simulation results have shown that the suggested defense mechanism increases the profit threshold up to 40% and decreases the revenue of selfish attackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Index Terms—Bitcoin, Selfish Mining, Defense Mechanism, Learning Automata, Distributed Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 1 INTRODUCTION B ITCOIN [1], a decentralized cryptocurrency, was intro- duced by Satoshi Nakamoto in 2009 [2], [3], [4], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' It has attracted much attention because of implementing a fully trustable decentralized financial system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In Bitcoin network, manipulating financial transactions is done us- ing blockchain technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In this technology, the system records all transactions between Bitcoin clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The secu- rity of the blockchain depends on a cryptographic puzzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Some of the participants in the blockchain’s network, called miners, try to solve a cryptographic puzzle for putting trans- actions in the newly discovered block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Each miner hopes to put the newly discovered block into the main chain to ob- tain a reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The mining process is incentive-compatible, meaning that a miner who solves the cryptographic puzzle gets a reward based on resource sharing in the mining process [6], [7], [8], [9], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In the next paragraph, the mining process and its challenges are highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In the mining process [6], [7], [8], [9], [10], miners com- pete collectively to discover and broadcast new blocks, but sometimes more than one block is ahead of the preceding block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' A fork has been created on top of the chain in this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' To resolve forks, the protocol suggests adopting and mining on the longest chain, which has the chain with the most work, or in the situation with the same chain length, the miner should choose the first received block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The following two paragraphs explain the challenges of the mining protocol of Bitcoin, which leads to selfish mining attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Ali Nikhalat Jahromi, Ali Mohammad Saghiri, and Mohammad Reza Meybodi are with the Department of Computer Engineering, Amirkabir University of Technology, Tehran, Iran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' E-mail: {ali.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='nikhalat,a m saghiri,mmeybodi}@aut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='ir The main drawback of Bitcoin protocols is that the system works under some assumptions that are not true in all situations [11], [12], [13], as explained as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The Bitcoin protocols require more than half of miners to be honest, which means that they should follow the mining process without any changes, But Eyal and Sirer [14] have shown that this assumption might not be correct in some situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In their work, they introduced a new mining strategy, denoting the selfish mining strategy version 1, which can be abbreviated as SM1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In the SM1’s strategy, miners try to selfishly increase their revenue by keeping newly discovered blocks private and creating a new fork.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Honest miners continue to mine on the public chain, while selfish miners continue to mine on the private chain they started.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' If selfish miners discover more blocks than the oth- ers, they will try to keep newly discovered blocks private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' This effort aims to develop a more extended chain basis on the current public chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' When the public chain approaches the selfish miners’ private chain in length, they will reveal blocks from their private chain to the public [14], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The selfish mining attack might threaten the decentral- ization and fairness capabilities of the bitcoin network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' By increasing the mining power threshold, selfish miners can obtain more revenue than their fair share.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Also, honest miners persuade to leave Bitcoin’s honest mining protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Honest miners prefer to join selfish miners by following selfish mining protocols;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' joining selfish miners causes them to increase selfish mining power quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' If selfish miners’ computational power reaches the majority threshold, it can lead to a new protocol that discards other miners’ discov- ered blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In this paper, we propose the Nik defense mechanism, the first artificial intelligence-based defense against selfish arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='11463v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='CR] 26 Jan 2023 2 mining based on one of the reinforcement learning methods called learning automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The proposed defense mechanism manipulates the fork-resolving policy, leading to a novel resolving approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In the proposed mechanism, for the first time in the literature, a defense algorithm based on learning automata theory and a novel weighting mecha- nism are suggested for managing operations on the chain in the blockchain network in a self-organized manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In other words, the system can reorganize itself against selfish mining attacks to decrease the profit of selfish miners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In addition, another policy will be suggested to change the fail-safe parameter by learning automaton on every miner in the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' This change in the fail-safe parameter also makes it difficult for selfish miners to decide between publishing or keeping newly discovered blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' To show the effectiveness of a new proposed defense, we compare the proposed defense with the most famous defense, called the tie-breaking defense [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The preliminaries related to the selfish mining attack are given in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In section 3, related work is being discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In section 4, we describe the proposed defense mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In section 5, the modified version of SM1 for evaluation is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Section 6 reports the result of the experiments, and section 7, discusses the pa- per’s limitations and problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Finally, section 8 concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 2 PRELIMINARIES In this section, required information about the proposed algorithm is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' We summarize information about blockchain, selfish mining strategies, and learning au- tomata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In the rest of this section, at first, an overview of the Bitcoin and its transaction is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Then the mining process, selfish mining attack, learning automata, and properties of an ideal defense are explained, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Bitcoin is a distributed and decentralized cryptocur- rency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The users of Bitcoin can transfer Bitcoins by creating a new transaction [16], [17] and sending it to a ledger based on the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The blockchain is an append-only ledger protected by a group of miners in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Miners are rewarded for their effort to protect the blockchain against tampering data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' A transaction in the blockchain is made of at least one input and one output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The difference between the total amount of inputs and outputs in a transaction is called a transaction fee [18], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The transaction fee goes to the miner, who includes the transaction in the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The following subsection gives more details about the min- ing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='1 Mining Process The state of the blockchain is changed through transactions [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Transactions are grouped into blocks that are appended to the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' A typical block in blockchain consists of two major parts: header and body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The block’s header contains the hash of the previous block, the hash of the current block, the Merkle root of all transactions included in this block, and a number called the nonce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The block’s body contains transactions that the miner decided to include in the block [2], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' A valid block contains a solution to a puzzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' To solve the puzzle, miners try to put the correct nonce in the block’s header so that the block’s hash is smaller than the block difficulty target [20], [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The block difficulty is dynamically adjusted such that blocks are generated at an average rate of one every ten minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' If a miner solves the puzzle and puts mined block in the longest chain, it will be rewarded with Bitcoins that did not exist before and the transaction fees of the newly created block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The probability of mining a new block is proportional to the computational resources used for solving the associ- ated puzzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Due to the nature of the mining process, the interval between mining events exhibits high variance from the point of view of a single miner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Consequently, miners typically organize themselves into mining pools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' All pool members work together to mine each block and share their revenues when one of them successfully mines a block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' While joining a pool does not change a miner’s expected revenue, it decreases the variance and makes the monthly revenue more predictable [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='2 Selfish Mining Attacks Bitcoin’s doc illustrates the approach of releasing blocks after mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' A miner who discovers a new valid block should release it immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Eyal and Sirer [14] showed that some miners could gain revenue more than their fair share by deviating from Bitcoin’s main mining rules, which they called ”selfish mining.” As noted, miners try to form a pool to decrease revenue variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' So a selfish pool with more than 1/3 computational power unfairly changes Bit- coin’s rewarding system by keeping a private chain and withholding blocks that have been mined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Saphirshtein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' [22] used Markov Decision Process (MDP) to investigate the profit threshold (the minimal frac- tion of resources required for a profitable attack).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' They find the bound under which the system can be considered secure against such attacks and modify the protocol to assess their susceptibility to selfish mining by computing the optimal attack under different variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' They showed circumstances under which selfish miners can hold selfish chain even public chain is longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' New research areas of selfish mining in machine learning have evolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Most of these researches have used reinforce- ment learning methods to improve the optimality of the attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' [23], [24] improved MDP-based solution of [22] by applying reinforcement learning algorithms to gain more revenue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' [25] developed a new deep reinforcement learning framework to analyze the incentives of a rational miner in various conditions and upper bound the security threshold of proof-of-work based blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='3 Learning Automata Learning automata are adaptive decision-making devices that operate in unknown random environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' A Learning automaton has a finite set of actions, and each action has a certain probability (unknown to the automaton) of getting rewarded by its environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The aim is to learn to choose the optimal action (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=', the action with the highest prob- ability of being rewarded) through repeated interactions with the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' If the learning algorithm is appropriately 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Learning automaton (LA) chosen, then the iterative process of interacting with the environment can result in selecting the optimal action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The interaction between the learning automaton and the envi- ronment is shown in Fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Learning Automata (LAs) can be classified into two main families, fixed and variable structure learning automata [26], [27], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Variable action set learning automata used in this paper is a sub-set of variable structure learning automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Variable action set learning automata can be represented by a sextuple < β, φ, α, P, G, T >, where β a set of inputs actions, φ is a set of internal states, α is a set of outputs, P denotes the state probability vector governing the choice of the state at each stage t, G is the output mapping, and T is the learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The learning algorithm is a recurrence relation used to modify the state probability vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Such learning automata have a finite set of r actions denoting < α1, α2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=', αr >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' At each stage t, the action subset ˆα ⊆ α is available for the learning automata to choose from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Selecting the elements of ˆα is made randomly by an external agency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Selecting an action and updating the action probability vector in these automata are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Let S(t) = � αi∈ˆα(t) Pi(t) presents the sum of probabilities of the available actions in subset ˆα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Before choosing an action, the available actions probability vector is scaled as Equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' ˆPi(t) = pi(t) S(t) ∀αi (1) The crucial factor affecting the performance of the vari- able action set learning automata is the learning algorithm for updating the action probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Let αi be the action chosen at time t as a sample realization from distribution p(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Let a and b the reward and penalty parameters, and m denotes the number of available actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Equations for updating the probability vector are defined by Equation 2 for action chosen by learning automata (i = j) and Equation 3 for other actions(i ̸= j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Pj(n + 1) = Pj(n) + aβ(1 − Pj(n)) − b(1 − β)Pj(n) (2) Pj(n+1) = Pj(n)−aβPj(n)+b(1−β)[ 1 m − 1 −Pj(n)] (3) If a=b, the learning algorithm is called the linear reward penalty(LRP );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' if b = ϵa with ϵ < 1, then the learn- ing algorithm is called linear reward ϵ-penalty (LRϵ−P ), if b=0, the learning algorithm is called the linear reward inaction(LRI), if a=0, the learning algorithm is called the penalty inaction(LP I)and finally, if a=b=0, the learning al- gorithm is called pure chance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Learning automata have found applications in many areas, such as defense mechanisms in network attacks [29], [30], peer-to-peer networks [31], Internet of Things (IoT) [32], and neural networks [33], to mention a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In this paper, for the first time, learning automata is used to create a defense mechanism against one of the blockchain attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='4 Properties of an Ideal Defense By explaining of problems and weaknesses of existing de- fenses, as [34] suggested, we can enumerate the desirable properties of an ideal defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Decentralization: Introducing a trusted server would open a new single point of failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Moreover, it violates Bitcoin’s fundamental philosophy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Incentive Compatibility: The expected relative rev- enue of a miner should be proportional to mining power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Backward compatibility: Non-miners who cannot upgrade their clients can still participate in the net- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' This is important for hardware products such as Bitcoin ATMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Specifically, the following rules should not be changed: – Block validity rules: A valid block in the current Bitcoin protocol should also be valid within the defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' – Reward distribution policy: All blocks in the main chain and no other block receive block rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' – Eventual consensus: Even when an attack happens, old and new clients should eventu- ally reach a consensus on the main chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 3 RELATED WORK In this section, existing defenses are summarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Summa- rizing of these defenses is based on the classification of the similarity of methods used in the proposed defenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The most popular defenses are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Some of the defenses need to make fundamental changes in blockchain structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Generally, they require significant updates in blockchain nodes, which are incompatible with previous versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' First, Bahack [35] proposed a defense with the punishment rule for all miners, including honest miners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In this defense, all miners who fork the blockchain will be punished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The problem with this defense is the punishment of honest miners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Solat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' [36] introduced Zeroblock, where miners are forced to release their blocks within an expected time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' If miners withhold their blocks for selfish mining and do not broadcast them within the scheduled time, the peers in the network create their own dummy blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' These defenses need block validity and reward distribution changes, so network nodes should up- date their clients to be familiar with the new protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' [37] Environment Action Response Learning Automaton4 introduced another defense that will change the structure of transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In this defense, the transaction has an extra parameter named Expected Confirmation Height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' A com- parison of the Expected Confirmation Height and expected value for published block height will use to detect selfish mining attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The following paragraph will introduce de- fenses that operate when a new fork is seen in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Defenses will decrease selfish miners’ chances when they create a fork.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The most accepted solution against selfish mining is the tie-breaking defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The tie-breaking defense was proposed by Eyal and Sirer [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' When a miner learns of computing branches of the same length, it should propagate all of them and choose which one to mine on uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' As they showed in their paper, the minimum mining power needed to start selfish mining will be about 25%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Heilman [38] proposed another backward-compatible defense against selfish mining called Freshness Preferred (FP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In the FP solution, Heilman suggested that each miner use an unforgeable timestamp parameter to penalize miners that withhold blocks by comparing the latest value of the unforgeable timestamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' A trusted party in the network generates an unforgeable timestamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Heilmen claimed that the lower bound threshold of selfish mining would increase from 25% to 32%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The problem with this solution is using a trusted party in the network, which conflicts with the phi- losophy of Bitcoin’s decentralization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The following para- graph will introduce fork-resolving policy-based defenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Defenses that will work based on fork-resolving policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In these defenses, protocols are changed so that when the selfish chain is longer than the public chain, the defense mechanism will work, unlike the tie-breaking defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The first solution in this category was published by Zhang and Preneel [34], called Publish or Perish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In Publish or Perish, the authors suggested neglecting blocks not published in time and appreciating blocks that incorporate links to com- peting blocks of their predecessors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Consequently, a block kept secure until a competing block is published contributes to neither or both branches;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' hence it confers no advantage in winning the block race.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The following paragraph will introduce a machine learning-based algorithm to detect selfish mining attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Researches have been done to identify factors that detect selfish mining attacks [39], [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' These researches use exist- ing data of selfish mining attacks to create training and test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' This research examines factors and tries to find future research areas in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In this paper, the suggested defense algorithm utilizes machine learning to manage parameters and decisions of mining process in a self-adaptive manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' From this per- spective, it cannot be compared with all defense mecha- nisms reported in the literature because there is no machine learning-based defense mechanism in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' From an- other perspective, the proposed mechanism suggests a self- adaptive algorithm that is matched with the dynamic and distributed nature of blockchain-based systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 4 PROPOSED ALGORITHM: NIK DEFENSE In this section, the proposed algorithm is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' At first, the system model in which selfish mining and the proposed Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Structure of the miner in the system model defense will work is described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Then, the required defini- tions for the proposed algorithm are explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Finally, the proposed algorithm is demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='1 System Model In this subsection, a model of the system will be presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In the proposed model, we consider miner nodes in the blockchain network, so other nodes like super nodes, light nodes, and others will be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Miner nodes in the selfish mining attack form a mining pool to obtain revenue more than their fair share.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' We will consider two groups of miners to create the most appropriate and potent model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' A group of miners follows the selfish mining strategy, which has less than 50% of total computing power, and a group of miners follows Bitcoin’s mining protocol without any deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Fig 2 shows a snapshot of mining pools in the proposed defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In this figure, selfish miners which form a mining pool are colored red, and other miners form an honest pool colored blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The following paragraph will present the distribution of computing power in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' To explain the distribution of computing power in the proposed model, we assume that the selfish mining pool has a α proportion of total computing power and other miners have a 1 − α proportion of total computing power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Therefore a newly discovered block with a probability of α belongs to the selfish pool, and a probability of 1−α belongs to other miners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The following paragraph will explain the connectivity of nodes in the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' We will ignore block propagation delay to clarify net- work connectivity in our proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' This assumption is rational because miners in Bitcoin’s network try to send and receive blocks as fast as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' If they have a delay in the process of sending and receiving, their mining pro- cess schedule will disrupt, so they can’t find a new block efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Another critical reason to ignore propagation delay is the effort of network researchers and developers to decrease this delay in Bitcoin’s network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' We see this result in published papers and Bitcoin Improvement Proposals [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The following paragraph will discuss the arrangement of blocks in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Blocks in every node of Bitcoin form a tree structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Each block refers to the previous block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' For simplicity, in ith miner I 4 I 1 1 1 1 1 : 1 1 15 our model, we don’t investigate all blocks in every branch of the block’s tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Instead, we can consider only two branches: The main chain or the longest chain, which results from consensus between nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Another branch is the private branch which selfish miners have created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' None of the honest nodes can’t distinguish these two branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The following paragraphs will discuss the model of miners in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In networks like Bitcoin with a proof-of-work consensus mechanism, we can assume creating a new block as an event in the mining process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' On the other hand, creating a new block doesn’t relate to time passing, so we can consider the mining process a discrete memoryless model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In this model, every miner, either honest or selfish, decides on the time of finding a new block, and their chosen action continues until the next event finds a new one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In our proposed model, the selfish mining attacker uses computational power to create a private chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In arbitrary time t, the selfish mining attacker must choose which block of the main chain to extend for its private chain and which block to release to increase selfish pool revenue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' If the selfish miner realizes that the honest miner finds a new block, it will try to substitute its private block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Parameter γ is defined as an advertisement factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Nodes with γ proportion of computing power would accept selfish miner block instead of honest miner block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In terms of block’s height, if Bitcoin’s network is in h height, the block of the selfish miner in h height with the probability of γ(1 − α) would accept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='Each miner node in the network is equipped with a learning automaton like in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Based on ith miner in the network, the required definitions of the proposed algorithm will be explained in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='2 Required Definitions To explain the proposed algorithm, the required definitions of the algorithm are defined here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' After related definitions, there is an example for explaining definitions in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Competing blocks from ith miner’s perspec- tive are defined in this definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Two blocks compete for being in the main chain if both are valid in obeying protocol rules in creating transactions and blocks and both have the same height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Usually, these blocks form a fork in the block’s tree structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Weight calculation from ith miner’s per- spective is defined in this definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Blocks in competing forks with the same height are compared to calculate the weight of the forks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Between blocks with the same height but in a different fork, the block created recently or has the most timestamp wins the competition, and the related fork’s weight will increase one unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' If two blocks have the same timestamp, one of them will be chosen randomly as the competition’s winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The weight of the fork from ith miner’s perspective is defined in this definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The weight of a fork will be the sum of the assigned weight after calculating the weight of blocks of the same height in different forks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In the proposed method, the fork’s weight is denoted by W, and the length is denoted by L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In Fig 3, an example is designed to illustrate definitions 1, 2, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In the scenario of Figure 3, 70 blocks were mined, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Structure of the block in the system model (TS denotes the timestamp of each blocks) and the network nodes had consent about these blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' At height 71, three forks were created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' At first, blocks with a height of 71 will be investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' At this height, fork 2 will win the race, and the weight of fork 2 will increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Because B71 of fork 2 has the most recent timestamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In height 72, another time, fork 2 will win.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' At height 73, the only fork with the block is fork 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' So, without any competition, fork 1 will win.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' By using these definitions, fork 1 has L = 3, which is the longest chain and fork 2 has W = 2, which is the heaviest chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' There should be a decision between these forks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In the proof-of-work consensus, fork 1 will be chosen because the height of the fork (L = 3) is longer than the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' But, using the proposed definitions, the weight of forks should be considered in decision making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' This definition defines how to choose a fork from ith miner’s perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Since the miner should choose between different forks that are created by competition of blocks in the same height and maybe one of them is created as a result of the selfish mining attack, need to decide between the length of fork or weight of fork as a base of decision for choosing a fork between different forks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' A new parameter is proposed in the proposed method, like [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' This parameter is called fail-safe and denoted by K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' When the length of one fork’s chain is no longer by K blocks than other forks, the miner should select weight as a base for choosing a chain among forks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In the Fig 3 example, a decision should be made to choose between the weight and height of the chain as a base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' If K = 1, the length of fork 1 is longer than the length of fork 2, so the decision is made by Length, and fork 1 will be chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' On the other hand, if K > 1, the length of fork 1 is no longer than the other by K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In this situation, the weight of the chains should be considered, and fork 2 will be chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Fork decision making time from ith miner’s perspective is defined in this definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' There should be a time for the miner to check if any forks exist and, if they exist, decide between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In the proposed method, this time parameter is denoted by τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Choosing a fail-safe parameter from ith Fork1 B71 B72 B73 TS:709 TS:718 TS:731 Fork2 B70 B71 B72 TS:700 TS:711 TS:720 Fork3 B71 TS:7106 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' An example of how the miner should calculate reinforcement ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='signal for its learning automaton ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='TABLE 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='Table of Notations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='Notation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='Description ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='The ratio of total computational power that selfish miner has ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='The ratio of honest miners that prefer to mine on selfish miners block ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='W ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='Total weight of chain in the created fork ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='Length of chain in the created fork ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='Fail-safe parameter for selecting between weight or height as a base for decision ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='Time parameter for fork checking ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='The Time Window for changing τ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='Reinforcement signal for miner’s learning automata ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='miner’s perspective is defined in this definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' To defend against selfish mining effectively, the miner must change its fail-safe parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' This change is done using an installed learning automaton on the miner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Learning automaton has three actions (Grow-Shrink-Stop).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' By selecting one of these actions, the K parameter will change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' By evaluating net- work parameters periodically, one of these actions will be chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In the proposed method, this period is called Time Window, denoted by θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' θ is a positive integer coefficient of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Calculating reinforcement signal from ith miner’s perspective is defined in this definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Learning automaton should calculate reinforcement signal, which is denoted by β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The reinforcement signal is the feedback from the environment, which the learning automaton applied used for updating the learning automaton’s probability vec- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In the proposed method, this signal is calculated from the analysis of each τ decision in one θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Equation 4 shows how to calculate β after one θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' β = Number of Weight Decision Number of {Height + Weight} Decision (4) Since the sum of height and weight decisions in a θ equals the Number of τ, Equation 4 can convert to Equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' β = Number of Weight Decision Number of τ in θ (5) The following example is designed to illustrate defini- tions 5, 6, and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In this example shown in Figure 4, the defined θ has 5 ∗ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Based on past decisions, the miner should calculate the reinforcement signal for its learning automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Since in τ number 3, and 5, miners had decisions based on weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Therefore β equals 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' To summarize all notations used in the proposed method, Table 1 shows the definition for each notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='3 Proposed Algorithm After modeling the system and defining the necessary prerequisites for the proposed algorithm, this section will present the proposed algorithm in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' At first, the pro- posed algorithm is explained using events that happened while running the defense method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' then sub algorithms will be described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The proposed algorithm can respond to the occurrence of these events: 1) One Block Receive Event a) If a fork exists, the miner will check the re- lation of a new block with existing forks by the previous hash parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' If the miner needs to create a fork, it will create.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 2) Decision Making Time (τ) Event a) Existence of a fork will be checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The miner must select weather by height or weight if a fork exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 3) Time Window Event a) Existence of a fork will be checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The miner must select whether by height or weight if a fork exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' b) Reinforcement signal will be created for updating learning automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' c) Learning automaton will choose the next action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' This action uses for updating the K parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In Algorithm 1, the proposed algorithm is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The proposed algorithm consists of five sub-algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' These five sub-algorithms relate to: Length Calculation: Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='1 explains this algo- rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Weight Calculation: Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='2 explains this algo- rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Chain Selection: Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='3 explains this algo- rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Action Selection by LA: Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='4 explains this algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Updating Reinforcement Signal of LA: Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='5 explains this algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Algorithm 1 NikDefense(event) Notation: event denotes event enum that triggered miner to do action {TimeWindowEvent, ForkDecisionMaking- TimeEvent, BlockReceiveEvent}, Kmin denotes the min value for K, Kmax denotes the max value for K, 1: Begin 2: switch (event) 3: case TimeWindowEvent: 4: ForkCreationChecking() 5: β=CalculateUpdateSignal() 6: LA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='update(β) 7: UpdateFailSafe(Kmin, Kmax) 8: case ForkDecisionMakingTimeEvent: 9: ForkCreationChecking() 10: case BlockReceiveEvent: 11: /*Just put the block on the correct fork’s chain, and If needed, use ForkSelection algorithm()*/ 12: end switch 13: End Time Window (0) T Number 1 T Number 2 T Number3 TNumber 4 TNumber 5 Height Decision HeightDecision Weight Decision Height Decision Weight Decision7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='1 Length Calculation To calculate the length of every chain created by the fork: 1- Getting the height of the last block before the creation of the fork 2-Calculating difference of the last block’s height from the height of the last block before the creation of the fork.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In Algorithm 2, the length calculation algorithm is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Algorithm 2 ForkChainsLengthCalculation(CH) Notation: N denotes the number of chains in a fork, Chains[i] denotes an ith element of chains array in recent τ time i ∈ [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='N], ChainsLength[i] denotes an ith element of chains array length in the fork i ∈ [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='N], CH denotes the last block’s height of the main chain before the creation of the fork, LastBlockHeight denotes the Height of last block in i th fork chains 1: Begin 2: for i ← 1 to N do 3: ChainsLength[i] = Chains[i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='LastBlockHeight-CH 4: end for 5: return ChainsLength 6: End 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='2 Weight Calculation To calculate the weight of every chain created by fork: 1- Calculating max length among available chains 2-According to max length, blocks of the different chains but of the same height will be evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Between blocks with the same height, the chain which has the block with the most recent timestamp will win the race 3-Calculation in part 2 will be continued until max length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' If one chain is shorter than the others, it will not conclude in comparisons of blocks with higher heights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In Algorithm 3, the weight calculation algorithm is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Algorithm 3 ForksWeightCalculation(LM) Notation: N denotes the number of chains in a fork, Blocks[j][i] denotes an ith block of jth chain in recent τ time i ∈ [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='N], LM denotes the max length of a fork in a chains array, MaxTimestampIndex denotes the index of max times- tamp in blocks with the same height in different forks 1: Begin 2: for i ← 1 to LM do 3: MaxTimestampIndex = 1 4: for j ← 1 to N do 5: if (Blocks[j][i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='Timestamp > Blocks[MinTimestampIndex][i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='Timestamp) then 6: MaxTimestampIndex = j 7: end if 8: ChainsWeigth[MaxTimestampIndex] += 1 9: end for 10: end for 11: return ChainsLength 12: End 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='3 Chain Selection To choose a chain among created chains by fork condition, miner needs to make a decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' So chain selection algorithm of the proposed defense can be described as below: 1- Calculating length of chains 2-Sorting chain based on length in descending order 3-If one chain is longer than the others by K, it will select for the next mining event 4-If no chain longer than the others by K, the weight of all chains will calculate by the algorithm described in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='2 5- Sorting chain base on weight in descending order 6-Heaviest chain will be chosen for the next mining event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In Algorithm 4, the chain selection algorithm is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Algorithm 4 ChainSelection() Notation: N denotes the number of chains in a fork, Chains[i] denotes an ith element of chains array in recent τ time i ∈ [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='N], LM denotes the max length of a fork in a chains array, CH denotes the last block’s height of the main chain before the creation of the fork, ChainsWeight[i] denotes an ithchain’s weight in a fork, i ∈ [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='N], ChainsLength[i] denotes an ith element of chains array length in the fork, i ∈ [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='N], LastBlockHeight denotes the Height of last block in ith fork chains, ChosenChain denotes chosen chain after defense 1: Begin 2: if (N > 1) then 3: ChainsLength = ForkChainsLengthCalculation(CH) 4: SortDescendingly(Chains, ChainsLength) 5: if (ChainsLength [0] - ChainsLength [1] > K) then 6: /*Decide based on Chain’s Height*/ 7: ChosenChain = Chains[0] 8: CH = ChosenChain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='LastBlockHeight 9: return ChosenChain 10: else 11: /*Decide based on Chain’s Weight*/ 12: LM=ChainsLength[0] 13: ChainsWeight = ChainsWeightCalculation(LM) 14: SortDescendingly(Chains, ChainsWeight) 15: ChosenChain = Chains[0] 16: CH = ChosenChain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='LastBlockHeight 17: return ChosenChain 18: end if 19: end if 20: End 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='4 Action Selection by LA θ consists of τ time intervals, and in any of the τ time inter- vals particular event can occur;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' when the θ has finished, it is needed to make a decision about the occurrence of events in any τ time intervals of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Learning automaton should make this decision as an AI tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Learning automaton will choose the next K parameter by selecting an action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Chosen action by Learning automaton can be one of these values: 1-Grow: Choosing this action shows that the network was under attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' So Learning automaton increase the K value by one unit to make chain selection harder 2-Shrink: Choosing this action indicates that the network was not under attack or the attack was ineffective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' So learning automaton decrease the K value by one unit 3-Stop: Choosing this action by learn- 8 ing automaton shows that the previous chosen action in θ was correct, and there is no need to change the K value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The reason for selecting variable action set learning automaton is that when the K parameter reaches the max value (Kmax), Grow action should omit from learning automaton’s action selection options, and when the K parameter reaches the min value (Kmin), Shrink action should omit from learning automaton’s action selection options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In Algorithm 5, the action selection algorithm has been shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Algorithm 5 UpdateFailSafe(LA, K, Kmin, Kmax) Notation: N denotes the number of chains in a fork, Chains[i] denotes an ith element of chains array in recent τ time i ∈ [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='N], ChainsLength[i] denotes an ith element of chains array length in the fork i ∈ [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='N], CH denotes the last block’s height of the main chain before the creation of the fork, LastBlockHeight denotes the Height of last block in i th fork chains 1: Begin 2: if (K = Kmax) then 3: L = LA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='choose action([’Stop’, ’Shrink’]) 4: else if (K = Kmin) then 5: L = LA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='choose action([’Grow’, ’Stop’]) 6: else 7: L = LA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='choose action([’Grow’, ’Stop’, ’Shrink’]) 8: end if 9: switch (L) 10: case ’Grow’: 11: K = K + 1 12: case ’Shrink’: 13: K = K − 1 14: case ’Stop’: 15: /*Do nothing about K*/ 16: end switch 17: return ChainsLength 18: End 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='5 Updating Reinforcement Signal of LA By ending θ and before choosing the next action, learning automaton needs to know how the previous action was by receiving a reinforcement signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' If the previous action was effective, learning automaton would be rewarded, and if the previous action was not effective, learning automaton would be punished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Equation 4 shows how the reinforce- ment signal will help learning automaton to improve itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In this equation, every τ interval in θ is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' At the end of τ, If the chain is selected by height (it means that one chain is longer than the other by K and does not need to use the weight calculation algorithm), the counter of height selection will increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Otherwise, the counter of weight calculation will increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 5 PROPOSED ALGORITHM: SM1 STRATEGY MOD- IFICATION The SM1 strategy is modified in this section to evaluate the proposed defense in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Since the pro- posed defense has modified the mining protocol of Bitcoin to create the first AI defense against selfish mining using learning automata, the effort is needed to change the SM1 strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The proposed defense uses θ and τ interval pa- rameters, so these parameters should be applied to modify the SM1 strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Also, in every θ, the fail-safe parameter will change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' To make an effective attack, the attacker should approximate the K parameter using learning automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In this paper, approximated K will be shown by ˜K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The procedure of approximating ˜K is similar to updating the fail-safe parameter algorithm in Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In Algorithm 6, the modified version of SM1 is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' There are two scenarios in selfish mining attacks that the selfish miner should make a decision about it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Both of these scenarios depend on the length difference between the selfish chain kept by the selfish miner and the honest chain that the honest miner creates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Another factor that can be effective in modified SM1 strategy is approximated ˜K parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The first scenario is making a decision on the selfish pool as a decision-maker succeeds in finding a new block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' If the private chain and the public chain have one block before finding a new block by the selfish miner, the new block will be added to the selfish chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In this situation, the selfish chain is ahead, and by publishing its secret blocks, obtaining a reward for two blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The second scenario is making a decision on other min- ers which considered as honest and succeed in finding a new block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' If no private chain is maintained by the selfish node, by receiving an honest block, no action must be taken because the public chain wins the race, and the attack should reset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' If the private chain was one block ahead, receiving a new block from an honest node would lose weight calculation due to having a timestamp less than the new received block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' So to try its chance, it won’t release the private chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The next block (based on mining by the selfish miner or others) will determine the winner of the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' If the private chain was two blocks ahead, by receiving a new block mined by an honest node, the selfish miner would release its private chain because the new received block wins the first block race in weight calculation by timestamp parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In this situation, there is a race between the selfish and honest branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' If the private chain was ˜K+1 was ahead, by receiving a new block mined by an honest node, the selfish miner will release all of the private chain and win ˜K+1 blocks revenue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Finally, if a private chain was more than ˜K+1 was ahead, by receiving a new block mined by an honest node, the selfish miner will release the first unpublished block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 6 EVALUATION This section will describe an evaluation of the Nik defense against selfish mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' To evaluate the proposed defense, we first introduce evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Then, various exper- iments have been designed to assess the proposed defense and compare it with other defenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='1 Metric In this subsection, metrics for the evaluation of the proposed defense algorithm are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' These metrics will be 9 used in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='2 experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In the following items, necessary metrics are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Algorithm 6 Modified SM1() Notation: PublicChain denotes the public chain which all miners can see,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' PrivateChain denotes the chain mined by selfish miners,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' PrivateBranchLength denotes the length of the selfish chain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' ∆ denotes the difference of selfish and public chain’s length,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' ˜K denotes the approximate fail-safe parameter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Kmin denotes the min value for K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Kmax denotes the max value for K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='WeightCalculationCounter denotes the number of weight ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='calculations in one θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='1: Begin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='2: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='On Init: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='3: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='PublicChain ← Publicity Known Blocks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='4: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='PrivateChain ← Publicity Known Blocks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='5: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='PrivateBranchLength = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='6: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='Mine at the head of the private chain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='7: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='On ending ForkDecisionMakingTime: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='8: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='if (ReleaseByWeightCalculation() = true) then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='9: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='WeightCalculationCounter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='WeightCalculation- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='Counter + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='10: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='end if ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='11: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='On ending TimeWindow: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='12: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='β = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='CalculateUpdateSignal(WeightCalculationCounter) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='13: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='LA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='update(β) 14: ApproximateFailSafe(LA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' � K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='Kmin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='Kmax) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='15: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='WeightCalculationCounter = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='16: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='On SelfishPool found a block: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='17: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='∆prev=length(PrivateChain)-length(PublicChain) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='18: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='Append the new block to PrivateChain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='19: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='PrivateBranchLength = PrivateBranchLength + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='20: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='if (∆prev = 0 and PrivateBranchLength = 2) then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='21: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='Publish all of the PrivateChain //Try By Luck ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='22: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='PrivateBranchLength = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='23: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='end if ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='24: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='Mine at the head of the PrivateChain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='25: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='On other miners found a block: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='26: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='∆prev=length(PrivateChain)-length(PublicChain) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='27: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='Append the new block to PublicChain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='28: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='if (∆prev = 0) then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='29: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='PrivateChain ← PublicChain //Others win ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='30: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='PrivateBranchLength = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='31: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='else if (∆prev = 1) then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='32: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='/*Do Nothing*/ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='33: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='else if (∆prev = 2) then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='34: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='Publish all of the PrivateChain //Try By Luck ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='35: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='PrivateBranchLength = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='36: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='else if (∆prev = � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='K + 1) then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='37: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='Publish all of the PrivateChain//SelfishPool wins ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='38: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='PrivateBranchLength = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='39: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='else if (∆prev > � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='K + 1) then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='40: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='Publish the first unpublished block of the private ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='chain //SelfishPool wins ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='41: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='end if ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='42: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='Mine at the head of the PrivateChain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='43: End Relative Revenue: This metric is used to measure a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='miner’s revenue based on the revenue of the other ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='miners in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Relative revenue is obtained by the number of total accepted mined blocks by one miner divided by the total number of blocks in the main chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Equation 6 is used to show how to calculate the relative revenue of an arbitrary miner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Number of Mined Block by ith Miner Total Number of Mined Blocks (6) Followed by Equation 6, the relative revenue of the honest miner and the selfish miner can be shown in Equations 7 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In these equations, all selfish and honest miners group together and assume that only two groups of miners exist in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Honest Miner W in Block Honest Miner W in Block + Selfish Miner W in Block (7) Selfish Miner W in Block Honest Miner W in Block + Selfish Miner W in Block (8) Lower Bound Threshold: This metric shows the minimum computing power that selfish miners should provide to start the attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The lower bound threshold is obtained in simulations by the inter- section point of the selfish miner’s relative revenue diagram and ideal defense diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='2 Experiment Results Inspiring by previous defenses and their simulators [14], [22], [34] the proposed algorithm was simulated by con- verting the mining model into a Monte Carlo simulation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' This conversion makes it possible to distribute newly discovered blocks among selfish and honest miners without solving a cryptographic puzzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' For reproducibility, the developed simulator can be found on Github 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Ex- periments reported in this subsection have been run using Intel Core i7 with a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='5 GHz frequency clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Experiments reported in this section are divided into four parts: 1) Experiment 1: This experiment aims to compare the proposed defense with the tie-breaking defense as the previous proposed selfish mining defense using a different kind of learning automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Reward and penalty parameters were changed, and then results were reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 2) Experiment 2: This experiment aims to check the effect of changing the K on the proposed defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' By changing the K interval, results of changing this parameter have been reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 3) Experiment 3: This experiment aims to check the ef- fect of changing the τ time interval on the proposed defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' By changing the τ time interval, the result of changing this parameter has been reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 4) Experiment 4: This experiment aims to check the effect of changing the number of τ in one θ on the proposed defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' By changing the number of τ, results of changing this parameter have been reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' For all of the experiments, 10000 blocks were generated by the simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The main learning automaton used in these experiments is LRϵ−P with a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='1, b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='com/AliNikhalat/SelfishMining 10 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='1 Experiment 1 This experiment is conducted to study the impact of chang- ing reward and penalty parameters of learning automaton on the performance of the proposed algorithm with respect to relative revenue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The results of proposed algorithm are compared with the tie-breaking algorithm which is a well- known defense mechanism against selfish mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' For this purpose, parameter K was selected from [1,3] interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The value of τ time interval was about the time of mining five blocks, and one θ has ten τ time intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The results of tie- breaking are compared with those obtained for the proposed defense in five reward and penalty rate values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The results of this experiment are shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='a to Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='e, all of them perform better than the tie-breaking defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' This means that regardless of the type of learning automata, the defense mechanism can prevent the selfish attacker by decreasing the relative revenue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' By comparing relative rev- enue between different learning automata, LRϵ−P performs better than the others because the reward rate is more than the penalty rate, and the relative revenue is near the ideal defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' So, choosing the proper reward and penalty rate can affect the quality of defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' (a) Comparison with Tie breaking defense when a=0, b=0 (b) Comparison with Tie breaking defense when a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='01, b=0 (c) Comparison with Tie breaking de- fense when a=0, b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='01 (d) Comparison with Tie breaking defense when a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='01, b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='01 (e) Comparison with Tie breaking defense when a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='1, b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='01 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Comparison with tie-breaking defense using different penalty and reward parameters The lower bound threshold value for starting the selfish mining attack is shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' This parameter can be TABLE 2 The Threshold for Starting the Selfish Mining Attack P LR−I LP −I LR−P LRε−p a 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='01 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='1 b 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='01 Nik 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='45 Tie-Breaking 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='25 used to compare starting point of the selfish mining attack’s profitability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The lower threshold value is, the less selfish miner attacker is successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' This can be used to show the effectiveness of the defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' According to the results of this experiment, the proposed defense has a higher threshold value than the tie-breaking defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' This threshold increases from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='25 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' By increasing the threshold, the selfish miner needs to increase its pool size to start an effective selfish mining attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='2 Experiment 2 This experiment is conducted to study the impact of the parameter K on the performance of the proposed algorithm with respect to relative revenue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' For this purpose, the pa- rameter K was tested for three intervals [1,3], [2,4] and [1,5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The value of parameter τ time interval is about the time of mining five blocks, and one θ has ten τ time intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The results obtained from different values of K are shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' By comparing K intervals, the advantage of choosing bigger values for K is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Figure 6 shows [1, 5] interval for K value has the most powerful defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' This can lead to two important factors: 1-learning automaton can choose value 5 for K 2-learning automaton has the freedom to a choose value for K in a bigger interval(choosing four values for K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' It shows that the interval with bigger values of K and more options for K can reduce the relative revenue of the attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The impact of K intervals on the proposed defense On the other hand, the effect of changing the value of parameter K’s on the attack threshold can be seen in Figure 100 K=[1,3] K=[2,4] K=[1,5] 80 - .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='IdealDefense UpperBound Relative Revenue 60 40 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='50 Poolsize100 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' NikDefense Tie Breaking No Defense 80 IdealDefense Upper Bound Relative Revenue 60 40 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='50 Pool size100 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='. Nik Defense Tie Breaking NoDefense IdealDefense 80 Upper Bound Relative Revenue 60 40 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='50 Pool size100 NikDefense Tie Breaking NoDefense IdealDefense 80 Upper Bound Relative Revenue 60 40 20 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='50 Pool size100 NikDefense Tie Breaking No Defense 80 - IdealDefense UpperBound Relative Revenue 60 40 20- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='50 Pool size100 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='. Nik Defense Tie Breaking NoDefense 80 - IdealDefense Upper Bound Relative Revenue 60 40 20 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='50 Pool size11 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' [1, 5] has the most threshold for starting an attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The increasing threshold of starting attack for bigger values of K in the chosen interval can be the result of selecting a bigger K value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Overall, the result shows the bigger values of K leads to the less powerful attack in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='3 Experiment 3 This experiment is conducted to study the impact of the τ time interval parameter on the performance of the proposed algorithm with respect to relative revenue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' For this purpose, the parameter K was selected from [1,3] interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' One θ has ten τ time intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' τ time interval was tested for 5, 9, and 15 mining blocks time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' For example, if τ = 6, the τ time interval is about the time of mining six blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The results from different values of τ time intervals are shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' This figure doesn’t show noticeable changes in relative revenue metric after selecting different values for the τ time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' This means that small changes in the value of τ doesn’t have a meaningful impact on the performance of the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Another important factor shown in Figure 7 is the thresh- old for starting the attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' It shows that a lower value τ time interval causes an increment in the threshold for starting the attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' As expected, this happened because the lower τ value in a constant number of τ (ten time intervals in this experiment) in one θ can increase the number of θ in simulation for the specified number of blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Increasing the number of θ can force the learning automaton to examinate more decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' By increasing the number of decisions, the learning automaton can tune the β parameter effectively, and as a result, the threshold for starting defense will increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Overall, the result shows a lower value for τ improves the performance of the proposed defense mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The impact of τ time interval on the proposed defense 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='4 Experiment 4 This experiment is conducted to study the impact of the number of τ time intervals parameter in one θ on the performance of the proposed algorithm with respect to relative revenue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' For this purpose, the parameter K was in the [1,3] interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' τ time interval was about the time of mining five blocks, and one θ has 6, 12, and 18 τ time intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The results from the different numbers for τ in one θ are shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' By comparing the results of this experiment in Figure 8, the lower number of τ has a better impact on decreasing the relative revenue of selfish miner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' As predicted from the behavior of the learning automaton, the less value for τ in one θ, leads to more numbers of θ in simulation for the specified number of blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Increasing the number of θ can force the learning automaton to make more decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' By increasing the number of decisions, the learning automaton can tune the β parameter effectively, and as a result, the threshold for starting defense will increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' It means that the lower number of τ time intervals parameter in one θ will decrease the effect of the selfish mining attacker in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In addition, Figure 8 shows that the number of τ time intervals creates a little difference in the threshold of starting a selfish mining attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' τ = 6 performs better than the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' This happened because of the effective adjusting of the β parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Eventually, the result shows a lower number of τ time intervals in one θ prevents selfish mining efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The impact of a number of τ time intervals in one θ on the proposed defense 7 DISCUSSION Selfish mining attack shows how decentralization of blockchain and Bitcoin as one of the most important im- plementations of blockchain can be threatened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Consid- ering Bitcoin’s popularity, capabilities, and the existence of this threat to Bitcoin, a solution should find to solve this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Various solutions have been studied earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Existing solutions suffer a lack of self-adaptability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Using learning automata as one of the reinforcement learning tools for designing self-adaptive systems in developing the first AI-based defense, the profitability of selfish mining can decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' We know the problems and limitations of our proposed defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Our work is designed and implemented in perfect conditions and may not cover all of Bitcoin’s protocols, while Bitcoin’s network and its protocols are so complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In the following, some critical questions have been answered for better discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 100 Blocks in t = 5 Blocks in t = 9 Blocks in T = 15 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='Ideal Defense 80 Upper Bound Relative Revenue 60 40 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='50 Pool size100- .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='. T=6 t = 12 t= 18 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Ideal Defense 80 - Upper Bound Revenue 60 - Relative 40 - 20 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='50 Pool size12 Q1: What are the assumptions used in the proposed defense mechanism?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In the proposed defense mechanism, we assume that every new block has propagated to all nodes before discovering a new block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' There are communication delays in sending and receiving messages between nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' These delays are ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' This assumption leads to the highest propagation speed to synchro- nize all nodes as soon as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Q2: How to set the interval for τ time interval?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' As examined by experiments performed while writ- ing this paper, time interval has no significant im- pact on Nik defense before setting τ time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' It doesn’t mean that time interval is unimportant, but choosing a reasonable interval can help defense mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Q3: How to set the number of τ time intervals in one θ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Before setting the number of τ time intervals, some experiments should be done based on the type of learning automata as we did in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' On the other hand, by answering Q1 in this section, commu- nication delays in real situations should be consid- ered in choosing the number of τ time intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Q4: Can different fail-safe parameters collapse the Bitcoin network?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Choosing a fail-safe parameter can be hard as a syn- chronization mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Miners may have different values for fail-safe parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Our model assumes that all nodes in the network are synced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Different values of the fail-safe parameter in the synced net- work may lead to a large-scale voting system for choosing the winning branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' At last, miners can choose the winning branch, decreasing the selfish attack’s profitability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Q5: Why do we use AI for the Nik defense as a self-organized mechanism?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' These days, AI has been increasingly used for de- tecting and preventing attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In blockchain based networks, The events are executed very fast and proposing self-organized and reactive mechanisms that are able to execute appropriate operation after execution of critical events is vital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Therefore, we have decided to use learning automata based self- adaptive algorithm as an AI tool to prevent selfish mining attacks considering this potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Blockchain is an unknown space, and since we don’t have any information about forks, reinforcement learning of learning automata is a good choice for defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Hence learning automata were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Q6: What development hurdles can we face while adopting the proposed mechanism?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The main development hurdle that affect the pro- posed mechanism is the synchronization algorithms of blocks in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The assumption of the upper limit for propagation time of Nik defense can be challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' It should be noted that this is not limitation of our model and other defense models reported in [34], [35], [36] face with this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 8 CONCLUSION AND FUTURE RESEARCH DIREC- TIONS The selfish mining attack hurt Bitcoin’s incentive com- patibility by disrupting the rewarding system’s fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='In addition, the decentralization capability of Bitcoin can be threatened by it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' To solve this problem, a new defense based on learning automata as an AI tool was proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' The pro- posed method suggested a set of policies to reinforce chain selection algorithm in a self-organized manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Experiments have shown the superiority of the proposed defense in Bitcoin’s network in comparison with existing mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Compared to existing defenses, the proposed defense can noticeably decrease the profitability of the selfish mining attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' On the other hand, it can increase the threshold for starting the attack’s profitability from 25% to 40% of the network’s computational power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Since the proposed study is the first use of an AI-based algorithm for organizing a self-organized defense mecha- nism, several works in the area of AI to detect and prevent selfish mining attacks can be considered as future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In this paper, learning automata theory was used as a self- adaptive decision-maker, but many other models such as Q-learning can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' This means that other AI tools can modify the proposed algorithm to have a better solution against selfish mining attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' In addition, many open problems in the area of proof-of- work consensus need to be studied and explored to develop a more usable and efficient industrial consensus without the selfish mining attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Alternative consensus algorithms and integration with other systems and architectures are examples of several open issues that can be considered as future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} 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+page_content=' IEEE International Conference On Communications (ICC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 1-6 (2019) [16] Ron, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' & Shamir, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Quantitative analysis of the full bitcoin transaction graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' International Conference On Financial Cryptography And Data Security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 6-24 (2013) [17] Sompolinsky, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' & Zohar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Secure high-rate transaction process- ing in bitcoin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' International Conference On Financial Cryptography And Data Security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' pp.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=', Liew, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' & Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' When blockchain meets AI: Optimal mining strategy achieved by machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' International Journal Of Intelligent Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 36, 2183-2207 (2021) [24] Hou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=', Zhou, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=', Ji, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=', Daian, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=', Tramer, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=', Fanti, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' & Juels, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' SquirRL: Automating attack analysis on blockchain incentive mech- anisms with deep reinforcement learning.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Proceedings Of The 15th ACM International Conference On Systems And Storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 148-148 (2022) [26] Thathachar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' & Sastry, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Networks of learning automata: Techniques for online stochastic optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' (Springer Science & Business Media,2003) [27] Narendra, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' & Thathachar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Learning automata: an introduc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' (Courier corporation,2012) [28] Rezvanian, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=', Saghiri, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=', Vahidipour, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=', Esnaashari, M.' 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[38] Heilman, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' One weird trick to stop selfish miners: Fresh bitcoins, a solution for the honest miner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' International Conference On Financial Cryptography And Data Security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 161-162 (2014) [39] Chicarino, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=', Albuquerque, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=', Jesus, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' & Rocha, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' On the detec- tion of selfish mining and stalker attacks in blockchain networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Annals Of Telecommunications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 75, 143-152 (2020) [40] Peterson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=', Andel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' & Benton, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Towards Detection of Selfish Mining Using Machine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' International Conference On Cyber Warfare And Security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 17 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' 237-243 (2022) Ali Nikhalat Jahromi He received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='Sc de- gree in Electrical Engineering in 2018 and then the M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='Sc degree in Computer Engineering in 2021, both from Amirkabir University of Technol- ogy, Tehran, Iran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' His research interests include Software Systems, Parallel and Distributed Sys- tems, and Programming Languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Ali Mohammad Saghiri He received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' degree from University of Science and Culture, in 2007 and the M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' and Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' degrees from AmirKabir University of Technology, Tehran, Iran, in 2010 and 2017, respectively, all in computer engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' He published more than 40 sci- entific papers on international conferences and journals among which Journal of Network and Computer Applications(JNCA), Applied Intelli- gence, and international journal of communica- tion systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' His research interests include the Internet of Things, Blockchain, and Artificial Intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' Mohammad Reza Meybodi He received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' degrees in economics from Shahid Beheshti University, Tehran, Iran, in 1973 and 1977, respectively, the M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' and Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' de- grees in computer science from Oklahoma Uni- versity, Norman, OK, USA, in 1980 and 1983, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=',He was an Assistant Professor with Western Michigan University, Kalamazoo, MI, USA, from 1983 to 1985, and an Associate Pro- fessor with Ohio University, Athens, OH, USA, from 1985 to 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' He is currently a Full Profes- sor with the Computer Engineering Department, Amirkabir University of Technology, Tehran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} +page_content=' His research interests include wireless networks, fault tolerant systems, learning systems, parallel algorithms, soft com- puting, and software development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FJT4oBgHgl3EQfKCyb/content/2301.11463v1.pdf'} diff --git a/QdAzT4oBgHgl3EQfz_5Y/vector_store/index.faiss b/QdAzT4oBgHgl3EQfz_5Y/vector_store/index.faiss new file mode 100644 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a/UtFLT4oBgHgl3EQfRS82/content/tmp_files/2301.12036v1.pdf.txt b/UtFLT4oBgHgl3EQfRS82/content/tmp_files/2301.12036v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bbf349620bf8f44f3eb24f503b7c954b1ade49f1 --- /dev/null +++ b/UtFLT4oBgHgl3EQfRS82/content/tmp_files/2301.12036v1.pdf.txt @@ -0,0 +1,769 @@ + +1 +Analyzing Robustness of the Deep Reinforcement Learning Algorithm in Ramp Metering +Applications Considering False Data Injection Attack and Defense + + +Diyi Liu +Affiliations: Department of Civil and Environment Engineering +University of Tennessee, Knoxville, Tennessee, USA +Email: dliu27@vols.utk.edu + +Lanmin Liu +Affiliations: Department of Civil and Environment Engineering +University of Tennessee, Knoxville, Tennessee, USA +Email: lliu53@vols.utk.edu + +Lee D Han +Affiliations: Department of Civil and Environment Engineering +University of Tennessee, Knoxville, Tennessee, USA +Email: lhan@utk.edu + + + + + +2 +Abstract +Decades of practices of ramp metering, by controlling downstream volume and smoothing the +interweaving traffic, have proved that ramp metering can decrease total travel time, mitigate shockwaves, +decrease rear-end collisions, reduce pollution, etc. Besides traditional methods like ALIENA algorithms, +Deep Reinforcement Learning algorithms have been established recently to build finer control on ramp +metering. However, those Deep Learning models may be venerable to adversarial attacks. Thus, it is +important to investigate the robustness of those models under False Data Injection adversarial attack. +Furthermore, algorithms capable of detecting anomaly data from clean data are the key to safeguard Deep +Learning algorithm. In this study, an online algorithm that can distinguish adversarial data from clean +data are tested. Results found that in most cases anomaly data can be distinguished from clean data, +although their difference is too small to be manually distinguished by humans. In practice, whenever +adversarial/hazardous data is detected, the system can fall back to a fixed control program, and experts +should investigate the detectors status or security protocols afterwards before real damages happen. +Keywords: Ramp Metering, Reinforcement Learning, Deep Q-Learning, adversarial data attack, False +Data Injection, anomaly data detection + + + + +3 +1. Introduction +Ramp metering reduces overall freeway congestion by installing traffic signals on freeway on-ramps to +manage the amount of traffic entering the freeway. Ramp metering strategy has been proven to be an +effective method for decades to reduce traffic delays by decreasing speed variance, shockwaves, average +delays, etc. The process of ramp metering on ramps is (1) vehicle pulls up to stop bar; (2) vehicle +detected, and then signal turns green; (3) vehicle merges onto freeway. To realize this process, the traffic +signal usually has a fixed green time duration (2 seconds for one vehicle passing) and changeable red +duration. +In recent years, reinforcement learning has become very successful in tackling many useful tasks +including play video games, autonomous driving, etc. In transportation, reinforcement learning becomes +useful in many applications including signal control, connected/automated vehicle’s algorithms, etc. In +reinforcement learning, unlike supervised/unsupervised machine learning methods, data is created by +agents by observing the environment. These agents can then run algorithms to decide their actions. The +action, in turn, changes the environment to their own benefits. + +One important part of reinforcement learning is the rewards. Given state at time 𝑡, rewards are the +benefits gained of the state denoted as 𝑅!. Usually, given the state 𝑠, the long term expected rewards are +recorded as 𝑅 = 𝐸[∑ +𝛾!𝑟! +" +!#$ +|𝑠$ = 𝑠]. Different from rewards, Q-value measures the value given both +state and action. Among different methods of training reinforcement learning models, the Q-learning has +become widely adopted for training different models. Deep Q-learning, instead of using a query table to +check Q-value given state and action, uses a neural network to output Q value. Compared to traditional +query table, the input data is continuous values, making it possible to more accurately estimating the Q- +values. +The study compares and tests the robustness of reinforcement algorithms in ramp metering control. Many +countermeasures are checked to make the program robust and general. The major steps of this study are as +follows: (1) Develop Ramp metering Algorithm. Instead of training models in one environment, the +model is jointly trained in different environments. (2) Implement Adversarial samples for DRL. White- +box attack: fast Gradient Sign method. (FGSM). (3) Identifying online cyber-attacks using Machine +Learning: building up statistical profiles and identifying erroneous data. +2. Literature Review +A considerable amount of literature has been published on ramp metering strategies and +algorithms, which could generally be divided into three categories: fixed time, local control, and +system-wide control. Papageorgiou (1) concludes that the strategies used for ramp metering: (1) +fixed-time strategies; (2) Reactive strategies; (3) Nonlinear optimal Ramp metering strategies; +(4) integrated freeway network traffic control. In his study, a freeway simulation was conducted +to compare densities/queues results between no control and control. + +Agent +state +reward +action +'s +Rt +A, +St+1 +Environment +4 +Fixed time metering is the simplest approach with fixed cycle length, but it is also considered +low efficient because the metering rate couldn’t be adjusted according to the real-time freeway +traffic states. System-wide control is proper when it comes to system optimization, which is +responsive to corridor-wide real-time traffic conditions. And system-wide control is usually +based on local control except that multiple ramps along the corridor are considered at the same +time. ALINEA is one of the local control strategies proposed by Papageorgiou (2) in 1991. In this +paper, the metering rates are modeled as a control theory problem, which is determined based on +occupancy data collected from mainline loop detectors located downstream. The goal is to +maximize the mainline throughput. And an experimental study was implemented in Paris, +France. Although the ALINEA method becomes the most recognized one, the method does have +some limitations. Firstly, the downstream bottleneck cannot be too far away from the ramp’s site +suffering from the “poorly damped closed-loop behavior”. Secondly, the critical occupancy +needs to be estimated. Thirdly, the placement of the loop detector must be at the traffic +bottlenecks. Many methods are proposed based on ALINEA to overcome its drawbacks. Instead +of measuring the downstream location, AU-ALINEA (3) used the measurements from the +upstream site instead of the downstream. PI-ALINEA (4) is proposed to tackle different geometry +cases with satisfactory performance including an uphill case, a lane drop case, and an +“uncontrolled downstream on-ramp case”. +In addition to ALINEA, there are many other methods. For example, Gomesa (5) models the +problem using the cell transmission model (CTM). A lot of math derivation is involved. And Ma +(6) applies a statistical model to evaluate the effectiveness of the before/after the ramp. Recently, +with the advancement in computing powers, reinforcement learning becomes another useful +method to train algorithms in ramp metering. While many reinforcement learning algorithms +claimed to be powerful, the performance of the model is unknown as many data input +assumptions of such models are of doubt. For example, some claim that a camera can view the +density and location of every vehicle. While this is true in a simulation environment, it is not +feasible in practice. To the authors' best knowledge, there lacks a comparison between traditional +and reinforcement methods under the same assumptions. Rezaee (7) applies reinforcement +learning to ramp metering and uses the KNN-TD method to represent continuous state space. He +also compared many RL methods and built test beds to test the performance (8). Schmidt- +Dumont (9)uses reinforcement learning (Q-learning) for optimal control, in which state-action +values are presented by a neural network instead of a table. And a simple simulation case is used +to test the performance of the algorithm. Belletti (10) tested the reinforcement learning algorithm +by simulation, in which the effectiveness is demonstrated by generating a space time diagram +with “any” speed distribution. From a system-wide aspect, Lu (11) considered minimizing TTT +and penalty using the variation of variables for equity issues (e.g., queue length) to solve +multiple ramp metering problems and did a simulation using a real network layout. +As the penetration rate of autonomous vehicles and connected vehicles increases, the traffic +becomes mixed traffic in the foreseeable future. New and more questions come into play with +respect to the interaction between connected vehicles. Those questions are identified and +discussed recently. Vrbanic (12) has a Good and in-depth understanding of the traffic control + + +5 +problem by discussing VSL and RM together and asks the right questions considering the +involvement of autonomous vehicles, connected vehicles, etc. +Although effective in practice, the problem cannot be solved in a perfect way since: (1) the +bottleneck cannot be identified; (2) the geometry layout is complex with multiple on-ramps and +multiple bottlenecks; (3) new emerging autonomous/connected vehicles make the system more +diverse, bringing in different driving behaviors. In contrast, the opportunities of deploying more +complex methods are also emerging in the last few years: (1) more complicated algorithms are +available as detectors and computing devices become cheaper and cheaper. (2) the new +information flow from connected vehicles or videos might bring new data sources for more +detailed control maneuvers. Thus, this research topic remains an important one. Compared with +complicated intersection signals with 4 directions and many signal phases, the connections +between data and control are easier for humans to comprehend subjectively. +There are many adversarial machine learning technologies to generate adversarial samples. One +of the first established algorithms is the Fast Gradient Sign Method (FGSM). First, the attacker +decides the target of interest (e.g., block a specific traffic lane). Then, given output selected as +the target of interest, a partial derivative is taken with respect to the input data to decide the +gradient sign for each data input. A noise data is generated by taking a small step along each +gradient sign direction. The FGSM method, by injecting a small value over the clean sample, +generates the adversarial samples that trick the deep learning model to generate wrong outputs. +3. Methodology +3.1 ALINEA +ALINEA is a real-time ramp metering strategy that controls the ramp input traffic flow by +monitoring the traffic occupancy on the mainstream. ALINEA keeps calculating the metering +rate(r) in each cycle to keep the main road stream stable, and accordingly mitigate congestion. +The normal scenario for ALINEA requires a traffic signal that is installed on the ramp which is +to control the ramp input traffic and loop detectors that are installed downstream of the main +road, which is to measure the occupancy of the mainstream. The formula is shown below: +𝑟(𝑘) = 𝑟 (𝑘 − 1) + 𝐾% ∗ (𝑜6 − 𝑜&'!(𝑘 − 1)) +𝑖𝑓 𝑟(𝑘) > 𝑟𝑚𝑎𝑥, 𝑟(𝑘) = 𝑟𝑚𝑎𝑥 +𝑖𝑓 𝑟(𝑘) < 𝑟𝑚𝑖𝑛, 𝑟(𝑘) = 𝑟𝑚𝑖𝑛 +Where 𝑘 is the time index, 𝑘 = 1, 2, 3…, means at cycle 𝑘; 𝑟(𝑘) is the metering rate at 𝑘; K! is a +fixed parameter; 𝑜6 is the desired occupancy, which is the occupancy needs to be maintained; o"#$ +is the occupancy in downstream, which is detected by loop detectors; 𝑟𝑚𝑎𝑥 is the maximum for +𝑟(𝑘) which equals to 1600 veh/h or 1800 veh/h for single-lane ramps; 𝑟𝑚𝑖𝑛 is the minimum for +𝑟(𝑘) which is the admissible flow for single-lane ramps (200~400 vehicles/hour). + +3.2 Reinforcement Learning + + +6 +Reinforcement learning is the training of machine learning models to make a sequence of +decisions. The agent chooses the optimal behavior in an environment to maximize the expected +total reward. The reinforcement learning method derives from the problem of optimal control of +Markov Decision Processes (MDP). The main elements of an RL system are: the agent or the +learner, the environment the agent interacts with, the policy that the agent follows to take actions, +and the reward signal that the agent observes upon taking actions. + +The key to reinforcement learning is to solve the Bellman equation, the agent interacts with the +environment to sample different states and rewards by using Epsilon greedy algorithm, which is +the balance between exploration and exploitation. After sampling enough states and actions, the +value function of the MDP can be estimated. And the optimal strategy is always choosing the +highest value to get the maximum total rewards. +𝑉(𝑠) = 𝑚𝑎𝑥%(𝑅(𝑠, 𝑎) + 𝛾𝑉(𝑠&)) +State (𝑠): the current state where the agent is in the environment +Next State (𝑠&): After taking action (𝑎) at state (𝑠) the agent reaches 𝑠& +Value (𝑉): Numeric representation of a state which helps the agent to find its path. +Reward (𝑅): treat which the agent gets after performing an action (𝑎) +3.3 Deep Q-learning +Q-learning is one of the reinforcement learning methods. The main idea is to update Q values +which denote the value of choosing action given state 𝑠. Q values are calculated by some +constant parameters, rewards, and the Q value of the next time step. For each iteration, Q values +of all states and actions are calculated, which is called the Q table. And the optimal strategy is to +always choose the action that has the highest Q value. +𝑄(𝑆', 𝐴') = (1 − 𝛼)𝑄(𝑆', 𝐴') + 𝛼 ∗ (𝑅' + 𝜆 ∗ 𝑚𝑎𝑥%𝑄(𝑆'(), 𝑎)) +𝑆= State, the data collected from loop detectors +𝐴= Action the agent takes, red duration +𝑅= Reward from taking an action +t= Time step +𝛼= The learning rate +𝜆= The discount factor + +Agent +state +reward +action +'s +Rt +A, +St+1 +Environment +7 +Although this method is simple to implement, the method is unable to estimate values for unseen +states. To fix this problem, the Deep Q-learning method is proposed. Deep Q-Network (DQN) +uses Neural Networks to estimate Q-values, which overcomes the complexity of environments. +But DQN can only handle discrete, low-dimensional action spaces. In this study, the DQN is +applied to solve the local ramp metering problem. In the ramp metering context, states are the +traffic states, and actions are the lengths of the red duration of the signal. By building the +connections between the SUMO simulation and the DQN algorithm, the iteration process could +be implemented. +3.4 Constructing Rewards Function +As mentioned, rewards function, 𝑅(𝑠, 𝑎), is defined as the benefits gained from taking action 𝑎 +at state 𝑠. For ramp metering control problem, the final objective is to minimize the total travel +time of all vehicles within the system, regardless of whether they are originated from the +highway mainline or from the on ramps. Thus, the rewards function should the action’s impact +on total traveling time. Given the current traffic state, for the next 30 seconds, the expected total +travel time of new incoming vehicles. To simplify, all delays are generated from two region: (1) +Region 1: the waiting time along the on-ramp queue; (2) Region 2: the traveling time through the +highway’s downstream merging zone. On ramp vehicles would traveling through both Region 1 +and Region 2, whereas mainline vehicles would only travel through Region 2. Those regions are +shown in Figure 1. + +Figure 1. Definition of important variables in formulating rewards function + +Assuming the downstream loop detector generates an observation (𝑛', 𝑜') every 𝑇* time (e.g., 30 +seconds). Also, assume that the average vehicle length is 𝐿+. Then, the downstream speed in +Region 2 is: +R2, !!" +R1, !!,# +"$,#%$ +"&,#%$ +… +##, %# +"&,#%$: incoming vehicles on ramp between t and t+1 +N$,'%$: incoming vehicles on ramp between t and t+1 +"1: Region 1, the on-ramp queueing zone +7#,%: Number of vehicles stayed in on ramp’s queue at time t +8%&': signal’s control plan between t and t+1 +9%&': the ramp metering rate of A(&' (seconds per vehicle) +R2: Region 2, the mainline merging zone +;%: occupancy detected at downstream loop detector between t and t+1 +<%: number of detected at downstream loop detector vehicles at time t +)(&: length of the merging zone +*#%$, +#%$ + +100m +8 +𝑣,-,' = 𝑛' ⋅ 𝐿+ +𝑜' ⋅ 𝑇* + +Assuming the speed for the next 𝑇* are similar to current speed: +𝑣,-,'() ≈ 𝑣,-,' +Also, the region 2’s speed are about the same. Then, the travel time for one vehicle to travel +through downstream region can be estimated as: +𝑡𝑡,- = 𝑑- +𝑣,-,' + +where 𝑑- is the length of the merging zone. Similarly, one can estimate the expected travel time +of one incoming vehicle traveling through the ramp. That is, the total time spending on the +queues. As vehicles can only join the back of the queue, this time depends on both the metering +rate 𝐶% (in seconds/vehicle) and the current queue length 𝐿/: +𝑡𝑡,) = A𝐿/ ⋅ 𝐶%B +Assuming in the next 𝑇* seconds, the corresponding number of incoming vehicles on the +mainline and on the ramp are 𝑁) and 𝑁-, respectively. Then, the total time for those vehicles to +travel through both Region 1 and 2 becomes: +𝑇𝑇'() = 𝑁) ⋅ 𝑡𝑡,) + (𝑁) + 𝑁-) ⋅ 𝑡𝑡,- +For each vehicle, the averaged travel time becomes: +𝑇𝑇'() +DDDDDDD = +𝑁) +(𝑁) + 𝑁-) ⋅ 𝑡𝑡,) + 𝑡𝑡,- +As for the rewards function at time t, the reciprocal of total travel time is used: +𝑅'(𝑠, 𝑎) = +1 +𝑇𝑇'() +DDDDDDD +Indeed, although there are many assumptions and approximations, the reward function is simple +enough to be applied to a wide range of scenarios without requiring very detailed information +including the number of departure vehicles, the number of vehicles within the region (i.e., +densities), etc. +4. Experiment +The Experiment is divided into two phases. In phase 1, the simulation testbed is established with different +algorithms tested. The performance of no control (NC), ALINEA method, and deep Q-learning method +(DQL) are compared. Furthermore, DQL is attacked by generating adversarial data to see the extend a +model can be undermined. Two attack scenarios are executed. One attack targeted at blocking ramp, the +other targeted at blocking downstream traffic bottlenecks. During the process, all signals, no matter clean +data or adversarial data, are recorded for the next phase of analysis. The second phase tries to analyze the +statistical difference between the clean dataset using statistical patterns. Furthermore, whether an online + + +9 +algorithm can distinguish the attack signals from is tested. With that, the testbed and its configuration are +discussed in the next paragraph. +In this Experiment, a SUMO simulation testbed is built for analysis. The geometries are shown in Figure +2. A ramp is merged into highway with 3 upstream lanes. The merging zone has an extra acceleration lane +for vehicles to merge into highway spans over 300 meters before merged back to 3 lanes. As shown in the +red rectangles, three sets of loop detectors are available: the upstream detectors; the downstream +detectors, and the on-ramp loop detectors. Each detector generates a signal of traffic volume and traffic +occupancy every 30 seconds. Besides, it is assumed that the queue length can also be detected on the +ramp, either through loop detectors or a CCTV/video-based system. Finally, the orange rectangle shows +the position of the ramp metering signal to meter the ramp traffic. + +Figure 2 Geometry layouts and detector configurations of the simulation testbed + +During the training/testing process, the following traffic demands are configured in Figure 3. The upper +and lower diagram shows the volume settings for training and testing scenarios, respectively. Only DQL +model needs to the training phase. The settings are relatively simple as it is easier to learn control policies +in a “stationary environment”. In contrast, the test case covers traffic scenarios from medium to high +highway traffic volume ranges from 667-2000 vehicles/hour/lane. The ramp volume is fixed to 700 +vehicles/hour, relatively close to the maximal capacity of a signal-controlled lane. Note that there is also a +clearance phase at the end of each simulation. As long as there are running vehicles, the simulation will +continue. In such cases, the traffic volume at a low level to simulate the off-peak periods. + +: Loop detectors (upstream, downstream) +: Ramp metering control (traffic signal) + Merge zones (4 lanes to 3 lanes) +Merging zone +3 lanes +4 lanes +3 lanes +Upstream loop detectors +Downstream loop detectors +Signals +on ramp loop detectors +10 + +Figure 3 Traffic input volume used for training/testing different algorithms + +To simulate the one-green-per-vehicle policy, each green phase is fixed to 2 seconds. By changing the +length of red signal phase, the volume inject to highway from ramp can be metered. 6 gears are provided +for the ramp metering, as summarized in Table 1 below. For both ALINEA method and DQL method, the +same gears are applied. +Table 1 Eight different signalizing plans for ramp metering control +“Gear” +G0 +G1 +G2 +G3 +G4 +G5 +G6 +G7 +Green phase length (seconds) +0* +2 +Red phase length (seconds) +0 +1 +2 +3 +4 +5 +6 +7 +Total cycle length (seconds) +2 +3 +4 +5 +6 +7 +8 +9 + +Finally, to handle the case of over-spilling queue on ramp, a fixed control rule is applied: as soon as the +ramp’s queue length becomes greater than 40 vehicles, ramp metering policies are not applied anymore +for 20 seconds to clear the queue. This rule is fix programmed and of highest order to prevent overspill +from happening regardless of the applied algorithm. +For DQL, some results for the training process are summarized in Figure 4 below. The model is trained +for 100 episodes, and each episode is composed of 400 epochs of training. As shown below, the total +travel time has been consistently decreased over time. + +upstream highway traffic +ramp metering volume +upstream highwaytraffic +ramp metering volume +11 + +Figure 4. Performance of the system during training episode + +5. Results +5.1 Phase 1 results: evaluating the performance of the system +To compare their performance, all testbed is configured the same (e.g., geometry, traffic volume, detector +layouts) except the control algorithm. Among all different measures, the Total Travel Time (TTT) is the +most reliable metric to measure system-wise performance. Note that to compare TTT, the simulation +cannot be terminated until all vehicles reached at their destinations. As mentioned, five different scenarios +are checked, and the results for each scenario are summarized in Table 2 below. Given limited training +time and information detected, the performance of Deep Q-Learning is not as good as the no control. +However, the algorithm in general can maintain the tradeoff between ramp’s queue length and +downstream bottleneck’s travel speed. ALINEA method performs the best among all scenarios. +Table 2 Simulation Results +Scenario +TTT +(10^3) +Adversarial +data injected +Attack Target +Comments +1. NC +1,089 + +N.A. +No control +2. ALINEA +1,045 + +N.A. + +3. DQL +854 + +N.A. +Deep Q- +Learning +4. DQL+FGSM1 +903 +x +Block ramp +FDI Step=0.02 +5. DQL+ FGSM2 +900 +x +Block downstream bottleneck +FDI Step=0.02 + +1e6 +1.60 +(seconds) +1.55 +e +1.50 +I time + travel +1.45 +Total +1.40 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +Episode +12 + +More information during the simulation time can be directly plotted and visualized to gain more insights, +as shown in Figure 5. Each scenario has an upper subplot and a lower subplot. The red lines and blue +lines in the upper subplot show the downstream vehicle speed and ramp’s queue length, respectively. By +observation, there is a tradeoff between the two metrics: if one wants to increase the downstream speed, +then the interest of on-ramp users is compromised by waiting for a longer time over the queue. Both +ALINEA and DQL methods can automatically balance this tradeoff, revealing that both algorithms +worked as expected. Scenario 4 and 5 tries to undermine the system’s performance using by injecting +adversarial data using Fast Gradient Sign Method (FSGM) with a small step width of 0.02. This means +that if one input number is 0.50, then the adversarial number may become either 0.48 or 0.52. With those +settings, the performances of the system are severely undermined to the attacker’s benefits. + +Figure 5 Comparison of performance of different scenarios by visualizing information over time + +5.2 Phase 2 results: Distinguishing the difference between adversarial and clean data +In methodology, two methods are proposed to distinguish injected data (i.e., adversarial data) from the +clean data: GEM and PCA methods. The adversarial data gained from scenarios 4 and 5 are separately +tested against the clean data generated from scenario 1-3. The results are shown in Figure 6 below. In +general, although a small step is chosen by the FGSM in generating adversarial data, adversarial data still +have different GEM and PCA statistics (see red histograms) compared with that of clean data (see green +histograms). +(a) +(b) +1. No control +2.ALINEA algorithm +3. DQL algorithm +4. DQL algorithm (attack to block ramp) +5. DQL algorithm (attack to block downstream highway) +(d) +(e) +(f) +Traffic Demand +(c) + +ranp's queue length +downstream speedmp's queue lengthra np's queue length +downstream speedra np's queue length +downstream speedra np's queue length +downstream speedupstream highway traffic +ramp metering volume +13 + +Figure 6 The distribution of distance statistics between of clean data (in green) vs. adversarial data +(in red) +5.3 Using these distance metrices, the performance of online detection algorithm is evaluated. For +last five records, a majority vote is adopted if there are more than 3 records referred as polluted +ones. More results are summarized in the following An online detecting method in identifying +false injected data +The previous section shows the statistical difference between clean sample and polluted sample. In +practice, one needs to decide whether a stream of signals is polluted or not. For each record, the following +Equations define score to show the magnitude of deviation from the correct data. +𝑠()* = log C +𝛼 +𝑝()*,, +F , 𝛼 = 0.10 +𝑠-./ = log C +𝛼 +𝑝-./,, +F , 𝛼 = 0.10 +Given this definition of scores, a record falls as an outlier of clean samples tend to be assigned a large +score, making it suspect to be a FDI data instead of a clean sample. Furthermore, scores can be averaged +to an ensembled score to jointly consider two statistical traits using one metric. +𝑠)01 = 0.5 × (𝑠()* + 𝑠-./) +In such settings, if an extreme data is observed with 𝑝()*,, = 0.001, the score 𝑠()* becomes +log(10) ≈ 2.30. The higher the score, the more likely data is not generated from the clean sample. On the +other hand, for a new observed data with 𝑝()*,, = 0.5, the score 𝑠()* becomes log�(0.2) ≈ −1.60. +Thus, if the score over a time series of records is accumulated and becomes a large positive number, we +can indicate that the series of data is polluted instead of being clean data, as defined in the new +indicator 𝑔2,! below: +𝑔2,! = maxR0, 𝑔2,!34 + 𝑠2,!S , 𝑔2,$ = 0; 𝑠2,!: score of method 𝑚 at time 𝑡, 𝑚∈{GEM, PCA, ENS}, +𝐴2,! = 1, 𝑖𝑓 𝑔2,! ≥ ℎ + +(a) +GEM-based summary histograms +(c) +1200 +good reading (S2) +700 +good reading (S2) +0.10 + polluted reading (S3) +polluted reading (S3) +good reading (S2) +1000 +600 +polluted reading (S3) +0.08 +800 +500 +400 +600 +300 +400 +200 +0.02 +200 +100 +0.00 +0 +4 +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +i +2 +5 +2 +(d) +() +GEM statistics +(e) +GEM-based summary histograms +PCA-based summary histograms +1400 +good reading (S2) +1000 +good reading (S2) +0.10 +good reading (S2) +polluted reading (S3) +polluted reading (53) +1200 +polluted reading (S3) +800 +1000 +0.08 +800 +600 +600 +400 +0.04 +400 +200 +0.02 +200 +0.00 +0 +1 +2 +3 +4 +5 +6 +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +1 +2 +5 +6 +GEM statistics +14 +Given the score of one data, an empirical score 𝑔2,! can be used to measure the accumulated score. If +𝑔2,! becomes no less than a threshold ℎ, then a False Data Injection alarm with be generated at time 𝑡 +under method 𝑚, which can be one of the three methods. +The results of one experiment run on both clean samples and adversarial samples are visualized below. +Among three proposed methods, the PCA method gets the best performance in detecting false data +injection attack, whereas the GEM method doesn’t perform too well. The ensemble method combining +two scores are also good in detecting False Data Injection. The benefits of using ensemble method can be +viewed in the lower diagram where all methods generate many false alarms except the ensemble method. +Thus, the ensemble method is more preferred among three methods. + +Figure 7 Performance of anomaly detector machine on both adversarial samples and clean samples + +5.4 An online detecting method in identifying false injected data +The previous section shows the statistical difference between clean sample and polluted sample. In +practice, one needs to decide whether a stream of signals is polluted or not. For each record, the following +Equations define score to show the magnitude of deviation from the correct data. +𝑠()* = log C +𝛼 +𝑝()*,, +F , 𝛼 = 0.10 +𝑠-./ = log C +𝛼 +𝑝-./,, +F , 𝛼 = 0.10 +!! +(on adversarial sample) +(on clean sample) + +PCA +GEM +ensemble +control threshold h=10PCA +GEM +ensemble +control threshold h=10 +15 +Given this definition of scores, a record falls as an outlier of clean samples tend to be assigned a large +score, making it suspect to be a FDI data instead of a clean sample. Furthermore, scores can be averaged +to an ensembled score to jointly consider two statistical traits using one metric. +𝑠)01 = 0.5 × (𝑠()* + 𝑠-./) +In such settings, if an extreme data is observed with 𝑝()*,, = 0.001, the score 𝑠()* becomes +log(10) ≈ 2.30. The higher the score, the more likely data is not generated from the clean sample. On the +other hand, for a new observed data with 𝑝()*,, = 0.5, the score 𝑠()* becomes log(0.2) ≈ −1.60. Thus, +if the score over a time series of records is accumulated and becomes a large positive number, we can +indicate that the series of data is polluted instead of being clean data, as defined in the new indicator 𝑔2,! +below: +𝑔2,! = maxR0, 𝑔2,!34 + 𝑠2,!S , 𝑔2,$ = 0; 𝑠2,!: score of method 𝑚 at time 𝑡, 𝑚 ∈ {GEM, PCA, ENS}, +𝐴2,! = 1, 𝑖𝑓 𝑔2,! ≥ ℎ +Given the score of one data, an empirical score 𝑔2,! can be used to measure the accumulated score. If +𝑔2,! becomes no less than a threshold ℎ, then a False Data Injection alarm with be generated at time 𝑡 +under method 𝑚, which can be one of the three methods. +The results of one experiment run on both clean samples and adversarial samples are visualized below. +Among three proposed methods, the PCA method gets the best performance in detecting false data +injection attack, whereas the GEM method doesn’t perform too well. The ensemble method combining +two scores are also good in detecting False Data Injection. The benefits of using ensemble method can be +viewed in the lower diagram where all methods generate many false alarms except the ensemble method. +Thus, the ensemble method is more preferred among three methods. +6. Limitations +There are some limitations in this study. First, the ramp metering algorithm is applied in a simple testbed. +For future study, the team will run the simulation over the whole corridor system with different highway +layouts at different locations. Next, there are many more off the shelf algorithms to generate adversarial +data. Also, we only consider white-box (insider) attack, the likelihood of black-box (outsider) attack +should also be included in the experiment. Similarly, there are other techniques identifying outliers (i.e., +adversarial data) to be tested but not included in this paper. Note that this study mainly applies the +strategy to fall back to traditional fixed programmed methods when adversarial attacks are detected. The +problem of how to train deep learning models robust to adversarial data is not discussed. There may be +other unforeseen simulation details that might undermining the simulation’s correctness. Besides ramp +metering, the Variable Speed Limit (VSL) is also a good supplement control strategy that needs to be +discussed to further exploit the benefits of Ramp Metering control. +There are also some other miscellaneous bullets to point out. The vehicle’s behavior in SUMO simulation +may be different from that of realistic world. For example, we command vehicles to turn left on the +acceleration ramp to merge with the other vehicles. If not, the vehicles would delay their left turn +maneuver until reaching the end of the acceleration ramp. In SUMO, it seems that the vehicles lane +changing model may outperform the realistic case. Although the demand is high on both ramp and +upstream highway, the benefits of ramp metering look trivial. +7. Conclusion + + +16 +The major contributions of this study are listed as follows: +• +Build up a testbed to test the efficiency of different ramp metering algorithms +• +Compare the efficiency of the different algorithms with or without white-box attack using FGSM +algorithm +• +Evaluate different statistical methods in identifying adversarial samples from clean samples +• +Evaluate the effectiveness and robustness of detecting false data +Overall, the results are positive that even for the insider attack, it is hard to generate adversarial data to +fool the proposed detection algorithms. +Acknowledgement +The project is sponsored by the Secure Constrained Machine Learning for Critical Infrastructure +CPS project, a National Science Foundation project. We would like to thank Dr. Asad J. Khattak +and Dr. Iman Mahdinia for providing useful feedbacks on this paper. + + + + +17 +Reference +1. +Papageorgiou, M., and A. Kotsialos. Freeway Ramp Metering: An Overview. IEEE Transactions on +Intelligent Transportation Systems. 4. Volume 3, 271–281. +2. +Papageorgiou, M., H. Hadj-salem, and J. Blosseville. ALINEA: A Local Feedback Control Law for On- +Ramp Metering. +3. +Smaragdis, E., M. Papageorgiou, and E. Kosmatopoulos. A Flow-Maximizing Adaptive Local Ramp +Metering Strategy. Transportation Research Part B: Methodological, Vol. 38, No. 3, 2004, pp. +251–270. https://doi.org/10.1016/S0191-2615(03)00012-2. +4. +Kan, Y., Y. Wang, M. Papageorgiou, and I. Papamichail. Local Ramp Metering with Distant +Downstream Bottlenecks: A Comparative Study. Transportation Research Part C: Emerging +Technologies, Vol. 62, 2016, pp. 149–170. https://doi.org/10.1016/j.trc.2015.08.016. +5. +Gomes, G., and R. Horowitz. Optimal Freeway Ramp Metering Using the Asymmetric Cell +Transmission Model. Transportation Research Part C: Emerging Technologies, Vol. 14, No. 4, +2006, pp. 244–262. https://doi.org/10.1016/j.trc.2006.08.001. +6. +Ma, X., A. Karimpour, and Y. J. Wu. Statistical Evaluation of Data Requirement for Ramp Metering +Performance Assessment. Transportation Research Part A: Policy and Practice, Vol. 141, 2020, pp. +248–261. https://doi.org/10.1016/j.tra.2020.09.011. +7. +Rezaee, K., B. Abdulhai, and H. Abdelgawad. Application of Reinforcement Learning with +Continuous State Space to Ramp Metering in Real-World Conditions. 2012. +8. +Rezaee, K. Decentralized Coordinated Optimal Ramp Metering Using Multi-Agent Reinforcement +Learning. 2014. +9. +Schmidt-Dumont, T., and J. H. van Vuuren. Decentralised Reinforcement Learning for Ramp +Metering and Variable Speed Limits on Highways. 2015. +10. +Belletti, F., D. Haziza, G. Gomes, and A. M. Bayen. Expert Level Control of Ramp Metering Based +on Multi-Task Deep Reinforcement Learning. IEEE Transactions on Intelligent Transportation +Systems, Vol. 19, No. 4, 2018, pp. 1198–1207. https://doi.org/10.1109/TITS.2017.2725912. +11. +Lu, C., J. Huang, L. Deng, and J. Gong. Coordinated Ramp Metering with Equity Consideration +Using Reinforcement Learning. Journal of Transportation Engineering, Vol. 143, No. 7, 2017. +https://doi.org/10.1061/JTEPBS.0000036. +12. +Vrbanić, F., E. Ivanjko, K. Kušić, and D. Čakija. Variable Speed Limit and Ramp Metering for Mixed +Traffic Flows: A Review and Open Questions. Applied Sciences (Switzerland). 6. Volume 11. + + diff --git a/UtFLT4oBgHgl3EQfRS82/content/tmp_files/load_file.txt b/UtFLT4oBgHgl3EQfRS82/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..aced3b155a84a388de4aa0bb00f6ccca52b92e0b --- /dev/null +++ b/UtFLT4oBgHgl3EQfRS82/content/tmp_files/load_file.txt @@ -0,0 +1,539 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf,len=538 +page_content='1 Analyzing Robustness of the Deep Reinforcement Learning Algorithm in Ramp Metering Applications Considering False Data Injection Attack and Defense Diyi Liu Affiliations: Department of Civil and Environment Engineering University of Tennessee, Knoxville, Tennessee, USA Email: dliu27@vols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='utk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='edu Lanmin Liu Affiliations: Department of Civil and Environment Engineering University of Tennessee, Knoxville, Tennessee, USA Email: lliu53@vols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='utk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='edu Lee D Han Affiliations: Department of Civil and Environment Engineering University of Tennessee, Knoxville, Tennessee, USA Email: lhan@utk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='edu 2 Abstract Decades of practices of ramp metering, by controlling downstream volume and smoothing the interweaving traffic, have proved that ramp metering can decrease total travel time, mitigate shockwaves, decrease rear-end collisions, reduce pollution, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Besides traditional methods like ALIENA algorithms, Deep Reinforcement Learning algorithms have been established recently to build finer control on ramp metering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' However, those Deep Learning models may be venerable to adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Thus, it is important to investigate the robustness of those models under False Data Injection adversarial attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Furthermore, algorithms capable of detecting anomaly data from clean data are the key to safeguard Deep Learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' In this study, an online algorithm that can distinguish adversarial data from clean data are tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Results found that in most cases anomaly data can be distinguished from clean data, although their difference is too small to be manually distinguished by humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' In practice, whenever adversarial/hazardous data is detected, the system can fall back to a fixed control program, and experts should investigate the detectors status or security protocols afterwards before real damages happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Keywords: Ramp Metering, Reinforcement Learning, Deep Q-Learning, adversarial data attack, False Data Injection, anomaly data detection 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Introduction Ramp metering reduces overall freeway congestion by installing traffic signals on freeway on-ramps to manage the amount of traffic entering the freeway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Ramp metering strategy has been proven to be an effective method for decades to reduce traffic delays by decreasing speed variance, shockwaves, average delays, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' The process of ramp metering on ramps is (1) vehicle pulls up to stop bar;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' (2) vehicle detected, and then signal turns green;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' (3) vehicle merges onto freeway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' To realize this process, the traffic signal usually has a fixed green time duration (2 seconds for one vehicle passing) and changeable red duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' In recent years, reinforcement learning has become very successful in tackling many useful tasks including play video games, autonomous driving, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' In transportation, reinforcement learning becomes useful in many applications including signal control, connected/automated vehicle’s algorithms, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' In reinforcement learning, unlike supervised/unsupervised machine learning methods, data is created by agents by observing the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' These agents can then run algorithms to decide their actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' The action, in turn, changes the environment to their own benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' One important part of reinforcement learning is the rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Given state at time 𝑡, rewards are the benefits gained of the state denoted as 𝑅!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='. Usually, given the state 𝑠, the long term expected rewards are recorded as 𝑅 = 𝐸[∑ 𝛾!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='𝑟!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' " !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='#$ |𝑠$ = 𝑠].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Different from rewards, Q-value measures the value given both state and action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Among different methods of training reinforcement learning models, the Q-learning has become widely adopted for training different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Deep Q-learning, instead of using a query table to check Q-value given state and action, uses a neural network to output Q value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Compared to traditional query table, the input data is continuous values, making it possible to more accurately estimating the Q- values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' The study compares and tests the robustness of reinforcement algorithms in ramp metering control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Many countermeasures are checked to make the program robust and general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' The major steps of this study are as follows: (1) Develop Ramp metering Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Instead of training models in one environment, the model is jointly trained in different environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' (2) Implement Adversarial samples for DRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' White- box attack: fast Gradient Sign method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' (FGSM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' (3) Identifying online cyber-attacks using Machine Learning: building up statistical profiles and identifying erroneous data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Literature Review A considerable amount of literature has been published on ramp metering strategies and algorithms, which could generally be divided into three categories: fixed time, local control, and system-wide control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Papageorgiou (1) concludes that the strategies used for ramp metering: (1) fixed-time strategies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' (2) Reactive strategies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' (3) Nonlinear optimal Ramp metering strategies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' (4) integrated freeway network traffic control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' In his study, a freeway simulation was conducted to compare densities/queues results between no control and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=" Agent state reward action 's Rt A, St+1 Environment 4 Fixed time metering is the simplest approach with fixed cycle length, but it is also considered low efficient because the metering rate couldn’t be adjusted according to the real-time freeway traffic states." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' System-wide control is proper when it comes to system optimization, which is responsive to corridor-wide real-time traffic conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' And system-wide control is usually based on local control except that multiple ramps along the corridor are considered at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' ALINEA is one of the local control strategies proposed by Papageorgiou (2) in 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' In this paper, the metering rates are modeled as a control theory problem, which is determined based on occupancy data collected from mainline loop detectors located downstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' The goal is to maximize the mainline throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' And an experimental study was implemented in Paris, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Although the ALINEA method becomes the most recognized one, the method does have some limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Firstly, the downstream bottleneck cannot be too far away from the ramp’s site suffering from the “poorly damped closed-loop behavior”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Secondly, the critical occupancy needs to be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Thirdly, the placement of the loop detector must be at the traffic bottlenecks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Many methods are proposed based on ALINEA to overcome its drawbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Instead of measuring the downstream location, AU-ALINEA (3) used the measurements from the upstream site instead of the downstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' PI-ALINEA (4) is proposed to tackle different geometry cases with satisfactory performance including an uphill case, a lane drop case, and an “uncontrolled downstream on-ramp case”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' In addition to ALINEA, there are many other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' For example, Gomesa (5) models the problem using the cell transmission model (CTM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' A lot of math derivation is involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' And Ma (6) applies a statistical model to evaluate the effectiveness of the before/after the ramp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Recently, with the advancement in computing powers, reinforcement learning becomes another useful method to train algorithms in ramp metering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' While many reinforcement learning algorithms claimed to be powerful, the performance of the model is unknown as many data input assumptions of such models are of doubt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' For example, some claim that a camera can view the density and location of every vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' While this is true in a simulation environment, it is not feasible in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=" To the authors' best knowledge, there lacks a comparison between traditional and reinforcement methods under the same assumptions." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Rezaee (7) applies reinforcement learning to ramp metering and uses the KNN-TD method to represent continuous state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' He also compared many RL methods and built test beds to test the performance (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Schmidt- Dumont (9)uses reinforcement learning (Q-learning) for optimal control, in which state-action values are presented by a neural network instead of a table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' And a simple simulation case is used to test the performance of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Belletti (10) tested the reinforcement learning algorithm by simulation, in which the effectiveness is demonstrated by generating a space time diagram with “any” speed distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' From a system-wide aspect, Lu (11) considered minimizing TTT and penalty using the variation of variables for equity issues (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=', queue length) to solve multiple ramp metering problems and did a simulation using a real network layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' As the penetration rate of autonomous vehicles and connected vehicles increases, the traffic becomes mixed traffic in the foreseeable future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' New and more questions come into play with respect to the interaction between connected vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Those questions are identified and discussed recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Vrbanic (12) has a Good and in-depth understanding of the traffic control 5 problem by discussing VSL and RM together and asks the right questions considering the involvement of autonomous vehicles, connected vehicles, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Although effective in practice, the problem cannot be solved in a perfect way since: (1) the bottleneck cannot be identified;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' (2) the geometry layout is complex with multiple on-ramps and multiple bottlenecks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' (3) new emerging autonomous/connected vehicles make the system more diverse, bringing in different driving behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' In contrast, the opportunities of deploying more complex methods are also emerging in the last few years: (1) more complicated algorithms are available as detectors and computing devices become cheaper and cheaper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' (2) the new information flow from connected vehicles or videos might bring new data sources for more detailed control maneuvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Thus, this research topic remains an important one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Compared with complicated intersection signals with 4 directions and many signal phases, the connections between data and control are easier for humans to comprehend subjectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' There are many adversarial machine learning technologies to generate adversarial samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' One of the first established algorithms is the Fast Gradient Sign Method (FGSM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' First, the attacker decides the target of interest (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=', block a specific traffic lane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Then, given output selected as the target of interest, a partial derivative is taken with respect to the input data to decide the gradient sign for each data input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' A noise data is generated by taking a small step along each gradient sign direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' The FGSM method, by injecting a small value over the clean sample, generates the adversarial samples that trick the deep learning model to generate wrong outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Methodology 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='1 ALINEA ALINEA is a real-time ramp metering strategy that controls the ramp input traffic flow by monitoring the traffic occupancy on the mainstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' ALINEA keeps calculating the metering rate(r) in each cycle to keep the main road stream stable, and accordingly mitigate congestion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' The normal scenario for ALINEA requires a traffic signal that is installed on the ramp which is to control the ramp input traffic and loop detectors that are installed downstream of the main road, which is to measure the occupancy of the mainstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=" The formula is shown below: 𝑟(𝑘) = 𝑟 (𝑘 − 1) + 𝐾% ∗ (𝑜6 − 𝑜&'!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' (𝑘 − 1)) 𝑖𝑓 𝑟(𝑘) > 𝑟𝑚𝑎𝑥, 𝑟(𝑘) = 𝑟𝑚𝑎𝑥 𝑖𝑓 𝑟(𝑘) < 𝑟𝑚𝑖𝑛, 𝑟(𝑘) = 𝑟𝑚𝑖𝑛 Where 𝑘 is the time index, 𝑘 = 1, 2, 3…, means at cycle 𝑘;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' 𝑟(𝑘) is the metering rate at 𝑘;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' K!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' is a fixed parameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' 𝑜6 is the desired occupancy, which is the occupancy needs to be maintained;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' o"#$ is the occupancy in downstream, which is detected by loop detectors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' 𝑟𝑚𝑎𝑥 is the maximum for 𝑟(𝑘) which equals to 1600 veh/h or 1800 veh/h for single-lane ramps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' 𝑟𝑚𝑖𝑛 is the minimum for 𝑟(𝑘) which is the admissible flow for single-lane ramps (200~400 vehicles/hour).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='2 Reinforcement Learning 6 Reinforcement learning is the training of machine learning models to make a sequence of decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' The agent chooses the optimal behavior in an environment to maximize the expected total reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' The reinforcement learning method derives from the problem of optimal control of Markov Decision Processes (MDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' The main elements of an RL system are: the agent or the learner, the environment the agent interacts with, the policy that the agent follows to take actions, and the reward signal that the agent observes upon taking actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' The key to reinforcement learning is to solve the Bellman equation, the agent interacts with the environment to sample different states and rewards by using Epsilon greedy algorithm, which is the balance between exploration and exploitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' After sampling enough states and actions, the value function of the MDP can be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' And the optimal strategy is always choosing the highest value to get the maximum total rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' 𝑉(𝑠) = 𝑚𝑎𝑥%(𝑅(𝑠, 𝑎) + 𝛾𝑉(𝑠&)) State (𝑠): the current state where the agent is in the environment Next State (𝑠&): After taking action (𝑎) at state (𝑠) the agent reaches 𝑠& Value (𝑉): Numeric representation of a state which helps the agent to find its path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Reward (𝑅): treat which the agent gets after performing an action (𝑎) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='3 Deep Q-learning Q-learning is one of the reinforcement learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' The main idea is to update Q values which denote the value of choosing action given state 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Q values are calculated by some constant parameters, rewards, and the Q value of the next time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' For each iteration, Q values of all states and actions are calculated, which is called the Q table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' And the optimal strategy is to always choose the action that has the highest Q value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=" 𝑄(𝑆', 𝐴') = (1 − 𝛼)𝑄(𝑆', 𝐴') + 𝛼 ∗ (𝑅' + 𝜆 ∗ 𝑚𝑎𝑥%𝑄(𝑆'(), 𝑎)) 𝑆= State, the data collected from loop detectors 𝐴= Action the agent takes, red duration 𝑅= Reward from taking an action t= Time step 𝛼= The learning rate 𝜆= The discount factor Agent state reward action 's Rt A, St+1 Environment 7 Although this method is simple to implement, the method is unable to estimate values for unseen states." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' To fix this problem, the Deep Q-learning method is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Deep Q-Network (DQN) uses Neural Networks to estimate Q-values, which overcomes the complexity of environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' But DQN can only handle discrete, low-dimensional action spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' In this study, the DQN is applied to solve the local ramp metering problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' In the ramp metering context, states are the traffic states, and actions are the lengths of the red duration of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' By building the connections between the SUMO simulation and the DQN algorithm, the iteration process could be implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='4 Constructing Rewards Function As mentioned, rewards function, 𝑅(𝑠, 𝑎), is defined as the benefits gained from taking action 𝑎 at state 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' For ramp metering control problem, the final objective is to minimize the total travel time of all vehicles within the system, regardless of whether they are originated from the highway mainline or from the on ramps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Thus, the rewards function should the action’s impact on total traveling time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Given the current traffic state, for the next 30 seconds, the expected total travel time of new incoming vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' To simplify, all delays are generated from two region: (1) Region 1: the waiting time along the on-ramp queue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' (2) Region 2: the traveling time through the highway’s downstream merging zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' On ramp vehicles would traveling through both Region 1 and Region 2, whereas mainline vehicles would only travel through Region 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Those regions are shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=" Definition of important variables in formulating rewards function Assuming the downstream loop detector generates an observation (𝑛', 𝑜') every 𝑇* time (e." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=', 30 seconds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Also, assume that the average vehicle length is 𝐿+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Then, the downstream speed in Region 2 is: R2, !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='" R1, !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=',# "$,#%$ "&,#%$ … ##, %# "&,#%$: incoming vehicles on ramp between t and t+1 N$,\'%$: incoming vehicles on ramp between t and t+1 "1: Region 1, the on-ramp queueing zone 7#,%: Number of vehicles stayed in on ramp’s queue at time t 8%&\': signal’s control plan between t and t+1 9%&\': the ramp metering rate of A(&\' (seconds per vehicle) R2: Region 2, the mainline merging zone ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content="%: occupancy detected at downstream loop detector between t and t+1 <%: number of detected at downstream loop detector vehicles at time t )(&: length of the merging zone #%$, +#%$ 100m 8 𝑣,-,' = 𝑛' ⋅ 𝐿+ 𝑜' ⋅ 𝑇* Assuming the speed for the next 𝑇* are similar to current speed: 𝑣,-,'() ≈ 𝑣,-,' Also, the region 2’s speed are about the same." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=" Then, the travel time for one vehicle to travel through downstream region can be estimated as: 𝑡𝑡,- = 𝑑- 𝑣,-,' where 𝑑- is the length of the merging zone." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Similarly, one can estimate the expected travel time of one incoming vehicle traveling through the ramp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' That is, the total time spending on the queues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' As vehicles can only join the back of the queue, this time depends on both the metering rate 𝐶% (in seconds/vehicle) and the current queue length 𝐿/: 𝑡𝑡,) = A𝐿/ ⋅ 𝐶%B Assuming in the next 𝑇* seconds, the corresponding number of incoming vehicles on the mainline and on the ramp are 𝑁) and 𝑁-, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=" the total time for those vehicles to travel through both Region 1 and 2 becomes: 𝑇𝑇'() = 𝑁) ⋅ 𝑡𝑡," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=') + (𝑁) + 𝑁-) ⋅ 𝑡𝑡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='- For each vehicle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=" the averaged travel time becomes: 𝑇𝑇'() DDDDDDD = 𝑁) (𝑁) + 𝑁-) ⋅ 𝑡𝑡," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=') + 𝑡𝑡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='- As for the rewards function at time t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=" the reciprocal of total travel time is used: 𝑅'(𝑠," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=" 𝑎) = 1 𝑇𝑇'() DDDDDDD Indeed," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' although there are many assumptions and approximations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' the reward function is simple enough to be applied to a wide range of scenarios without requiring very detailed information including the number of departure vehicles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' the number of vehicles within the region (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=', densities), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Experiment The Experiment is divided into two phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' In phase 1, the simulation testbed is established with different algorithms tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' The performance of no control (NC), ALINEA method, and deep Q-learning method (DQL) are compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Furthermore, DQL is attacked by generating adversarial data to see the extend a model can be undermined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Two attack scenarios are executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' One attack targeted at blocking ramp, the other targeted at blocking downstream traffic bottlenecks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' During the process, all signals, no matter clean data or adversarial data, are recorded for the next phase of analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' The second phase tries to analyze the statistical difference between the clean dataset using statistical patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Furthermore, whether an online 9 algorithm can distinguish the attack signals from is tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' With that, the testbed and its configuration are discussed in the next paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' In this Experiment, a SUMO simulation testbed is built for analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' The geometries are shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' A ramp is merged into highway with 3 upstream lanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' The merging zone has an extra acceleration lane for vehicles to merge into highway spans over 300 meters before merged back to 3 lanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' As shown in the red rectangles, three sets of loop detectors are available: the upstream detectors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' the downstream detectors, and the on-ramp loop detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Each detector generates a signal of traffic volume and traffic occupancy every 30 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Besides, it is assumed that the queue length can also be detected on the ramp, either through loop detectors or a CCTV/video-based system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Finally, the orange rectangle shows the position of the ramp metering signal to meter the ramp traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Figure 2 Geometry layouts and detector configurations of the simulation testbed During the training/testing process, the following traffic demands are configured in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' The upper and lower diagram shows the volume settings for training and testing scenarios, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Only DQL model needs to the training phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' The settings are relatively simple as it is easier to learn control policies in a “stationary environment”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' In contrast, the test case covers traffic scenarios from medium to high highway traffic volume ranges from 667-2000 vehicles/hour/lane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' The ramp volume is fixed to 700 vehicles/hour, relatively close to the maximal capacity of a signal-controlled lane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Note that there is also a clearance phase at the end of each simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' As long as there are running vehicles, the simulation will continue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' In such cases, the traffic volume at a low level to simulate the off-peak periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' : Loop detectors (upstream, downstream) : Ramp metering control (traffic signal) Merge zones (4 lanes to 3 lanes) Merging zone 3 lanes 4 lanes 3 lanes Upstream loop detectors Downstream loop detectors Signals on ramp loop detectors 10 Figure 3 Traffic input volume used for training/testing different algorithms To simulate the one-green-per-vehicle policy, each green phase is fixed to 2 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' By changing the length of red signal phase, the volume inject to highway from ramp can be metered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' 6 gears are provided for the ramp metering, as summarized in Table 1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' For both ALINEA method and DQL method, the same gears are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Table 1 Eight different signalizing plans for ramp metering control “Gear” G0 G1 G2 G3 G4 G5 G6 G7 Green phase length (seconds) 0* 2 Red phase length (seconds) 0 1 2 3 4 5 6 7 Total cycle length (seconds) 2 3 4 5 6 7 8 9 Finally, to handle the case of over-spilling queue on ramp, a fixed control rule is applied: as soon as the ramp’s queue length becomes greater than 40 vehicles, ramp metering policies are not applied anymore for 20 seconds to clear the queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' This rule is fix programmed and of highest order to prevent overspill from happening regardless of the applied algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' For DQL, some results for the training process are summarized in Figure 4 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' The model is trained for 100 episodes, and each episode is composed of 400 epochs of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' As shown below, the total travel time has been consistently decreased over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' upstream highway traffic ramp metering volume upstream highwaytraffic ramp metering volume 11 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Performance of the system during training episode 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Results 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='1 Phase 1 results: evaluating the performance of the system To compare their performance, all testbed is configured the same (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=', geometry, traffic volume, detector layouts) except the control algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Among all different measures, the Total Travel Time (TTT) is the most reliable metric to measure system-wise performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Note that to compare TTT, the simulation cannot be terminated until all vehicles reached at their destinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' As mentioned, five different scenarios are checked, and the results for each scenario are summarized in Table 2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Given limited training time and information detected, the performance of Deep Q-Learning is not as good as the no control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' However, the algorithm in general can maintain the tradeoff between ramp’s queue length and downstream bottleneck’s travel speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' ALINEA method performs the best among all scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Table 2 Simulation Results Scenario TTT (10^3) Adversarial data injected Attack Target Comments 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' NC 1,089 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' No control 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' ALINEA 1,045 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' DQL 854 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Deep Q- Learning 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' DQL+FGSM1 903 x Block ramp FDI Step=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='02 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' DQL+ FGSM2 900 x Block downstream bottleneck FDI Step=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='02 1e6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='60 (seconds) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='55 e 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='50 I time travel 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='45 Total 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='5 Episode 12 More information during the simulation time can be directly plotted and visualized to gain more insights, as shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Each scenario has an upper subplot and a lower subplot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' The red lines and blue lines in the upper subplot show the downstream vehicle speed and ramp’s queue length, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' By observation, there is a tradeoff between the two metrics: if one wants to increase the downstream speed, then the interest of on-ramp users is compromised by waiting for a longer time over the queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Both ALINEA and DQL methods can automatically balance this tradeoff, revealing that both algorithms worked as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Scenario 4 and 5 tries to undermine the system’s performance using by injecting adversarial data using Fast Gradient Sign Method (FSGM) with a small step width of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' This means that if one input number is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='50, then the adversarial number may become either 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='48 or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' With those settings, the performances of the system are severely undermined to the attacker’s benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Figure 5 Comparison of performance of different scenarios by visualizing information over time 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='2 Phase 2 results: Distinguishing the difference between adversarial and clean data In methodology, two methods are proposed to distinguish injected data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=', adversarial data) from the clean data: GEM and PCA methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' The adversarial data gained from scenarios 4 and 5 are separately tested against the clean data generated from scenario 1-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' The results are shown in Figure 6 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' In general, although a small step is chosen by the FGSM in generating adversarial data, adversarial data still have different GEM and PCA statistics (see red histograms) compared with that of clean data (see green histograms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' (a) (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' No control 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='ALINEA algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' DQL algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' DQL algorithm (attack to block ramp) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=" DQL algorithm (attack to block downstream highway) (d) (e) (f) Traffic Demand (c) ranp's queue length downstream speedmp's queue lengthra np's queue length downstream speedra np's queue length downstream speedra np's queue length downstream speedupstream highway traffic ramp metering volume 13 Figure 6 The distribution of distance statistics between of clean data (in green) vs." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' adversarial data (in red) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='3 Using these distance metrices, the performance of online detection algorithm is evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' For last five records, a majority vote is adopted if there are more than 3 records referred as polluted ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' More results are summarized in the following An online detecting method in identifying false injected data The previous section shows the statistical difference between clean sample and polluted sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' In practice, one needs to decide whether a stream of signals is polluted or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' For each record, the following Equations define score to show the magnitude of deviation from the correct data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' 𝑠()* = log C 𝛼 𝑝()*,, F , 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='10 𝑠-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='/ = log C 𝛼 𝑝-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='/,, F , 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='10 Given this definition of scores, a record falls as an outlier of clean samples tend to be assigned a large score, making it suspect to be a FDI data instead of a clean sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Furthermore, scores can be averaged to an ensembled score to jointly consider two statistical traits using one metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' 𝑠)01 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='5 × (𝑠()* + 𝑠-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='/) In such settings, if an extreme data is observed with 𝑝()*,, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='001, the score 𝑠()* becomes log(10) ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' The higher the score, the more likely data is not generated from the clean sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' On the other hand, for a new observed data with 𝑝()*,, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='5, the score 𝑠()* becomes log�(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='2) ≈ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Thus, if the score over a time series of records is accumulated and becomes a large positive number, we can indicate that the series of data is polluted instead of being clean data, as defined in the new indicator 𝑔2,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' below: 𝑔2,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' = maxR0, 𝑔2,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='34 + 𝑠2,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='S , 𝑔2,$ = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' 𝑠2,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' : score of method 𝑚 at time 𝑡, 𝑚∈{GEM, PCA, ENS}, 𝐴2,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' = 1, 𝑖𝑓 𝑔2,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' ≥ ℎ (a) GEM-based summary histograms (c) 1200 good reading (S2) 700 good reading (S2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='10 polluted reading (S3) polluted reading (S3) good reading (S2) 1000 600 polluted reading (S3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='08 800 500 400 600 300 400 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='02 200 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='00 0 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='10 i 2 5 2 (d) () GEM statistics (e) GEM-based summary histograms PCA-based summary histograms 1400 good reading (S2) 1000 good reading (S2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='10 good reading (S2) polluted reading (S3) polluted reading (53) 1200 polluted reading (S3) 800 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='08 800 600 600 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='04 400 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='02 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='00 0 1 2 3 4 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='10 1 2 5 6 GEM statistics 14 Given the score of one data, an empirical score 𝑔2,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' can be used to measure the accumulated score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' If 𝑔2,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' becomes no less than a threshold ℎ, then a False Data Injection alarm with be generated at time 𝑡 under method 𝑚, which can be one of the three methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' The results of one experiment run on both clean samples and adversarial samples are visualized below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Among three proposed methods, the PCA method gets the best performance in detecting false data injection attack, whereas the GEM method doesn’t perform too well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' The ensemble method combining two scores are also good in detecting False Data Injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' The benefits of using ensemble method can be viewed in the lower diagram where all methods generate many false alarms except the ensemble method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Thus, the ensemble method is more preferred among three methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Figure 7 Performance of anomaly detector machine on both adversarial samples and clean samples 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='4 An online detecting method in identifying false injected data The previous section shows the statistical difference between clean sample and polluted sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' In practice, one needs to decide whether a stream of signals is polluted or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' For each record, the following Equations define score to show the magnitude of deviation from the correct data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' 𝑠()* = log C 𝛼 𝑝()*,, F , 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='10 𝑠-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='/ = log C 𝛼 𝑝-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='/,, F , 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='10 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' (on adversarial sample) (on clean sample) PCA GEM ensemble control threshold h=10PCA GEM ensemble control threshold h=10 15 Given this definition of scores, a record falls as an outlier of clean samples tend to be assigned a large score, making it suspect to be a FDI data instead of a clean sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Furthermore, scores can be averaged to an ensembled score to jointly consider two statistical traits using one metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' 𝑠)01 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='5 × (𝑠()* + 𝑠-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='/) In such settings, if an extreme data is observed with 𝑝()*,, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='001, the score 𝑠()* becomes log(10) ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' The higher the score, the more likely data is not generated from the clean sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' On the other hand, for a new observed data with 𝑝()*,, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='5, the score 𝑠()* becomes log(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='2) ≈ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Thus, if the score over a time series of records is accumulated and becomes a large positive number, we can indicate that the series of data is polluted instead of being clean data, as defined in the new indicator 𝑔2,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' below: 𝑔2,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' = maxR0, 𝑔2,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='34 + 𝑠2,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='S , 𝑔2,$ = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' 𝑠2,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' : score of method 𝑚 at time 𝑡, 𝑚 ∈ {GEM, PCA, ENS}, 𝐴2,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' = 1, 𝑖𝑓 𝑔2,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' ≥ ℎ Given the score of one data, an empirical score 𝑔2,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' can be used to measure the accumulated score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' If 𝑔2,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' becomes no less than a threshold ℎ, then a False Data Injection alarm with be generated at time 𝑡 under method 𝑚, which can be one of the three methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' The results of one experiment run on both clean samples and adversarial samples are visualized below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Among three proposed methods, the PCA method gets the best performance in detecting false data injection attack, whereas the GEM method doesn’t perform too well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' The ensemble method combining two scores are also good in detecting False Data Injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' The benefits of using ensemble method can be viewed in the lower diagram where all methods generate many false alarms except the ensemble method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Thus, the ensemble method is more preferred among three methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Limitations There are some limitations in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' First, the ramp metering algorithm is applied in a simple testbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' For future study, the team will run the simulation over the whole corridor system with different highway layouts at different locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Next, there are many more off the shelf algorithms to generate adversarial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Also, we only consider white-box (insider) attack, the likelihood of black-box (outsider) attack should also be included in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Similarly, there are other techniques identifying outliers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=', adversarial data) to be tested but not included in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Note that this study mainly applies the strategy to fall back to traditional fixed programmed methods when adversarial attacks are detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' The problem of how to train deep learning models robust to adversarial data is not discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' There may be other unforeseen simulation details that might undermining the simulation’s correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Besides ramp metering, the Variable Speed Limit (VSL) is also a good supplement control strategy that needs to be discussed to further exploit the benefits of Ramp Metering control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' There are also some other miscellaneous bullets to point out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' The vehicle’s behavior in SUMO simulation may be different from that of realistic world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' For example, we command vehicles to turn left on the acceleration ramp to merge with the other vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' If not, the vehicles would delay their left turn maneuver until reaching the end of the acceleration ramp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' In SUMO, it seems that the vehicles lane changing model may outperform the realistic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Although the demand is high on both ramp and upstream highway, the benefits of ramp metering look trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Conclusion 16 The major contributions of this study are listed as follows: Build up a testbed to test the efficiency of different ramp metering algorithms Compare the efficiency of the different algorithms with or without white-box attack using FGSM algorithm Evaluate different statistical methods in identifying adversarial samples from clean samples Evaluate the effectiveness and robustness of detecting false data Overall,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' the results are positive that even for the insider attack,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' it is hard to generate adversarial data to fool the proposed detection algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Acknowledgement The project is sponsored by the Secure Constrained Machine Learning for Critical Infrastructure CPS project, a National Science Foundation project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' We would like to thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Asad J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Khattak and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Iman Mahdinia for providing useful feedbacks on this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' 17 Reference 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Papageorgiou, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=', and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Kotsialos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Freeway Ramp Metering: An Overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' IEEE Transactions on Intelligent Transportation Systems.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='1109/TITS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content='2725912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Lu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Huang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Deng, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Gong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Coordinated Ramp Metering with Equity Consideration Using Reinforcement Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Journal of Transportation Engineering, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' 143, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' 7, 2017.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Ivanjko, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Kušić, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Čakija.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Variable Speed Limit and Ramp Metering for Mixed Traffic Flows: A Review and Open Questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' Applied Sciences (Switzerland).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtFLT4oBgHgl3EQfRS82/content/2301.12036v1.pdf'} +page_content=' 6.' metadata={'source': 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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf,len=531 +page_content='On doubly symmetric periodic orbits Urs Frauenfelder, Agustin Moreno Abstract In this article, for Hamiltonian systems with two degrees of freedom, we study doubly symmetric periodic orbits, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' those which are symmet- ric with respect to two (distinct) commuting antisymplectic involutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' These are ubiquitous in several problems of interest in mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' We show that, in dimension four, doubly symmetric periodic orbits cannot be negative hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' This has a number of consequences: (1) all cov- ers of doubly symmetric orbits are good, in the sense of Symplectic Field Theory [6];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' (2) a non-degenerate doubly symmetric orbit is stable if and only if its CZ-index is odd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' (3) a doubly symmetric orbit does not un- dergo period doubling bifurcation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' and (4) there is always a stable orbit in any collection of doubly symmetric periodic orbits with negative SFT- Euler characteristic (as coined in [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' The above results follow from: (5) a symmetric orbit is negative hyperbolic if and only its two B-signs (introduced in [10]) differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Contents 1 Introduction 1 2 Examples of doubly symmetric periodic orbits 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='1 The direct and retrograde periodic orbit in Hill’s lunar problem .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='2 The Levi-Civita regularization .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' 10 3 Real couples 10 4 Doubly symmetric periodic orbits 13 5 The reduced monodromy 17 1 Introduction This article deals with the study of doubly symmetric periodic orbits in dimen- sion four, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' for Hamiltonian systems with two degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' These are 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='01803v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='SG] 4 Jan 2023 t=0 t= /2 symmetric points τ L=Fix(ρ) L =Fix(ρ ) L =Fix(ρ ) 1 1 2 2 Figure 1: Left: A symmetric orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Right: A doubly symmetric orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' ubiquitous in problems of interest in mechanics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' we give several examples in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Let us introduce the basic concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Symmetric orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Consider a symplectic manifold (M, ω) endowed with an antisymplectic involution ρ : M → M (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' ρ2 = id, ρ∗ = ω = −ω), also referred to as a real structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Its fixed point set L = Fix(ρ) is a Lagrangian submanifold of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Given a Hamiltonian H : M → R, we say that ρ is a symmetry of the Hamiltonian system induced by H, if H ◦ ρ = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' In this situation, a symmetric periodic orbit is a periodic orbit v : S1 = R/τR → M satisfying ρ(v(−t)) = v(t) for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' A symmetric periodic orbit can also be thought of as a chord starting and ending in L, where the endpoints coincide with v(0), v(τ/2) (the symmetric points), see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Now suppose we have two distinct antisymplectic involutions ρ1 and ρ2 which commute with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' In this case we have two Lagrangian submanifolds L1 = Fix(ρ1) and L2 = Fix(ρ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Given a chord from L1 to L2 we can apply ρ2 to it to get a chord from L1 to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Now apply ρ1 to this chord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' The resulting periodic orbit is then doubly symmetric, as it is symmetric with respect to both ρ1, ρ2, see again Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' We provide a more formal definition of the notion of a doubly symmetric periodic orbits in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Reduced monodromy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Suppose that (M, ω) is a four-dimensional sym- plectic manifold, H : M → R is a smooth Hamiltonian, and v is a nonconstant periodic orbit of the Hamiltonian vector field XH of H of period τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' By preser- vation of energy H is constant along v, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=', v lies for all times on a level set Σ = H−1(c) for some c ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' The differential of the flow φt H induces a map on the two-dimensional quotient vector space Mv := dφτ H(v(0)): Tv(0)Σ/⟨XH(v(0))⟩ → Tv(0)Σ/⟨XH(v(0))⟩, referred to as the reduced monodromy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' The reduced monodromy is a two- dimensional symplectic transformation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=', det Mv = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Depending on the 2 trace of its reduced monodromy, periodic orbits on a four-dimensional sym- plectic manifold are now partitioned into three classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Positive hyperbolic: tr(Mv) > 2, in which case the reduced monodromy has two positive, real eigenvalues inverse to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Negative hyperbolic: tr(Mv) < 2, in which case the reduced monodromy has two negative, real eigenvalues inverse to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Elliptic: −2 ≤ tr(Mv) ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' If the trace is precisely two, the reduced mon- odromy has one as an eigenvalue with algebraic multiplicity two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' If the trace is precisely minus two, it has minus one as an eigenvalue with al- gebraic multiplicity two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Otherwise it has two nonreal eigenvalues on the unit circle conjugated to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' In the language of Symplectic Field Theory, an even cover of a negative hyper- bolic orbit is called bad;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' otherwise a periodic orbit is called good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Here we prove the following: Theorem A: For a Hamiltonian system with two degrees of freedom, a doubly symmetric periodic orbit cannot be negative hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' In particular, it follows from Theorem A that all covers of a doubly symmetric periodic orbit are good periodic orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' While elliptic periodic orbits are stable, hyperbolic ones are un- stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' On the other hand, elliptic and negative hyperbolic orbits have odd Conley-Zehnder index, while positive hyperbolic ones have even Conley-Zehnder index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' For the second statement it is better to exclude the degenerate case where the trace of the reduced monodromy is two, since in this case there are different conventions on how to define the Conley-Zehnder index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' We see from this that if we can exclude negative hyperbolic orbits, the question of stability of a periodic orbit can be answered in terms of the parity of its Conley-Zehnder index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' In particular, we have the following Corollary of Theorem A: Corollary B: Suppose that v is a nondegenerate doubly symmetric periodic orbit of a Hamiltonian system with two degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Then it it stable if and only if its Conley-Zehnder index is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Overview of proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' The proof of Theorem A uses a real ver- sion of Krein theory for the reduced monodromy of a symmetric periodic orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Given a symmetric orbit v, the differential of the antisymplectic involution at v(0) ∈ L = Fix(ρ) induces an antisymplectic involution R: Tv(0)Σ/⟨XH(v(0))⟩ → Tv(0)Σ/⟨XH(v(0))⟩, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' an orientation reversing involution on the two-dimensional vector space Tv(0)Σ/⟨XH(v(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' The involution R conjugates the reduced monodromy with 3 its inverse, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' RMvR = M −1 v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' (1) We choose a symplectic basis on Tv(0)Σ/⟨XH(v(0)) such that the involution R gets identified with the matrix R = � 1 0 0 −1 � and the reduced monodromy is given by a matrix Mv = � a b c d � satisfying the determinant condition ad − bc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' It follows from (1) that a = d so that Mv = � a b c a � , a2 − bc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' In particular, the question to which class the periodic orbit belongs is completely answered by the entry a of the reduced monodromy matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' For fixed a, if an off- diagonal entry is not zero, then it completely determines the other one in view of the determinant condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' On the other hand, the off-diagonal entries depends on the choice of the symplectic basis used to identify the reduced monodromy with a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Since the symplectic basis vectors are required to be eigenvectors of the antisymplectic involution R, such a symplectic basis is determined up to a scaling factor, so that the identification of the reduced monodromy with a matrix is unique up to conjugation by a matrix of the form � µ 0 0 1 µ � , µ ∈ R \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' In particular, while the value of b is not an invariant, its sign is an invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Following [10] we refer to sign(b) as the B-sign of the reduced monodromy, see also [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' In the case elliptic case, by [10, Appendix B], the B-sign gives the same information as the Krein type of the eigenvalues of the reduced monodromy (as introduced in [15, 16, 17, 18, 19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' In the hyperbolic case the eigenvalues have no Krein type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Therefore the B-sign in the hyperbolic case is an additional invariant of the real structure ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' A symmetric periodic orbit intersects the Lagrangian L = Fix(ρ) in its two symmetric points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' From the reduced monodromies of each symmetric point we obtain a B-sign, so that a symmetric periodic orbit is actually endowed with two B-signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' The main observation to prove Theorem A is the following: Theorem C: A symmetric periodic orbit of a Hamiltonian system with two degrees of freedom is negative hyperbolic if and only if its two B-signs are dif- ferent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' 4 If the symmetric periodic orbit is elliptic it is actually clear that the two B-signs have to agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Indeed, as already mentioned, in the elliptic case the B-sign is just determined by the Krein sign of the eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Since reduced monodromy matrices of a periodic orbit for different starting points are all conjugated to each other, Theorem C follows in the elliptic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' What remains to be exam- ined is the hyperbolic case, namely that in the positive hyperbolic case the two B-signs agree, while in the negative hyperbolic case they disagree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' To address this, in Section 3 we introduce the notion of real couples, so that Theorem C becomes a consequence of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' The strategy to prove Theorem A is now rather obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' One shows that the additional real structure for a doubly symmetric periodic orbit forces the two B-signs to agree, so that, in view of Theorem C, a doubly periodic orbit cannot be negative hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' This is carried out in Section 5 where Theorem A is referred to as Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Period doubling bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' When considered in families, periodic orbits may undergo bifurcation, by which a non-degenerate orbit becomes degenerate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' 1 becomes an eigenvalue of its monodromy), and new orbits may appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Generic bifurcations in dimension four are well understood, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' [1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' 599].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' However, the presence of symmetry, and in particular the presence of doubly symmetric orbits, is non-generic, and hence one expects new phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' And indeed, what follows aligns well with this expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' As a particular case of bifurcations, the transition from an elliptic periodic orbit to a negative hyperbolic orbit leads to a period doubling bifurcation, by which a new orbit appears, whose period is close to double the period of the original orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' In the case where the negative hyperbolic orbit is symmetric, its two different B-signs can actually be useful to figure out where the new periodic orbit of double period bifurcates, see [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Namely, bifurcation happens near the symmetric point where the B-sign does not jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Moreover, a consequence of Theorem A is the following, which emphasizes the non-generic nature of sym- metry: Corollary D: In dimension four, doubly symmetric periodic orbits do not undergo period doubling bifurcation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Indeed, as in period doubling bifurcation the orbit itself does not bifurcate (its double cover does), the orbit after such a bifurcation would have to be dou- bly symmetric if the orbit before bifurcation is, thus contradicting Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' We remark that Corollary D fails in dimension six, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' for systems with three degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Indeed, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' [11, Section 6] for a numerical example of a planar-to-spatial period doubling bifurcation of doubly symmetric orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' SFT-Euler characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' In order to address the situation of more general bifurcations than period doubling bifurcation (in the presence of symmetry), we consider a Floer numerical invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Namely, following [10], the SFT-Euler 5 characteristic of a periodic orbit v is by definition the Euler characteristic of its local Floer homology, given by χSF T (v) = #{good positive hyperbolic} − #{elliptic, negative hyperbolic}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Here, one counts each type of orbit that appears after a generic perturbation of the orbit v, so that it bifurcates into a collection of non-degenerate orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' We remark that bad orbits do not contribute to this number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Note also that this number is ±1 in the case where v is itself non-degenerate, depending on its type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' The remarkable fact, which follows from Floer theory, is that χSF T (v) is independent of the perturbation, and so in particular it remains invariant under bifurcations of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' It is therefore very useful in order to study non-generic bifurcations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Moreover, given a collection of periodic orbits (which may not necessarily arise from a bifurcation, but e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' as critical points of an action functional, with a priori fixed homotopy class) one can also consider the same number computed via the above formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Its invariance under arbitrary homotopies will of course not be guaranteed, and will depend on the particular situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' An example of interest, for which a suitable homotopy invariance holds, are frozen planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' These are periodic orbits for the Helium problem which we discuss in more detail in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Due to the interaction between the two electrons in Helium, frozen planets cannot be approached by perturbative methods but instead one can replace the instantaneous interaction of the two electrons by a mean interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' If one interpolates between mean and instantaneous interaction one obtains a homotopy of a frozen planet problem for which one has compactness in the symmetric case [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' This allows one to define a version of the Euler characteristic for frozen planets which is invariant under this homotopy [5], and which agrees with the SFT-Euler charactersitic χSF T for the instantaneous interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' The Euler characteristic for this problem is −1, see the remark after Corollary B in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' For each negative energy, this implies the existence of a symmetric frozen planet orbit for the instantaneous interaction, see Corollary C in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' This follows by homotopy invariance of the Euler characteristic, and the existence (proved analytically in [7]) of a unique nondegenerate symmetric orbit for the mean interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' With these motivations in mind, the following is again a consequence of Theorem A: Corollary E: In dimension four, suppose that a collection of doubly sym- metric periodic orbits has negative SFT-Euler characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Then a stable periodic orbit exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Indeed, Theorem A and the formula defining χSF T imply the existence of an elliptic orbit, and one needs to recall that elliptic orbits are precisely the stable orbits for a Hamiltonian system in dimension four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Moreno is supported by the National Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' DMS-1926686, and by the Sonderforschungsbereich TRR 191 Symplectic Structures in Geometry, Algebra and Dynamics, funded by the DFG (Projektnummer 281071066 – TRR 191).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' 6 2 Examples of doubly symmetric periodic orbits 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='1 The direct and retrograde periodic orbit in Hill’s lunar problem Hill’s lunar Hamiltonian goes back to Hill’s groundbreaking work on the orbit of our Moon [14], describing its motion around the Earth and the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' The Earth lies in the center of the frame of reference, while the Sun, assumed to be infinitely much heavier than the Earth, lies at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' The Hamiltonian reads H : T ∗(R2 \\ {0}) → R, (q, p) �→ 1 2 � (p1 + q2)2 + (p2 − q1)2� − 1 |q| − 3 2q2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' It is invariant under the two commuting antisymplectic involutions ρ1, ρ2 : T ∗R2 → T ∗R2 given, for (q, p) ∈ T ∗R2, by ρ1(q1, q2, p1, p2) = (q1, −q2, −p1, p2), ρ2(q1, q2, p1, p2) = (−q1, q2, p1, −p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' The fixed point sets of the two antisymplectic involutions are the conormal bun- dles of the x-axis and the y-axis, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' If one studies a doubly symmetric periodic orbit in configuration space R2 \\ {0}, this means that it starts perpen- dicularly at the x-axis, after a quarter period hits the y-axis perpendicularly, then gets reflected at the y-axis for the next quarter period, and finally gets reflected at the x-axis for the second half of the period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Such periodic orbits can be found by a shooting argument where one shoots perpendicularly from the x-axis for a varying starting point at the x-axis, until one hits the y-axis perpendicularly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Birkhoff used in [2] this shooting argument to prove the ex- istence of the retrograde periodic orbit for all energies below the first critical value, see also [12, Chapter 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Although the retrograde periodic orbit looks simpler than the direct one [13], astronomers are actually often more interested in the direct one, since our Moon and actually most moons in our solar system are direct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' However, there are prominent counterexamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Triton, the largest moon of the planet Neptun, is for example retrograde.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='2 The Levi-Civita regularization Hill’s lunar problem arises as a limit case of the restricted three-body problem, see for instance [12, Chapter 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' In the restricted three-body problem the masses of the Sun and the Earth are comparable and their distance is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Different from the Hill’s lunar problem, the restricted three-body problem is only invariant under the antisymplectic involution ρ: T ∗R2 → T ∗R2, (q1, q2, p1, p2) �→ (q1, −q2, −p1, p2) obtained from reflection at the x-axis, but not anymore under the antisymplec- tic involution corresponding to reflection at the y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' 7 We identify R2 with the complex plane C and denote by C∗ := C \\ {0} the complex plane pointed at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' We consider the squaring map ℓ: C∗ → C∗, z �→ z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Note that the squaring map is a two-to-one covering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' The contragradient (or symplectic lift) of the squaring map is the symplectic map L: T ∗C∗ → T ∗C∗, (z, w) �→ � z2, w 2¯z � , where ¯z is the complex conjugate of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' This map was used by Levi-Civita to regularise two-body collisions [21] and therefore it is known under the name of Levi-Civita regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' On T ∗C we have the two commuting antisymplectic involutions σ1, σ2 : T ∗C → T ∗C which are given, for (z, w) ∈ C × C = T ∗C, by σ1(z, w) = (¯z, − ¯w), σ2(z, w) = (−¯z, ¯w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' The Levi-Civita regularization lifts the restriction of the antisymplectic involu- tion ρ to T ∗C∗ to the restriction of σ1 and σ2 to T ∗C∗, so that we have L ◦ σ1 �� T ∗C∗ = ρ �� T ∗C∗ ◦ L, L ◦ σ2 �� T ∗C∗ = ρ �� T ∗C∗ ◦ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Now suppose that v = (q, p) is a periodic orbit in T ∗C∗ which is symmetric with respect to ρ, and such that it has odd winding number around the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Then v lifts under the Levi-Civita regularisation to a periodic orbit on T ∗C∗ which is doubly symmetric with respect to σ1 and σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' On the other hand, retrograde and direct orbits exist as well in the re- stricted three-body problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Different from Hill’s lunar problem, they are just symmetric, but not doubly symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' However, the lifts under the Levi-Civita regularisation are doubly symmetric, as the retrograde and direct periodic orbit have winding number one around the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='3 Langmuir’s periodic orbit Langmuir’s periodic orbit is a periodic orbit for the Helium problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' It was first discovered by Langmuir [20] numerically as a candidate for the ground state of the Helium atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' For an analytic existence proof we refer to [3], and for its role in the semiclassical treatment of Helium, to [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' In the Helium atom, there is a nucleus of positive charge plus two at the ori- gin, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' there are two protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' It attracts two electrons of charge minus one according to Coulomb’s law, which looks formally the same as Newton’s law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' 8 Moreover, the two electrons repel each other, again according to Coulomb’s law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' We abbreviate by ∆ := � (q, q) : q ∈ C∗� ⊂ C∗ × C∗ the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' The Hamiltonian for the planar Helium problem is then a smooth function H : T ∗� C∗ × C∗ \\ ∆ � → R given by H(q1, q2, p1, p2) = 1 2|p1|2 + 1 2|p2|2 − 2 |q1| − 2 |q2| + 1 |q1 − q2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' The Hamiltonian is invariant under the symplectic involution σ: T ∗� C∗ × C∗ \\ ∆ � → T ∗� C∗ × C∗ \\ ∆ � given by σ(q1, q2, p1, p2) = (¯q2, ¯q1, ¯p2, ¯p1), consisting of the combination of particle interchange and reflection at the x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' The Langmuir Hamiltonian is the restriction of H to the fixed point set of σ Hσ := H �� Fix(σ) : Fix(σ) → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' The fixed points set consists of points (q1, q2, p1, p2) ∈ T ∗(C∗ × C∗ \\ ∆) which satisfy q1 = ¯q2 =: q, p1 = ¯p2 =: p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' It therefore suffices to consider the Langmuir Hamiltonian on the cotangent bundle of the upper halfplane H = � q = q1 + iq2 ∈ C : q2 > 0 � where it is given by Hσ(q, p) = |p|2 − 4 |q| + 1 2q2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' On the cotangent bundle of the uper halfplane we have the two antisymplectic involutions ρ1, ρ2 : T ∗H → T ∗H given by ρ1(q, p) = (−¯q, ¯p), ρ2(q, p) = (q, −p), under both of which Hσ is invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' The fixed point set of ρ1 is the conormal bundle of the positive imaginary axis, while the fixed point set of ρ2 consists of brake points, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' at which the velocity is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' The Langmuir orbit for the first electron e− 1 starts perpendicularly at the imaginary axis and brakes at a quarter of the period, and is therefore a doubly symmetric periodic orbit with respect to ρ1 and ρ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' The second electron e− 2 similarly has an associated Langmuir orbit, obtained by conjugation of that of e− 1 , see Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' 9 brake points +2 e- e- 1 2 Figure 2: Langmuir’s doubly symmetric orbit, and its symmetric version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' +2 e- 1 e- 2 brake points Figure 3: A frozen planet configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='4 Symmetric frozen planets Other examples of periodic orbits for the Helium problem are frozen planet orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' In this examples both electrons lie on a line on the same side of the nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' The inner electron makes consecutive collisions with the nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' The outer electron, the actual “frozen planet”, which is attracted by the nucleus but repelled by the inner electron, stays almost stationary but librates slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Frozen planet orbits were discovered by physicists [22, 23] in the context of semiclassics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' They recently attracted the interest of mathematicians [4, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' A frozen planet orbit is called symmetric if the two electrons brake at the same time, and at the time the inner electron collides with the nucleus the outer electron brakes again, see Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' If one applies the Levi–Civita regularization to a symmetric frozen planet one obtains a doubly symmetric periodic orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' 3 Real couples A real symplectic vector space is a triple (V, ω, R) consisting of a symplectic vector space (V, ω) and a linear antisymplectic involution R: V → V , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' R2 = Id, R∗ω = −ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='1 Assume that (V1, ω1, R1) and (V2, ω2, R2) are real symplectic 10 vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' A real couple (Ψ, Φ) is a tuple of linear symplectic maps Ψ: (V1, ω1) → (V2, ω2), Φ: (V2, ω2) → (V1, ω1) which are related by R2ΨR1 = Φ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' (2) Note that if (Ψ, Φ) is a real couple, then (Φ, Ψ) is one as well, since it follows from (2) that R1ΦR2 = R1R−1 1 Ψ−1R−1 2 R2 = Ψ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' If (Ψ, Φ) is a real couple then its composition ΦΨ: (V1, ω1) → (V1, ω1) is a linear symplectic map from the fixed symplectic vector space (V1, ω1) into itself which has the special property that it is conjugated to its inverse via the antisymplectic involution R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Indeed, R1ΦΨR1 = R1ΦR2R2ΨR1 = Ψ−1Φ−1 = (ΦΨ)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' (3) We now consider more closely the two-dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Note that every two- dimensional real symplectic vector space is conjugated to R2, endowed with its standard symplectic structure and antisymplectic involution R = � 1 0 0 −1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' After such conjugation, a real couple then consists of a pair of matrices (A, B) ∈ SL(2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' R) × SL(2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' R) such that RAR = B−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' (4) Writing A = � a b c d � , ad − bc = 1 we have RAR = � 1 0 0 −1 � � a b c d � � 1 0 0 −1 � = � 1 0 0 −1 � � a −b c −d � = � a −b −c d � and therefore B = (RAR)−1 = � d b c a � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' 11 Hence their products are given by the following matrices AB = � a b c d � � d b c a � = � ad + bc 2ab 2cd ad + bc � (5) and BA = � d b c a � � a b c d � = � ad + bc 2bd 2ac ad + bc � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' (6) Since BA = B(AB)B−1 the two product are conjugated to each other in SL(2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Moreover, they both belong to the subspace SLR(2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' R) := � M = � α β γ α � : α2 − βγ = 1 � of SL(2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' If M ∈ SLR(2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' R) satisfies tr(M) ̸= ±2 we define its real Krein sign as κ(M) := sign(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Note that the trace condition implies that α ̸= ±1 so that, in view of the determinant condition α2 − βγ, we have that β ̸= 0, and so its sign is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' The following proposition is now straightforward to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='2 The real Krein signs of AB and BA differ, if and only if tr(AB) = tr(BA) < −2, (7) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=', if and only if AB and therefore as well BA are negative hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Proof: By (5) and (6) the trace condition (7) is equivalent to the inequality ad + bc < −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' In view of the determinant condition ad − bc = 1 this in turn is equivalent to the inequality ad < 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=', the requirement that the signs of a and d are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Having once more a look at (5) and (6), we see that this happens if and only if the real Krein signs of AB and BA disagree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' This proves the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' □ In the following we assume that (Ψ, Φ) is a real couple between real symplectic vector spaces (V1, ω1, R1) and (V2, ω2, R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='3 The real couple (Ψ, Φ) is called symmetric if there exists a linear map S : V1 → V2 12 which is antisymplectic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=', S∗ω2 = −ω1 and satisfies Ψ = SΨ−1S, Φ−1 = SΦS, R2SR1 = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' (8) For a symmetric real couple T := SR1 = R2S : (V1, ω1) → (V2, ω2) is a linear symplectic map which in view of TR1 = S = R2T interchanges the two real structures, so that T leads to an identification of the two real symplectic vector spaces (V1, ω1, R1) and (V2, ω2, R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' In the two- dimensional case if we identify this further with R2 endowed with its standard symplectic form and standard real structure R, then not only R1 and R2 are identified with R, but so is S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' The real tuple becomes identified with a pair (A, B) of SL(2, R)-matrices which not only satisfy (4) but due to (8) also satisfy RAR = A−1, RBR = B−1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=', both matrices are conjugated to their inverse via R and therefore lie in the subspace SLR(2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' R) of SL(2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' This implies that A = B = � a b c a � , a2 − bc = 1 and therefore AB = BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' In particular, AB and BA have the same real Krein sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Therefore we obtain the following corollary from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='4 Suppose that (Ψ, Φ) is a two-dimensional symmetric real couple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Then neither ΦΨ nor ΨΦ are negative hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' 4 Doubly symmetric periodic orbits Suppose that (M, ω) is a symplectic manifold and H : M → R is a smooth Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' The Hamiltonian vector field XH of H is implicitly defined by the condition dH = ω(·, XH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' We abbreviate by S1 = R/Z the circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' A simple periodic orbit is a bijective map v: S1 → R for which there exists τ > 0 such that v solves the ODE ∂tv(t) = τXH(v(t)), t ∈ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' 13 Since for a simple periodic orbit the map is bijective the Hamiltonian vector field XH is nonvanishing along v and therefore τ is uniquely determined by v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' We refer to τ as the period of the simple periodic orbit v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' We abbreviate by PH ⊂ C∞(S1, M) the set of simple periodic orbits of the Hamiltonian vector field XH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' A real symplectic manifold is a triple (M, ω, ρ) where (M, ω) is a symplectic manifold and ρ ∈ Diff(M) is an antisymplectic involution on M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=', ρ2 = id, ρ∗ω = −ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' If H : M → R is a smooth function on a real symplectic manifold which is invariant under the antisymplectic involution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=', H ◦ ρ = H, then its Hamiltonian vector field is anti-invariant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=', ρ∗XH = −XH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' We then obtain an involution I : PH → PH, v �→ ρ ◦ v− where v− is the orbit traversed backwards, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=', v−(t) = v(−t), t ∈ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' A simple symmetric periodic orbit is a fixed point of I, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='e, v ∈ PH satisfying I(v) = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' We abbreviate by PI H := Fix(I) ⊂ PH the set of simple symmetric periodic orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' We remark that the fixed point set of an antisymplectic involution L := Fix(ρ) is a Lagrangian submanifold of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Note that if v ∈ PI H then v � 0 � , v � 1 2 � ∈ L so that v[0, 1 2 ] can be interpreted as a chord from L to L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' A doubly real symplectic manifold is a quadruple (M, ω, ρ1, ρ2) where (M, ω) is a symplectic manifold and ρ1, ρ2 ∈ Diff(M) are two distinct antisymplectic 14 involutions which commute with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Note since ρ1 and ρ2 commute their composition σ := ρ1 ◦ ρ2 = ρ2 ◦ ρ1 is a symplectic involution on (M, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Suppose that (M, ω, ρ1, ρ2) is a doubly real symplectic manifold and H : M → R is a smooth map which is invariant under both involutions ρ1 and ρ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' We then have on the set of simple periodic orbits PH two involutions I1 : PH → PH, v �→ ρ1 ◦ v−, I2 : PH → PH, v �→ ρ2 ◦ v−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Moreover, we have two Lagrangian submanifolds of M L1 = Fix(ρ1), L2 = Fix(ρ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='1 Suppose that (M, ω, ρ1, ρ2) is a doubly real symplectic manifold and H : M → R is a smooth function invariant under both involutions ρ1 and ρ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' A simple symmetric periodic orbit v ∈ PI1 H of ρ1 is called doubly symmetric if ρ2 ◦ v � 0 � = v � 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' (9) Observe that since for a symmetric periodic orbit v(1/2) lies in the fixed point set of ρ1 condition (9) is equivalent to σ ◦ v � 0 � = v � 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Doubly symmetric periodic orbits with respect to ρ1 are in natural one-to-one correspondence with double symmetric periodic orbits with respect to ρ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' For r ∈ S1 and v ∈ PH we denote by r∗v ∈ PH the reparametrized simple periodic orbit r∗v(t) = v(r + t), t ∈ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' We have the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='2 An orbit v ∈ PI1 H is doubly symmetric with respect to ρ2 if and only if � 1 4 � ∗v ∈ PI2 H is doubly symmetric with respect to ρ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Proof: Suppose that v ∈ PI1 H is doubly symmetric with respect to ρ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' After reparametrization a simple periodic orbit is still a simple periodic orbit so that we have � 1 4 � ∗v ∈ PH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Since H is invariant under ρ2 we have that I2 �� 1 4 � ∗v � ∈ PH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' 15 Using (9) we compute I2 �� 1 4 � ∗v �� 1 4 � = ρ2 ◦ �� 1 4 � ∗v �−� 1 4 � = ρ2 �� 1 4 � ∗v �� − 1 4 � = ρ2 ◦ v � 1 4 − 1 4 � = ρ2 ◦ v(0) = v � 1 2 � = �� 1 4 � ∗v �� 1 4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' That means that � 1 4 � ∗v and I2 �� 1 4 � ∗v � are solutions of the same first order ODE which at time 1 4 go through the same point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Therefore from the uniqueness of the initial value problem of first order ODE’s we deduce that I2 �� 1 4 � ∗v � = � 1 4 � ∗v and hence � 1 4 � ∗v ∈ PI2 H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' It remains to check its double symmetry with respect to ρ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' For that we compute ρ1 ◦ �� 1 4 � ∗v � (0) = ρ1 ◦ v � 1 4 � = v � − 1 4 � = v � 3 4 � = �� 1 4 � ∗v �� 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Here we have used in the second equation that v is symmetric with respect to ρ1 and in the third equation that it is one-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' This shows that � 1 4 � ∗v is doubly symmetric with respect to ρ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' It remains to check that if � 1 4 � ∗v ∈ PI2 H is doubly symmetric with respect to ρ1 it follows that v ∈ PI1 H is doubly symmetriy with respect to ρ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Interchanging in the previous discussion the roles of ρ1 and ρ2 we obtain that � 1 4 � ∗ � 1 4 � ∗v = � 1 2 � ∗v ∈ PI1 H 16 is doubly symmetric with respect to ρ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' The fact that � 1 2 � ∗v is invariant under I1 implies that I1v(t) = ρ1 ◦ v−(t) = ρ1 ◦ v(−t) = ρ1 ◦ �� 1 2 � ∗v �� − t − 1 2 � = ρ1 ◦ �� 1 2 � ∗v �−� t + 1 2 � = I1 �� 1 2 � ∗v �� t + 1 2 � = �� 1 2 � ∗v �� t + 1 2 � = v � t + 1) = v(t), so that v ∈ PI1 H is as well invariant under I1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Since � 1 2 � ∗v is doubly symmetric with respect to ρ2 we obtain further that ρ2 ◦ v(0) = ρ2 ◦ �� 1 2 � ∗v �� − 1 2 � = ρ2 ◦ �� 1 2 � ∗v �� 1 2 � = ρ2 2 ◦ �� 1 2 � ∗v �� 0 � = �� 1 2 � ∗v �� 0 � = v � 1 2 � , so that v is doubly symmetric with respect to ρ2 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' This finishes the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' □ 5 The reduced monodromy Suppose that (M, ω) is a symplectic manifold and H : M → R is a smooth function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' We denote by φt H the flow of the Hamiltonian vector field of H, characterized by φ0 H(x) = x, d dtφt H(x) = XH(φt H(x)), x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' If v is a simple periodic orbit of XH of period τ we have φτ H(v(0)) = v(0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=', v(0) is a fixed point of φτ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' The differential of the flow dφτ H(v(0)): Tv(0)M → Tv(0)M 17 is a linear symplectic map of the symplectic vector space (Tv(0)M, ωv(0)) into itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' This map is referred to as the unreduced monodromy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Since H is au- tonomous, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=', does not depend on time, we have dφτ H(v(0))XH(v(0)) = XH(v(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Moreover, by preservation of energy the Hamiltonian H is preserved along the flow of its Hamiltonian vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' In particular, if c is the energy of v, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=', the value H attains along v, the differential of the flow maps the tangent space Tv(0)Σ of the energy hypersurface Σ = H−1(c) back to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Therefore the unreduced monodromy induces a linear map Mv := dφτ H(v(0)): Tv(0)Σ/⟨XH(v(0))⟩ → Tv(0)Σ/⟨XH(v(0))⟩ which is still symplectic for the symplectic structure on Tv(0)Σ/⟨XH(v(0))⟩ in- duced from ωv(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' This map is referred to as the reduced monodromy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Instead of restricting our attention to 0 we could consider the reduced monodromy M t v := dφτ H(v(t)): Tv(t)Σ/⟨XH(v(t))⟩ → Tv(t)Σ/⟨XH(v(t))⟩ for any t ∈ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Note that for different times t the reduced monodromies are symplectically conjugated to each other by the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Suppose now in addition that ρ is a real structure on (M, ω) under which H is invariant and v ∈ PI H is a symmetric periodic orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Since both points v(0) and v � 1 2 � lie in the fixed point set of ρ the differential of ρ gives rise to linear antisymplectic involutions dρ � v � 0 �� : Tv(0)M → Tv(0)M, dρ � v � 1 2 �� : Tv(1/2)M → Tv(1/2)M which induce real structures on the quotient spaces Tv(0)Σ/⟨XH(v(0))⟩ respec- tively Tv(1/2)Σ/⟨XH(v(1/2))⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Since the Hamiltonian vector field is anti-invariant, the antisymplectic involution ρ conjugates the forward flow to the backward flow ρφt Hρ = φ−t H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' In particular, differentiating this identity we have dρ � v � 1 2 �� dφτ/2 H � v � 0 �� dρ � v � 0 �� = � dφτ/2 H � v � 1 2 ���−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Therefore the induced maps Ψ := dφτ/2 H � v � 0 �� : Tv(0)Σ/⟨XH(v(0))⟩ → Tv(1/2)Σ/⟨XH(v(1/2))⟩ and Φ := dφτ/2 H � v � 1 2 �� : Tv(1/2)Σ/⟨XH(v(1/2))⟩ → Tv(0)Σ/⟨XH(v(0))⟩ 18 give rise to a real couple (Ψ, Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Note that the compositions coincide with the reduced monodromies at times 0 and 1 2 ΦΨ = dφτ H � v � 0 �� , ΨΦ = dφτ H � v � 1 2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Now we even assume that the symplectic manifold (M, ω) is doubly real with real structures ρ1 and ρ2 under both of which H is invariant and v ∈ PI1 H is doubly symmetric with respect to ρ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' The differential of ρ2 gives rise to a linear antisymplectic map dρ2(v(0)): Tv(0)M → Tv(1/2)M which induces an antisymplectic map on the quotient spaces S : Tv(0)Σ/⟨XH(v(0))⟩ → Tv(1/2)Σ/⟨XH(v(1/2))⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Since ρ1 commutes with ρ2 this map interchanges the real structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='2 we have that � 1 4 � ∗v ∈ PI2 H and therefore S makes the real cou- ple (Ψ, Φ) symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Therefore we obtain, as a consequence of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='4, the following 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='de A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content=' Moreno, Institute for Advanced Study, Princeton NJ, USA/ Heidelberg Uni- versit¨at, Heidelberg, Germany E-mail address: agustin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='moreno2191@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} +page_content='com 21' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQf1v5O/content/2301.01803v1.pdf'} diff --git a/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf b/WdE1T4oBgHgl3EQfvgUI/content/2301.03399v1.pdf new file mode 100644 index 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Livingston McPherson, and Katherine Driggs-Campbell +University of Illinois at Urbana-Champaign +Urbana, United States +{neeloyc2,aamirh2,sliu105,tj12,weihang2,dlivm,krdc}@illinois.edu +ABSTRACT +In autonomous driving, detection of abnormal driving behaviors is +essential to ensure the safety of vehicle controllers. Prior works in +vehicle anomaly detection have shown that modeling interactions +between agents improves detection accuracy, but certain abnor- +mal behaviors where structured road information is paramount +are poorly identified, such as wrong-way and off-road driving. We +propose a novel unsupervised framework for highway anomaly +detection named Structural Attention-based Recurrent VAE (SABeR- +VAE), which explicitly uses the structure of the environment to aid +anomaly identification. Specifically, we use a vehicle self-attention +module to learn the relations among vehicles on a road, and a sep- +arate lane-vehicle attention module to model the importance of +permissible lanes to aid in trajectory prediction. Conditioned on the +attention modules’ outputs, a recurrent encoder-decoder architec- +ture with a stochastic Koopman operator-propagated latent space +predicts the next states of vehicles. Our model is trained end-to-end +to minimize prediction loss on normal vehicle behaviors, and is +deployed to detect anomalies in (ab)normal scenarios. By combin- +ing the heterogeneous vehicle and lane information, SABeR-VAE +and its deterministic variant, SABeR-AE, improve abnormal AUPR +by 18% and 25% respectively on the simulated MAAD highway +dataset. Furthermore, we show that the learned Koopman operator +in SABeR-VAE enforces interpretable structure in the variational +latent space. The results of our method indeed show that model- +ing environmental factors is essential to detecting a diverse set of +anomalies in deployment. For code implementation, please visit +https://sites.google.com/illinois.edu/saber-vae. +KEYWORDS +Anomaly Detection, Autonomous Vehicles, Unsupervised Learning, +Human Behavior Modeling +1 +INTRODUCTION +Autonomous vehicles have the potential to realize a fast, safe, and +labor-free transportation system. A trustworthy self-driving vehi- +cle should have the ability to operate reliably in normal situations +and, more importantly, to perceive and react to anomalous driving +scenarios (e.g., skidding and wrong-way driving of surrounding +human vehicles) promptly and robustly. The detection of such ab- +normal situations can help identify traffic accidents and dangerous +driving behaviors of road participants, and thus provide high-level +guidance for vehicle controllers to act safely. +* denotes equal contribution. +Deep-learning based Anomaly Detection (AD) algorithms have +shown great promise in intelligent vehicle applications [7]. Many +previous works utilize vehicle trajectories as an anomaly signal [2, +11, 40]. However, only a few vehicle trajectory datasets with suf- +ficient anomaly labels exist for supervised learning methods [17, +40, 44]. To leverage the larger store of unlabeled driving data, re- +searchers like Yao and Wiederer have employed unsupervised learn- +ing methods [39, 41, 42]. Specifically, a neural network, which +generally follows an encoder-decoder architecture for trajectory +reconstruction or prediction, learns an underlying distribution of +normal vehicle trajectories in the latent space. An anomaly is then +detected whenever the trajectory is out of distribution and produces +a large reconstruction or prediction error. In interactive driving +scenarios, Wiederer et al. [39] showed that modeling interactions +between agents can largely improve the reconstruction accuracy +and subsequently the AD performance. However, such interaction- +aware methods still ignore the effect of road structures on vehicle +behaviors, and thus can miss abnormal scenarios like wrong-way +driving trajectories that appear normal when environmental con- +text is overlooked. +Alongside performance accuracy, the decisions made by AD +algorithms need to be interpretable to stakeholders. Deep neural +networks are black boxes by nature. However, the decisions of +deep networks impact various stakeholders such as policy makers +and end users. Designing methods with interpretable features for +stakeholders is a key challenge in AD, and the field of machine +learning overall [5, 15, 34, 36]. In vehicle AD more specifically, in- +terpretable algorithms need to account for the wide distribution +of human drivers who act according to their own policies [6]. For +example, different drivers may choose to overtake other vehicles at +different times and speeds. To ensure interpretability, we use vari- +ational autoencoder (VAE) to cluster useful features from similar +behaviors together in a continuous and stochastic latent space. Our +results indicate that vehicle trajectories transitioning to an abnor- +mal state are explicitly encoded by interpretable transformations +in the learned latent space. +In this paper, we present our novel unsupervised Structural +Attention-based Recurrent Variational Autoencoder (SABeR-VAE) +for highway vehicle anomaly detection. Since contemporary vehi- +cles have map information available to them regarding their nearby +environment and lanes, we make use of the environmental infor- +mation that prior works [26, 33, 39] have ignored to explicitly +model the effect of lane structure on normal vehicle behaviors. +Specifically, we treat a highway scenario as a structured interac- +tion graph where nodes represent vehicles and lane positions, and +edges connect nearby vehicles, and permissible lanes. Two separate +arXiv:2301.03634v1 [cs.RO] 9 Jan 2023 + +attention modules learn relations between vehicles (vehicle-vehicle +self-attention) and legal permissible route trajectories (lane-vehicle +attention) respectively. A sequence of embeddings from the vehicle- +vehicle attention module are encoded into a Gaussian latent space +to capture the randomness of vehicle trajectories with a recur- +rent network, and cluster similar behaviors close together in an +interpretable fashion. Our work is more computationally efficient +than STGAE-KDE [39], which has a deterministic latent space and +requires the expensive process of fitting a Kernel Density Estima- +tor (KDE) to learn a meaningful distribution of normal behaviors. +We then use a learned Koopman operator to propagate the cur- +rent latent distributions forward in time conditioned on the useful +lane embeddings. We show that the Koopman operator explicitly +enforces interpretable transformations in the latent space that stan- +dard autoencoders like STGAE are unable to incorporate, and is able +to model the complex, non-linear dynamics of drivers. Finally, we +decode a sampled point from the propagated distribution to predict +next states of vehicles. We train our method to predict trajecto- +ries from normal scenarios in the Multi-Agent Anomaly Detection +(MAAD) dataset [39], and compare accuracy metrics against linear, +recurrent, and graph convolutional approaches on anomalous tra- +jectories [31, 33, 39]. Our SABeR-VAE improves AUPR-Abnormal +and wrong-way driving detection over the STGAE-KDE by 18% and +35% respectively, and has an interpretable latent space over driving +behaviors. +Our contributions can be summarized as follows: (1) We present a +novel unsupervised variational approach for anomaly detection con- +ditioned on structured lane information; (2) We quantitatively show +that incorporating the structured information increases anomaly +detection accuracy, compared with state-of-the-art baselines and +ablations using the MAAD dataset; (3) We show that the stochas- +tic Koopman operator learns interpretable features of (ab)normal +behaviors in the latent space. +Our paper is organized as follows: Section 2 discusses relevant +works in the areas of structured modeling and anomaly detection. +Our problem formulation and methods are presented in Section 3. +We discuss results in Section 4. Finally, we conclude the paper and +discuss future directions in Section 5. +2 +RELATED WORKS +2.1 +Exploiting Map Information +The quality of information about an environment provided by High +Definition maps (HD-maps) has dramatically increased and led to +their ubiquitous use due to recent advancements in autonomous +driving [27, 43]. Currently, most state-of-the-art methods for vehicle +trajectory prediction, motion forecasting, and anomaly detection, +do not make effective use of the rich information provided in these +HD-maps, and only rely on modeling the interactions between +vehicles on the road [9, 24, 35]. Hence, these methods ignore vital +information such as the plausible movement of vehicles in the +environment, which can be paramount in identifying anomalies +such as wrong-way driving. +However, trajectory prediction methods such as those proposed +by Deo et al. and Liang et al. do exploit the information in these +HD-maps and significantly outperform their counterparts [16, 25]. +In proposing LaneGCN, Liang et al. encode different types of inter- +actions between agents on the road with lane information extracted +from maps. [25]. They show that attention-based models can be +used to encode interactions between vehicles and lanes, which are +learned by constructing a graph representation of the road. PGP, +proposed by Deo et al. , further produces scene-compliant trajecto- +ries by sampling from a distribution of driving profiles conditioned +on environment and vehicle interactions [16]. We corroborate the +usefulness of these vehicle and lane attention-based representa- +tions and show that such embeddings do in fact provide meaningful +insights in detecting highway vehicle anomalies in SABeR-VAE. +2.2 +Variational Autoencoders for Sequences +Variational autoencoders (VAE) have been applied to sequential +data combined with recurrent neural networks (RNN) in fields such +as speech and image synthesis and autonomous driving [8, 12– +14, 18, 26, 31]. Liu et al. attempt to infer the traits of drivers from +trajectories encoded in a variational latent space [26]. However, +only two classes of traits and a restricted set of defined trajectories +were considered, while real drivers have a much wider range of +behaviors on the road. Furthermore, they do not utilize map infor- +mation in their learning process, which provide relevant context +for traits. Conditional VAE formulations have also been found to +be able to generate trajectories with different driving styles, but +fail to consistently produce feasible trajectories without necessary +environment context [21, 32, 37]. Recurrent VAEs have also been +applied to robot anomaly detection, but are limited by the simplicity +of the single agent problem statement [31]. These sequential gen- +erative modeling approaches perform reasonably on their simple +tasks, but fail to generate realistic samples from points in the latent +space in more complex areas, due to the limitations of their RNN +components [12, 13, 19]. +To bridge the gap between complex human behaviors and the +structured environment, and overcome the hurdles of the tempo- +ral propagation in simplistic RNNs, we propose the use of a lane- +conditioned Koopman Operator to model the temporal relations in +the latent space. We were specifically inspired to use the Koopman +operator to propagate the latent space due to its capability to model +the dynamics of complex, non-linear data, including fluid dynamics, +battery properties, and control tasks [1, 3, 4, 28]. +2.3 +Anomaly Detection +Anomaly detection is well studied in diverse research areas and +application domains [10, 29]. In robotics and automated vehicles, +AD has been used to detect abnormal patterns such as robot fail- +ures [23, 30] and dangerous driving scenarios [39, 42]. +Park et al. propose a long short-term memory based variational +autoencoder (LSTM-VAE) to reconstruct the expected distribution +of robot sensor signals. A reconstruction-based anomaly score is +then used for anomaly detection [31]. Furthermore, Ji et al. adopt +an attention mechanism to fuse multi-sensor signals for robust +anomaly detection in uncertain environments [22]. While these +approaches focus on AD for single agent problem statements, our +highway scenarios consist of complex multi-agent social interac- +tions among vehicles, and need to be modeled as such. + +In the domain of traffic anomaly detection using multi-agent tra- +jectories, the most similar work to ours is the spatio-temporal graph +auto-encoder (STGAE) proposed with the MAAD dataset [39]. The +architecture follows an encoder-decoder structure to reconstruct +vehicle trajectories, where vehicle interactions and motions are con- +sidered using spatial graph convolution and temporal convolution +layers, respectively. The method has been shown to be effective +by modeling interactions among vehicles to detect anomalous ma- +neuvers in traffic. However, such a network ignores the constraints +imposed by road structures on vehicle trajectories and the variabil- +ity of human driver behaviors. In this work, we explicitly model +both vehicle-to-vehicle interactions and lane-to-vehicle interac- +tions to boost performance, and use an interpretable variational +architecture to learn a continuous distribution over behaviors. +3 +METHODOLOGY +In this section, we first introduce our problem formulation of anom- +aly detection from vehicle trajectories, and then explain our pro- +posed SABeR-VAE framework. +3.1 +Problem Formulation +Suppose 𝑛𝑡 ∈ [1, 𝑁] vehicles are on a road segment at any time 𝑡, +and each vehicle takes an acceleration and steering action every +timestep according to unknown policies. Let 𝑐 (𝑖) +𝑡 += +� +𝑥 (𝑖) +𝑡 ,𝑦(𝑖) +𝑡 +� +be +the 2𝐷 coordinates of the 𝑖th vehicle at time 𝑡, where 𝑖 ∈ [1, ...,𝑛𝑡]. +Each vehicle also has a set of corresponding permissible lane po- +sitions in front, to the left, and to the right of the vehicle, pro- +vided in the form of a discretized map representation shown in +Fig. 1. At every timestep, each vehicle’s discretized position within +the map is used to identify their corresponding front, left, and +right lane nodes. We define a tuple 𝑙 (𝑖) +𝑡 += (front, left, right)(𝑖) +𝑡 +of three 2𝐷 coordinates containing the lane information for ve- +hicle 𝑖 at time 𝑡. Altogether, the observed information of each +vehicle at any time is the relative displacement of coordinates +𝑜 (𝑖) +𝑡 += +� +𝑐 (𝑖) +𝑡 +− 𝑐 (𝑖) +𝑡−1,𝑙 (𝑖) +𝑡 +− 𝑐 (𝑖) +𝑡 +� += +� +𝑋 (𝑖) +𝑡 +, 𝐿(𝑖) +𝑡 +� +. A trajectory of length +𝑇 for any vehicle is represented as +� +𝑜 (𝑖) +0 ,𝑜 (𝑖) +1 , ...,𝑜 (𝑖) +𝑇−1 +� +. We assume +that any vehicle 𝐴 that is within a distance 𝑑 to another vehicle 𝐵 +at time 𝑡 can accurately detect and track the relative coordinates +𝑐 (𝐵) +𝑡 +−𝑐 (𝐴) +𝑡 += 𝑅(𝐴,𝐵) +𝑡 +. The purple arrow between the green and blue +vehicle in Fig. 1 represents this vehicle interaction type. For the +𝑖-th car, the number of observable cars is 𝑚𝑖 +𝑡 ∈ [0,𝐶]. Given all +vehicle trajectories in a scene, our goal is to provide an anomaly +score AS𝑡 ∈ R≥0 for each time 𝑡. +3.2 +Architecture +Figure 2 contains the complete architecture diagram of SABeR-VAE, +which we discuss in this section. +3.2.1 +Vehicle-Vehicle Self-Attention Network. Our goal is to +learn a representation of spatial interactions among vehicles. Rather +than using convolutional methods like those in prior works [25, 39], +we encode the positions of vehicles on the road at each time with +scaled dot-product multi-head self-attention, which allows each +head to learn different features of the data [38]. +Figure 1: Map discretization and interaction edges. We model +the vehicle AD problem as an interaction graph with vehicle and +lane nodes. A continuous map of the road is discretized into blocks. +Directed lane edges between lane nodes encode permissible routes +for vehicles. The red vehicle has a directed edge toward the lane +nodes in front and to its left because the driver can legally continue +forward or merge left. Conversely, the green vehicle has no edge +connecting to a left lane node since it cannot cross the road divider. +Vehicle edges, shown in purple, exist for vehicles that are close +enough to interact with each other. +We embed the coordinates of each car 𝑋𝑡 with a multi-layer +perceptron (MLP) 𝑓 𝑉𝑉 +𝑄 +to obtain queries 𝑄𝑉𝑉 +𝑡 +: +𝑓 𝑉𝑉 +𝑄 +(𝑋𝑡) = 𝑄𝑉𝑉 +𝑡 +∈ R1×𝐷, +(1) +where 𝐷 is the attention size. Let R𝑡 = +� +𝑅(𝑖,1) +𝑡 +, ..., 𝑅(𝑖,𝑚) +𝑡 +�⊺ +be the +displacements of all neighboring cars for the 𝑖-th car. We use two +other MLPs, 𝑓 𝑉𝑉 +𝐾 +and 𝑓 𝑉𝑉 +𝑉 +, to embed R𝑡 to obtain keys 𝐾𝑉𝑉 +𝑡 +and +values 𝑉𝑉𝑉 +𝑡 +respectively: +𝑓 𝑉𝑉 +𝐾 +(R𝑡) = 𝐾𝑉𝑉 +𝑡 +∈ R𝑚×𝐷 +𝑓 𝑉𝑉 +𝑉 +(R𝑡) = 𝑉𝑉𝑉 +𝑡 +∈ R𝑚×𝐷 +(2) +The final encoding of each vehicle position from this self-attention +layer for time 𝑡 is calculated as: +softmax +��� +� +𝑄𝑉𝑉 +𝑡 +� +𝐾𝑉𝑉 +𝑡 +�⊺ +√ +𝐷 +��� +� +𝑉𝑉𝑉 +𝑡 += 𝑝𝑉𝑉 +𝑡 +∈ R1×𝐷 +(3) +Nonexistent or unobserved vehicles further than a distance 𝑑 +cannot be allowed to contribute to the attention score of other +vehicles. Thus, we use a mask to set the score contributed from +unobserved vehicles to −∞. +3.2.2 +Lane-Vehicle Attention Network. We use available map +and lane information in a separate lane-vehicle attention layer +to model legal maneuvers in structured environments. Similar to +vehicle-vehicle attention, the query 𝑄𝐿𝑉 +𝑡 +is an embedding of 𝑋𝑡. +Lane information of each vehicle 𝐿𝑡 is used to produce keys 𝐾𝐿𝑉 +𝑡 + +and values 𝑉 𝐿𝑉 +𝑡 +: +𝑓 𝐿𝑉 +𝑄 +(𝑋𝑡) = 𝑄𝐿𝑉 +𝑡 +∈ R1×𝐷 +𝑓 𝐿𝑉 +𝐾 +(𝐿𝑡) = 𝐾𝐿𝑉 +𝑡 +∈ R3×𝐷 +𝑓 𝐿𝑉 +𝑉 +(𝐿𝑡) = 𝑉 𝐿𝑉 +𝑡 +∈ R3×𝐷 +(4) +The lane-conditioned vehicle embeddings are calculated as: +softmax +��� +� +𝑄𝐿𝑉 +𝑡 +� +𝐾𝐿𝑉 +𝑡 +�⊺ +√ +𝐷 +��� +� +𝑉 𝐿𝑉 +𝑡 += 𝑝𝐿𝑉 +𝑡 +∈ R1×𝐷 +(5) +Note that all three lane nodes may not always be permissible to +a vehicle. For example, a car in the left-most lane of a road is unable +to legally turn left. As such, we mask out impermissible lane nodes +like in the self-attention layer. +3.2.3 +Recurrent Encoder. A gated recurrent unit (GRU) network +encodes the sequence of self-attention features for each vehicle +� +𝑝𝑉𝑉 +0 +, 𝑝𝑉𝑉 +1 +, ..., 𝑝𝑉𝑉 +𝑇−1 +� +into a sequence of Gaussian distributions in +the latent space with temporal correlation. Thus, the latent space +captures the stochastic nature of human behaviors. Specifically, af- +ter embedding the vehicle-vehicle attention feature with a network +𝑓𝑒, we pass the embedding through the GRU to get the hidden state +of each vehicle for the current timestep: +ℎ𝑡 = GRU +� +ℎ𝑡−1, 𝑓𝑒 +� +𝑝𝑉𝑉 +𝑡 +�� +(6) +Mean and variance neural networks 𝑓𝜇 and 𝑓𝜎 produce parame- +ters for a latent normal distribution of dimension 𝑗 conditioned on +a vehicle’s hidden state at any time: +𝜇𝑉𝑉 +𝑡 += 𝑓𝜇 (ℎ𝑡) , +𝜎𝑉𝑉 +𝑡 += 𝑓𝜎 (ℎ𝑡) . +(7) +3.2.4 +Latent Propagation with Koopman Operator. While the +GRU encoder encodes vehicle behaviors into the latent space solely +conditioned on past and current vehicle interactions, we need a +method to propagate the latent distributions in time to predict the +future states of vehicles. To this end, we learn a stochastic Koopman +operator conditioned on the lane-vehicle embeddings to perform +this task, like Balakrishnan and Upadhyay [4]. +In Koopman operator theory, a discrete time system evolves +according to potentially nonlinear dynamics𝑥𝑡+1 = 𝐹 (𝑥𝑡). However, +a function 𝑔 maps the state 𝑥𝑡 into a space where dynamics evolve +linearly with the Koopman operator K [4]: +K𝑔 (𝑥𝑡) = 𝑔 (𝐹 (𝑥𝑡)) = 𝑔 (𝑥𝑡+1) +(8) +Similarly, the inverse of function 𝑔 translates an observable of 𝑥 +back into the original dynamics space [4]: +𝑔−1 (K𝑔 (𝑥𝑡)) = 𝑥𝑡+1 +(9) +In our case, function 𝑔 is represented by the GRU encoder and +neural networks 𝑓𝜇 and 𝑓𝜎, which altogether, produce a latent dis- +tribution N (𝜇𝑡, 𝜎𝑡) conditioned on inter-vehicle embeddings 𝑝𝑉𝑉 +𝑡 +. +Like the Stochastic Adversarial Koopman (SAK) model [4], we +use auxiliary neural networks 𝑓 𝜇 +aux and 𝑓 𝜎 +aux to predict tridiagonal +Koopman matrices 𝐾𝜇,𝑡 and 𝐾𝜎,𝑡, rather than solving for their closed +form solution. The outputs of 𝑓 𝜇 +aux and 𝑓 𝜎 +aux are conditioned on the +current latent distributions N +� +𝜇𝑉𝑉 +𝑡 +, 𝜎𝑉𝑉 +𝑡 +� +and the lane features +𝑝𝐿𝑉 +𝑡 +, so that the Koopman operators capture legal route maneuvers +in the latent space propagation: +𝐾𝜇,𝑡 = 𝑓 𝜇 +aux +� +𝜇𝑉𝑉 +𝑡 +, 𝑝𝐿𝑉 +𝑡 +� +𝐾𝜎,𝑡 = 𝑓 𝜎 +aux +� +𝜎𝑉𝑉 +𝑡 +, 𝑝𝐿𝑉 +𝑡 +� +(10) +The predicted Koopman matrices are applied to the inter-vehicle +distributions to linearly propagate the mean and variance of the +latent distributions forward in time: +𝜇𝐿𝑉 +𝑡+1 = 𝐾𝜇,𝑡 𝜇𝑉𝑉 +𝑡 ++ 𝜇𝑉𝑉 +𝑡 +(11a) +𝜎𝐿𝑉 +𝑡+1 = 𝐾𝜎,𝑡 𝜎𝑉𝑉 +𝑡 ++ 𝜎𝑉𝑉 +𝑡 +(11b) +Intuitively, we can interpret the GRU encoder as predicting a +distribution of vehicle behaviors from their current trajectories, and +the Koopman operator propagates to a one-step future distribution +of behaviors based on lane information. +3.2.5 +The Decoder Network. At this point, we have two sets of +distributions in the 𝑗-dimensional latent space for the current states +and future predictions of vehicles at each time: N +� +𝜇𝑉𝑉 +𝑡 +, 𝜎𝑉𝑉 +𝑡 +� +and +N +� +𝜇𝐿𝑉 +𝑡+1, 𝜎𝐿𝑉 +𝑡+1 +� +. We utilize the reparameterization trick to sample a +point from each of the distributions: +𝜖𝑉𝑉 +𝑡 +∼ N (0, 1) +𝑧𝑉𝑉 +𝑡 += 𝜇𝑉𝑉 +𝑡 ++ 𝜖𝑉𝑉 +𝑡 +𝜎𝑉𝑉 +𝑡 +𝜖𝐿𝑉 +𝑡+1 ∼ N (0, 1) +𝑧𝐿𝑉 +𝑡+1 = 𝜇𝐿𝑉 +𝑡+1 + 𝜖𝐿𝑉 +𝑡+1𝜎𝐿𝑉 +𝑡+1 +(12) +A multi-layer perceptron 𝑓dec is used as a decoder network, sim- +ilar to 𝑔−1 in Eq. 9, to predict a vehicle coordinate change from the +sampled latent points: +ˆ𝑋𝑡 = 𝑓dec +� +𝑧𝑉𝑉 +𝑡 +� +, +ˆ𝑋𝑡+1 = 𝑓dec +� +𝑧𝐿𝑉 +𝑡+1 +� +. +(13) +3.3 +Training and Evaluation +3.3.1 +End-to-End Training. To fairly compare our method with +prior convolutional approaches, we utilize a similar sliding window +training approach performed by Wiederer et al. [39]. Specifically, +whole trajectories of length 𝑇 are divided into small overlapping +segments, or windows, of constant length 𝑇 ′. +In our training objective, we minimize the current reconstruc- +tion loss and one-step future prediction loss of the model by split- +ting our input ground truth trajectories into current states 𝑋 − = +𝑋0:𝑇 ′−2 and one-step future states 𝑋 + = 𝑋1:𝑇 ′−1. We also regularize +the current distributions N +� +𝜇𝑉𝑉 +𝑡 +, 𝜎𝑉𝑉 +𝑡 +� +and propagated distribu- +tions N +� +𝜇𝐿𝑉 +𝑡+1, 𝜎𝐿𝑉 +𝑡+1 +� +to follow a standard normal distribution. Let +𝐷KL(𝜇, 𝜎) be the KL divergence between any Gaussian distribution +N (𝜇, 𝜎) and the standard normal distribution N (0, 1). Then the +regularized prediction and reconstruction losses are: +Lpred = 𝛽1 · 𝐷KL +� +𝜇𝐿𝑉 , 𝜎𝐿𝑉 � ++ ∥𝑋 + − ˆ𝑋 +∥2 +Lrecon = 𝛽2 · 𝐷KL +� +𝜇𝑉𝑉 , 𝜎𝑉𝑉 � ++ ∥𝑋 − − ˆ𝑋 −∥2, +(14) +where 𝛽1 and 𝛽2 are tunable weights applied to the regularization +of the latent distributions similar to beta-VAE [20]. +The overall objective we optimize is: +L = Lpred + Lrecon +(15) + +GRU Encoder +Decoder +Attention Modules +Koopman Propagation +Vehicle +Self-Attention +GRU +Lane-Vehicle +Attention +Figure 2: SABeR-VAE architecture. The SABeR-VAE architecture attempts to predict the one-step future states of vehicles conditioned on +current vehicle positions and structural lane information. Vehicle interactions are modeled by the self-attention module while permissible +routes are encoded by the lane-vehicle attention module. A GRU encoder processes the self-attention embeddings through time to produce a +latent distribution. Then the Koopman operator conditioned on the lane embeddings propagates the latent distributions forward, which +finally get decoded to predict next states. The 𝑓dec network shares parameters for reconstruction and prediction. +We again mask out coordinates of unobserved vehicles so they do +not contribute to the loss. +While SAK [4] applies maximum mean discrepancy (MMD) to +synchronize the current and propagated distributions of their Koop- +man model to any general distribution, we explicitly encourage the +latent space distributions to follow the standard gaussian. We leave +experimentation of various Koopman synchronization methods for +the anomaly detection task as a future direction of research. +3.3.2 +Anomaly Detection Evaluation. At test time, we follow +the same sliding window practice as performed in training. First, +we calculate the one-step future prediction loss Lpred,𝑡+1 = ∥𝑋 (𝑖) +𝑡+1 − +ˆ𝑋 (𝑖) +𝑡+1∥2 for every vehicle at each timestep, within every window of +a complete trajectory. +Then, we average the prediction loss of overlapping timesteps +among all windows in the sequence, for each vehicle separately. +Suppose W (𝑡,𝑖) is the set of all windows in the complete trajec- +tory containing time 𝑡 where vehicle 𝑖 is observed. The averaged +prediction error for car 𝑖 at 𝑡 is: +¯L(𝑖) +pred,𝑡 = +� +𝑤∈W (𝑡,𝑖) Lpred,𝑤(𝑖) +𝑡 +|W (𝑡,𝑖) | +, +(16) +where Lpred,𝑤 (𝑖) +𝑡 +is the prediction error of time 𝑡 for vehicle 𝑖 in +window 𝑤 of the set W (𝑡,𝑖). +Finally, we choose the anomaly score AS to be the maximum +averaged prediction loss over all vehicles at a given timestep 𝑡: +AS𝑡 = +max +𝑖=1,...,𝑛𝑡 +¯L(𝑖) +pred,𝑡 +(17) +4 +EXPERIMENTAL SETUP AND RESULTS +In this section, we first describe the MAAD dataset on which we +performed experiments and detail baselines and ablations. We also +present our quantitative results and latent space interpretations. +4.1 +MAAD Dataset and Augmentation +The MAAD dataset [39] consists of 2𝐷 trajectories of two vehicles +on a straight two-lane highway with a divider separating the two +Figure 3: Trajectories and SABeR-VAE anomaly scores. (top +row) Examples of a normal overtaking (a,) abnormal off-road driv- +ing (b,) and wrong-way driving (c) scenarios in the MAAD dataset. +White arrows point toward direction of normal traffic flow. (bot- +tom row) Predicted anomaly score curves for each scenario above. +Colors of lines within the curves show the ground-truth labels of +normal (green,) ignored (yellow,) & abnormal (red) timesteps. +possible directions, as visualized in the top row of Fig. 3. There are +80 training and 66 test-split trajectories ranging from a length of 25 +to 127 timesteps. These dynamic length trajectories are subsampled +to produce approximately 6.3K training and 3.1K testing windows +of constant length𝑇 ′ = 15. As the original dataset sequences did not +come with map or lane details, we augmented the data to include +this information. Specifically, we discretized the highway in the +𝑥-coordinate direction into blocks of length five meters as shown +in Fig. 1, and stored the 2𝐷 coordinates of the front, left, and right +blocks for each vehicle at every timestep in all trajectories. All +the training sequences consist of normal vehicle behaviors like +driving side-by-side, overtaking, following, and driving in opposite +directions. In contrast, the test-split contains both normal and 11 +anomalous behavior classes like aggressive overtaking, pushing +aside, tailgating, off-road, and wrong-way driving. +4.2 +Baseline Methods +We compare against baselines implemented by Wiederer et al. that +depend on reconstruction loss rather than future prediction er- +ror [39]. (1) The Constant Velocity Model (CVM) is a standard +baseline that predicts the next states of vehicles assuming each + +1.0 +: 10 +40 +0.8- +30 +5 +0.6- +201 +:1.0- +1.0 T +0.4 - +: 0.5 +; 0.5 / +0.2 +0.0 1 +i0.0 +10.0 +20 +40 +60 +80 +10 +15 +20 +25 +5 +10 +15 +20 +25 +0 +Timestep +Timestep +Timestep +(a) +(b) +(c)Table 1: Accuracy results of baselines, ablations, and SABeR methods over ten runs. +Method +Detection Type +AUROC ↑ +AUPR-Abnormal ↑ +AUPR-Normal ↑ +FPR @ 95%-TPR ↓ +CVM +Reconstruction Loss +83.1 ± 0.0 +54.5 ± 0.0 +96.0 ± 0.0 +74.6 ± 0.0 +RAE-Recon* +Reconstruction Loss +56.2 ± 0.7 +16.9 ± 1.0 +89.5 ± 0.1 +84.6 ± 0.3 +STGAE* +Reconstruction Loss +74.8 ± 5.1 +37.8 ± 7.2 +94.1 ± 1.3 +77.8 ± 9.8 +STGAE-KDE* +One Class +86.3 ± 1.7 +55.2 ± 7.7 +97.2 ± 0.5 +50.0 ± 7.9 +RAE-Pred +Prediction Loss +72.5 ± 15.3 +43.5 ± 17.4 +92.9 ± 4.4 +75.8 ± 10.6 +VV-RAE +Prediction Loss +54.2 ± 4.9 +14.8 ± 0.9 +89.5 ± 2.4 +77.1 ± 7.3 +SABeR-AE +Prediction Loss +87.2 ± 0.4 +69.0 ± 0.5 +96.9 ± 0.2 +64.1 ± 5.0 +SABeR-VAE +Prediction Loss +87.0 ± 1.5 +65.5 ± 2.9 +96.9 ± 0.5 +57.7 ± 7.6 +* These results are presented in [39]. +vehicle travels at the same velocity as the last timestep, without +modeling any inter-vehicle relations. (2) Recurrent Autoencoder +(RAE-Recon) uses an LSTM network to encode and decode a se- +quence of coordinates from an unregularized latent space, attempt- +ing to minimize reconstruction loss. (3) Spatio-Temporal Graph +Autoencoder (STGAE) is a convolutional method that models inter- +vehicle behaviors, and outputs parameters for a bi-variate distri- +bution describing the estimated state of the reconstructed pose +of vehicles, and is trained to maximize the log-likelihood of the +estimated probability distribution. Finally, (4) the STGAE-KDE +baseline fits a Kernel Density Estimator (KDE) to the unregularized +latent space of a trained STGAE model to predict the one-class +probability of a set of points originating from a normal behavior +window. Unlike the STGAE-KDE, our SABeR-VAE does not require +an expensive KDE fitting procedure since our anomaly score solely +relies on prediction error, and we still model inter-vehicle relations +unlike CVM and RAE-Recon. +We additionally train ablation models with future prediction +loss to identify the impact of different components in our method. +We train (5) an unregularized Recurrent Autoencoder (RAE-Pred,) +using a standard deterministic MLP to propagate latent points for- +ward in time, without explicitly modeling any inter-vehicle be- +haviors, like the RAE-Recon. (6) A Recurrent Autoencoder with a +vehicle-vehicle Self-Attention module (VV-RAE) minimizes predic- +tion error while modeling inter-vehicle relations. We also train (7) +a deterministic variant of SABeR-VAE without a regularized latent +space, SABeR-AE. SABeR-AE utilizes both vehicle self-attention +and lane-vehicle attention like SABeR-VAE, but encodes trajectories +into an unregularized (uninterpretable) latent space. +4.3 +Quantitative Evaluation Metrics +We quantitatively evaluate the effectiveness of models on the MAAD +dataset using four metrics. (1) Area Under Receiver-Operating +Characteristic curve (AUROC) is calculated by plotting the False- +Positive Rate (FPR) and True-Positive Rate (TPR) of a model over +several decision thresholds, and computing the area under the +curve. A model with greater AUROC performs better, and a perfect +classifier has an AUROC of 100%. Though, AUROC is skewed in +datasets where there are very few positive labels, like in the field +of outlier identification. As such, FPR may be misleadingly low, +producing an optimistic AUROC value. We compute (2) the Area +Under Precision-Recall Curve (AUPR) with the anomalous points +being the positive class (AUPR-Abnormal) and (3) with normal +points being positive (AUPR-Normal). The AUPR metric adjusts +for skewed dataset distributions, and we evaluate model effective- +ness of classifying anomalies, and not mis-classifying normal points +with AUPR-Abnormal and AUPR-Normal respectively. Finally, we +use (4) FPR @ 95%-TPR to check the rate of mis-labeling normal +points when TPR is high. +4.4 +Training +Several runs of SABeR-VAE were trained with learning rates rang- +ing from 5𝑒 − 5 to 1𝑒 − 3, batch sizes from 32 to 128, KL-Divergence +𝛽1 = 𝛽2 weighting from 1𝑒 − 6 to 1𝑒 − 3, latent dimension size from +2 to 64, attention embedding sizes 32 and 64, and an inter-vehicle +distance 𝑑 of 45 meters for the vehicle-vehicle attention mask. The +RAE-Pred, VV-RAE, and SABeR-AE models were similarly trained, +but without 𝛽, attention embedding sizes, and 𝑑 where irrelevant. +Multi-head attention modules were instantiated with 8 heads each. +Every model was trained for 500 epochs on the training split with +a Tesla V100 GPU. Like Wiederer et al. , we calculate metrics for +each hyperparameter choice on a 20% validation split of the whole +test data, and choose to evaluate the best set of hyperparameters +for each respective method on the complete test split [39]. +4.5 +Accuracy Results +Table 1 holds accuracy results of baselines, ablations, and the SABeR +methods on the test split of the MAAD dataset. Each method, be- +sides CVM, was trained ten separate times with the same hyperpa- +rameters, and we report the average and standard deviation of each +methods’ results over the ten runs. Figure 4 plots the ROC curves +for each method. +Amongst baselines, the simple CVM model already performs +well as an anomaly detector since its AUROC is only 3% less than +that of the STGAE-KDE method. CVM also has no variation of +results since it is a deterministic model that is not trained. The +LSTM-based RAE-Recon model is unable to effectively distinguish +between anomalies and normal scenarios using reconstruction loss, +since it does not model vehicle or lane information. While recurrent +models encode current timestep features based solely on previous +timesteps, temporal convolution methods extract information from + +the whole trajectory, which helps to predict a more accurate re- +construction. Thus, the convolutional STGAE method drastically +improves AUROC and AUPR scores over RAE-Recon. +However, once we incorporate a latent propagation network and +predict future timesteps, the RAE-Pred ablation increases AUROC +over RAE-Recon by 29% and even outperforms STGAE in the AUPR- +Abnormal metric, without even modeling inter-vehicle behaviors. +This result hints to the idea that recurrent networks learn to model +normal behaviors more accurately with future prediction error, +than reconstruction error of observed timesteps, which assists in +AD performance. Furthermore, recurrent methods are capable of +reaching the same performance as convolutional methods, while +relying on fewer trajectory data points. Still, RAE-Pred is shown to +be unstable as it produces a high variance in results over the ten +trained models. This variance was caused by two of the ten runs +achieving only 45% AUROC. STGAE also has the highest variance in +results among baselines since it is a stochastic method reconstruct- +ing a distribution over states, rather than the deterministic CVM +and RAE-Recon approaches, but is more stable than RAE-Pred. +Surprisingly, we see that adding a vehicle-vehicle self-attention +layer to RAE-Pred to produce the VV-RAE model actually hinders +performance, and gives results similar to RAE-Recon. Effectively, +the vehicle-vehicle self-attention layer did not learn useful features +for the future prediction task, and confused the model generations. +This outcome could be a result of a low complexity neural network +or a potentially poor choice for masking distance 𝑑. +The one-class prediction model STGAE-KDE fits a KDE to the +latent space of the STGAE under the assumption that normal be- +havior latent points will be clustered together in the autoencoder. +As such, this one-class classification approach improves detection +rates and training stability over the STGAE such that it outperforms +other baselines. However, the fitting process of a KDE to a large +dimensional space is a computationally complex and constrictive +part the method. With gaussian regularization of the latent space, +our SABeR-VAE clusters similar behaviors together and learns a +latent distribution without fitting a KDE, which we discuss in 4.6. +Finally, SABeR-AE and SABeR-VAE incorporate a lane-vehicle +attention module to capture the effect of the structure of the envi- +ronment on normal behaviors. We see that SABeR-AE outperforms +all methods in AUROC and AUPR-Abnormal with low variance, +showcasing the importance of modeling environment structure in +this field. SABeR-VAE performs slightly better than STGAE-KDE in +AUROC, and significantly increases the AUPR-Abnormal score by +18%. However, the stochasticity of the SABeR-VAE method hinders +its reproducibility, and AUROC scores ranged from 84% to 89% over +the ten training runs. SABeR-VAE further decreases the average +FPR @ 95%-TPR of SABeR-AE by 10%. STGAE-KDE and the two +SABeR approaches have similar AUPR-Normal. +We present examples of SABeR-VAE scoring anomalous timesteps +in Fig. 3. There, a normal overtaking maneuver was scored very +low during the whole trajectory, whereas going off-road or driv- +ing in the wrong direction were scored high. We can also see in +Fig. 3.b that timesteps where vehicles are acting normally prior to +erratic behavior are still correctly scored low. Table 2 holds AU- +ROC by anomaly type for CVM, STGAE-KDE, and SABeR methods. +Figure 4: ROC curves of tested methods. +Table 2: AUROC (↑) of methods by anomaly type. +Anomaly Type +CVM +STGAE-KDE* +SABeR-AE +SABeR-VAE +Reeving +96.7 +94.6 +87.9 +89.5 +Pushing Aside +91.5 +90.4 +88.6 +91.3 +Right Spreading +87.7 +96.2 +86.7 +95.2 +Left Spreading +90.9 +96.6 +96.2 +95.9 +Off-Road +88.7 +98.2 +98.2 +99.7 +Skidding +96.9 +99.7 +∼100.0 +99.8 +Wrong-way +63.2 +73.2 +∼100.0 +99.3 +* These results are presented in [39]. +SABeR-VAE improves wrong-way driving AD by 35% over STGAE- +KDE, while performing comparably in other metrics. The complete +version of Table 2 is provided in the supplementary material. +4.6 +Latent Space Interpretation +SABeR-VAE is a variational model with a continuous latent space +such that observations with similar learned characteristics are clus- +tered closer together in the latent space. In Fig. 5.a, we plot the +test-split latent space of a SABeR-VAE model trained with a latent +dimension size of 2. Points are clustered into three distinct regions +in the latent space, which we will refer to as “bottom,” “middle,” and +“top” clusters respectively. We see from sampled trajectories that +the bottom and middle clusters encode vehicles that travel toward +the −𝑥 (left) direction in either of the top two lanes of the highway, +while the top cluster encodes vehicles traveling to the right in the +bottom two lanes. For example, points 1P, 2P, 3B, 4B, and 5B corre- +spond to blue (B) and pink (P) cars that travel to the left in the top +lanes. Similarly, the trajectory of the pink car driving to the right +in window 4 is encoded to point 4P in the top cluster of the latent +space. Vehicles that are also physically close and interacting with +each other are encoded closely in the latent space, as shown with + +Receiver Operating Characteristic Curves +1.0 +0.8 +True Positive Rate +0.6 +0.4 +CVM +RAE-Recon +STGAE + STGAE+KDE +0.2 +RAE-Pred +VV-RAE +SABeR-AE +SABeR-VAE +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +False Positive RateFigure 5: Koopman propagated latent space and corresponding trajectories. (a) We encode every window trajectory in the test-split +of the MAAD dataset and plot the 2𝐷 sampled latent positions of the final timestep of the windows. Blue points correspond to ground truth +normal windows while orange are abnormal. (1-5) Five scenario windows are encoded into the latent space, and are explicitly annotated +in (a.) Each of the five windows has two latent points for the pink and blue vehicles respectively. (e.g., annotations “1B” & “1P” are the +latent points of the blue and pink vehicles in road trajectory window 1.) Within the five trajectory windows, solid lines are the ground +truth trajectory of the vehicle while open circles are predicted by SABeR-VAE. White arrows denote direction of traffic flow. (b & d) The +prediction error curves for trajectories 4 and 5 respectively. (c & e) The trajectory of latent points for vehicle windows 4 and 5 respectively. +A heat map of the original latent space is plotted in orange in the background. Blue and pink circles are the latent trajectories of the blue and +pink car through time. The largest circles encode the initial timestep of the window, and they decrease in size as the window progresses. +latent points corresponding to windows 1 and 2. The middle cluster +embeds anomalous scenarios from the top lanes where vehicles are +close enough to interact with each other, like window 2. +Furthermore, anomalous, non-interactive trajectories that are +poorly predicted are encoded to the outskirts of the primary cluster +distributions. For example, the pink cars in trajectories 3 and 5 +are driving in the wrong direction. These trajectories have high +prediction error as visualized by the little overlap between the pre- +dicted open circle positions and ground truth trajectories. As such, +those poorly predicted points are encoded in the spaces between +the bottom and middle, and middle and top clusters respectively. +In contrast, trajectories 1 and 4 have low loss and are encoded to +positions within the primary clusters. Thus the latent space has +learned a correspondence between permissible lane routings and +expected vehicle behavior. +Finally, we visualize the transformation of the latent space over +time within one trajectory window to show the interpretable im- +pact of the learned Koopman operator. Figures 5.c and 5.e show the +transformation of the latent space as time progresses in trajectory +windows 4 and 5. We can see in Fig. 5.c that the blue and pink latent +trajectories stay in the bottom and top clusters respectively, since +the vehicles follow the correct direction on the road throughout +window 4. Conversely, we see in Fig. 5.e that the pink latent trajec- +tory begins in the top cluster since the pink vehicle in trajectory +5 is in one of the bottom two lanes on the road. But, at timestep +6, the pink car crosses the road divider into the wrong direction +lane. Thus, we see a jump in the pink car’s latent trajectory in +Fig. 5.e from the top cluster to the bottom and middle clusters that +correspond to the top two lanes. At the same time, Fig. 5.d has a +spike in the prediction loss of the pink vehicle. For the remainder +of the trajectory window, the pink car oscillates drastically in the +latent space around the middle cluster, since the model expects +the vehicle to be traveling to the left. Note, even though the pink +vehicle in trajectory 5 is acting abnormally, this does not effect the +latent trajectory of the blue vehicle in the same window, since the +vehicles are not close enough to impact each other. Overall, the +Koopman operator explicitly models this transition from normal to +anomalous states in the latent space, in an interpretable manner. +5 +CONCLUSIONS AND FUTURE WORK +In this paper, we propose a novel framework for anomaly detection +with an unsupervised recurrent VAE network conditioned on struc- +tured environment information and vehicle interactions. We show +that modeling this structured information is imperative to having +high accuracy over a wide range of anomaly types. Furthermore, +we have identified that our stochastic Koopman operator enforces +interpretable propagation of the latent space using the lane embed- +dings. Several areas for possible future study include: (1) applying +SABeR methods to real-world trajectory datasets containing anom- +alies, (2) combining the anomaly detector with a vehicle controller, +(3) using raw sensor data rather than processed ground truth po- +sitions of vehicles for anomaly detection, and (4) performing user +studies on preferences for notification and handling of anomalies. + +1 +Annotated Propagated Latent Space +normal +abnormal +1.0 +40 +3P +0.8 +30/ +2P +2B右 +0.6 +1.8* +0.4 +5P +2 +0.2 +0.5 +口 +0.0 +2 +10 +12 +10 +12 +Timestep +Timestep +H +5B +1B. + t=o +C +1P +t=6 +t= 12 +3B +3 +4B +(c) +(e)ACKNOWLEDGMENTS +We thank Julian Wiederer for providing access to the MAAD dataset +and assisting with baseline code. This work utilizes resources sup- +ported by the National Science Foundation’s Major Research In- +strumentation program, grant #1725729, as well as the University +of Illinois at Urbana-Champaign. +REFERENCES +[1] Hassan Arbabi and Igor Mezić. 2017. 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Finding Crit- +ical Scenarios for Automated Driving Systems: A Systematic Literature Review. +ArXiv abs/2110.08664 (2021). + diff --git a/YtE2T4oBgHgl3EQfEQa1/content/tmp_files/load_file.txt b/YtE2T4oBgHgl3EQfEQa1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5dba9badf54b63f3e31b969850f9f56e7f7f001f --- /dev/null +++ b/YtE2T4oBgHgl3EQfEQa1/content/tmp_files/load_file.txt @@ -0,0 +1,783 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf,len=782 +page_content='Structural Attention-Based Recurrent Variational Autoencoder for Highway Vehicle Anomaly Detection Neeloy Chakraborty, Aamir Hasan*, Shuijing Liu*, Tianchen Ji*, Weihang Liang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Livingston McPherson, and Katherine Driggs-Campbell University of Illinois at Urbana-Champaign Urbana, United States {neeloyc2,aamirh2,sliu105,tj12,weihang2,dlivm,krdc}@illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='edu ABSTRACT In autonomous driving, detection of abnormal driving behaviors is essential to ensure the safety of vehicle controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Prior works in vehicle anomaly detection have shown that modeling interactions between agents improves detection accuracy, but certain abnor- mal behaviors where structured road information is paramount are poorly identified, such as wrong-way and off-road driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' We propose a novel unsupervised framework for highway anomaly detection named Structural Attention-based Recurrent VAE (SABeR- VAE), which explicitly uses the structure of the environment to aid anomaly identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Specifically, we use a vehicle self-attention module to learn the relations among vehicles on a road, and a sep- arate lane-vehicle attention module to model the importance of permissible lanes to aid in trajectory prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Conditioned on the attention modules’ outputs, a recurrent encoder-decoder architec- ture with a stochastic Koopman operator-propagated latent space predicts the next states of vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Our model is trained end-to-end to minimize prediction loss on normal vehicle behaviors, and is deployed to detect anomalies in (ab)normal scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' By combin- ing the heterogeneous vehicle and lane information, SABeR-VAE and its deterministic variant, SABeR-AE, improve abnormal AUPR by 18% and 25% respectively on the simulated MAAD highway dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Furthermore, we show that the learned Koopman operator in SABeR-VAE enforces interpretable structure in the variational latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' The results of our method indeed show that model- ing environmental factors is essential to detecting a diverse set of anomalies in deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' For code implementation, please visit https://sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='com/illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='edu/saber-vae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' KEYWORDS Anomaly Detection, Autonomous Vehicles, Unsupervised Learning, Human Behavior Modeling 1 INTRODUCTION Autonomous vehicles have the potential to realize a fast, safe, and labor-free transportation system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' A trustworthy self-driving vehi- cle should have the ability to operate reliably in normal situations and, more importantly, to perceive and react to anomalous driving scenarios (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=', skidding and wrong-way driving of surrounding human vehicles) promptly and robustly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' The detection of such ab- normal situations can help identify traffic accidents and dangerous driving behaviors of road participants, and thus provide high-level guidance for vehicle controllers to act safely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' denotes equal contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Deep-learning based Anomaly Detection (AD) algorithms have shown great promise in intelligent vehicle applications [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Many previous works utilize vehicle trajectories as an anomaly signal [2, 11, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' However, only a few vehicle trajectory datasets with suf- ficient anomaly labels exist for supervised learning methods [17, 40, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' To leverage the larger store of unlabeled driving data, re- searchers like Yao and Wiederer have employed unsupervised learn- ing methods [39, 41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Specifically, a neural network, which generally follows an encoder-decoder architecture for trajectory reconstruction or prediction, learns an underlying distribution of normal vehicle trajectories in the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' An anomaly is then detected whenever the trajectory is out of distribution and produces a large reconstruction or prediction error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' In interactive driving scenarios, Wiederer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' [39] showed that modeling interactions between agents can largely improve the reconstruction accuracy and subsequently the AD performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' However, such interaction- aware methods still ignore the effect of road structures on vehicle behaviors, and thus can miss abnormal scenarios like wrong-way driving trajectories that appear normal when environmental con- text is overlooked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Alongside performance accuracy, the decisions made by AD algorithms need to be interpretable to stakeholders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Deep neural networks are black boxes by nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' However, the decisions of deep networks impact various stakeholders such as policy makers and end users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Designing methods with interpretable features for stakeholders is a key challenge in AD, and the field of machine learning overall [5, 15, 34, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' In vehicle AD more specifically, in- terpretable algorithms need to account for the wide distribution of human drivers who act according to their own policies [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' For example, different drivers may choose to overtake other vehicles at different times and speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' To ensure interpretability, we use vari- ational autoencoder (VAE) to cluster useful features from similar behaviors together in a continuous and stochastic latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Our results indicate that vehicle trajectories transitioning to an abnor- mal state are explicitly encoded by interpretable transformations in the learned latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' In this paper, we present our novel unsupervised Structural Attention-based Recurrent Variational Autoencoder (SABeR-VAE) for highway vehicle anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Since contemporary vehi- cles have map information available to them regarding their nearby environment and lanes, we make use of the environmental infor- mation that prior works [26, 33, 39] have ignored to explicitly model the effect of lane structure on normal vehicle behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Specifically, we treat a highway scenario as a structured interac- tion graph where nodes represent vehicles and lane positions, and edges connect nearby vehicles, and permissible lanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Two separate arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='03634v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='RO] 9 Jan 2023 attention modules learn relations between vehicles (vehicle-vehicle self-attention) and legal permissible route trajectories (lane-vehicle attention) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' A sequence of embeddings from the vehicle- vehicle attention module are encoded into a Gaussian latent space to capture the randomness of vehicle trajectories with a recur- rent network, and cluster similar behaviors close together in an interpretable fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Our work is more computationally efficient than STGAE-KDE [39], which has a deterministic latent space and requires the expensive process of fitting a Kernel Density Estima- tor (KDE) to learn a meaningful distribution of normal behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' We then use a learned Koopman operator to propagate the cur- rent latent distributions forward in time conditioned on the useful lane embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' We show that the Koopman operator explicitly enforces interpretable transformations in the latent space that stan- dard autoencoders like STGAE are unable to incorporate, and is able to model the complex, non-linear dynamics of drivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Finally, we decode a sampled point from the propagated distribution to predict next states of vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' We train our method to predict trajecto- ries from normal scenarios in the Multi-Agent Anomaly Detection (MAAD) dataset [39], and compare accuracy metrics against linear, recurrent, and graph convolutional approaches on anomalous tra- jectories [31, 33, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Our SABeR-VAE improves AUPR-Abnormal and wrong-way driving detection over the STGAE-KDE by 18% and 35% respectively, and has an interpretable latent space over driving behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Our contributions can be summarized as follows: (1) We present a novel unsupervised variational approach for anomaly detection con- ditioned on structured lane information;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' (2) We quantitatively show that incorporating the structured information increases anomaly detection accuracy, compared with state-of-the-art baselines and ablations using the MAAD dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' (3) We show that the stochas- tic Koopman operator learns interpretable features of (ab)normal behaviors in the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Our paper is organized as follows: Section 2 discusses relevant works in the areas of structured modeling and anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Our problem formulation and methods are presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' We discuss results in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Finally, we conclude the paper and discuss future directions in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' 2 RELATED WORKS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='1 Exploiting Map Information The quality of information about an environment provided by High Definition maps (HD-maps) has dramatically increased and led to their ubiquitous use due to recent advancements in autonomous driving [27, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Currently, most state-of-the-art methods for vehicle trajectory prediction, motion forecasting, and anomaly detection, do not make effective use of the rich information provided in these HD-maps, and only rely on modeling the interactions between vehicles on the road [9, 24, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Hence, these methods ignore vital information such as the plausible movement of vehicles in the environment, which can be paramount in identifying anomalies such as wrong-way driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' However, trajectory prediction methods such as those proposed by Deo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' and Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' do exploit the information in these HD-maps and significantly outperform their counterparts [16, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' In proposing LaneGCN, Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' encode different types of inter- actions between agents on the road with lane information extracted from maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' They show that attention-based models can be used to encode interactions between vehicles and lanes, which are learned by constructing a graph representation of the road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' PGP, proposed by Deo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' , further produces scene-compliant trajecto- ries by sampling from a distribution of driving profiles conditioned on environment and vehicle interactions [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' We corroborate the usefulness of these vehicle and lane attention-based representa- tions and show that such embeddings do in fact provide meaningful insights in detecting highway vehicle anomalies in SABeR-VAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='2 Variational Autoencoders for Sequences Variational autoencoders (VAE) have been applied to sequential data combined with recurrent neural networks (RNN) in fields such as speech and image synthesis and autonomous driving [8, 12– 14, 18, 26, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' attempt to infer the traits of drivers from trajectories encoded in a variational latent space [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' However, only two classes of traits and a restricted set of defined trajectories were considered, while real drivers have a much wider range of behaviors on the road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Furthermore, they do not utilize map infor- mation in their learning process, which provide relevant context for traits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Conditional VAE formulations have also been found to be able to generate trajectories with different driving styles, but fail to consistently produce feasible trajectories without necessary environment context [21, 32, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Recurrent VAEs have also been applied to robot anomaly detection, but are limited by the simplicity of the single agent problem statement [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' These sequential gen- erative modeling approaches perform reasonably on their simple tasks, but fail to generate realistic samples from points in the latent space in more complex areas, due to the limitations of their RNN components [12, 13, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' To bridge the gap between complex human behaviors and the structured environment, and overcome the hurdles of the tempo- ral propagation in simplistic RNNs, we propose the use of a lane- conditioned Koopman Operator to model the temporal relations in the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' We were specifically inspired to use the Koopman operator to propagate the latent space due to its capability to model the dynamics of complex, non-linear data, including fluid dynamics, battery properties, and control tasks [1, 3, 4, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='3 Anomaly Detection Anomaly detection is well studied in diverse research areas and application domains [10, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' In robotics and automated vehicles, AD has been used to detect abnormal patterns such as robot fail- ures [23, 30] and dangerous driving scenarios [39, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' propose a long short-term memory based variational autoencoder (LSTM-VAE) to reconstruct the expected distribution of robot sensor signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' A reconstruction-based anomaly score is then used for anomaly detection [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Furthermore, Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' adopt an attention mechanism to fuse multi-sensor signals for robust anomaly detection in uncertain environments [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' While these approaches focus on AD for single agent problem statements, our highway scenarios consist of complex multi-agent social interac- tions among vehicles, and need to be modeled as such.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' In the domain of traffic anomaly detection using multi-agent tra- jectories, the most similar work to ours is the spatio-temporal graph auto-encoder (STGAE) proposed with the MAAD dataset [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' The architecture follows an encoder-decoder structure to reconstruct vehicle trajectories, where vehicle interactions and motions are con- sidered using spatial graph convolution and temporal convolution layers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' The method has been shown to be effective by modeling interactions among vehicles to detect anomalous ma- neuvers in traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' However, such a network ignores the constraints imposed by road structures on vehicle trajectories and the variabil- ity of human driver behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' In this work, we explicitly model both vehicle-to-vehicle interactions and lane-to-vehicle interac- tions to boost performance, and use an interpretable variational architecture to learn a continuous distribution over behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' 3 METHODOLOGY In this section, we first introduce our problem formulation of anom- aly detection from vehicle trajectories, and then explain our pro- posed SABeR-VAE framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='1 Problem Formulation Suppose 𝑛𝑡 ∈ [1, 𝑁] vehicles are on a road segment at any time 𝑡, and each vehicle takes an acceleration and steering action every timestep according to unknown policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Let 𝑐 (𝑖) 𝑡 = � 𝑥 (𝑖) 𝑡 ,𝑦(𝑖) 𝑡 � be the 2𝐷 coordinates of the 𝑖th vehicle at time 𝑡, where 𝑖 ∈ [1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=',𝑛𝑡].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Each vehicle also has a set of corresponding permissible lane po- sitions in front, to the left, and to the right of the vehicle, pro- vided in the form of a discretized map representation shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' At every timestep, each vehicle’s discretized position within the map is used to identify their corresponding front, left, and right lane nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' We define a tuple 𝑙 (𝑖) 𝑡 = (front, left, right)(𝑖) 𝑡 of three 2𝐷 coordinates containing the lane information for ve- hicle 𝑖 at time 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Altogether, the observed information of each vehicle at any time is the relative displacement of coordinates 𝑜 (𝑖) 𝑡 = � 𝑐 (𝑖) 𝑡 − 𝑐 (𝑖) 𝑡−1,𝑙 (𝑖) 𝑡 − 𝑐 (𝑖) 𝑡 � = � 𝑋 (𝑖) 𝑡 , 𝐿(𝑖) 𝑡 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' A trajectory of length 𝑇 for any vehicle is represented as � 𝑜 (𝑖) 0 ,𝑜 (𝑖) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=',𝑜 (𝑖) 𝑇−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' We assume that any vehicle 𝐴 that is within a distance 𝑑 to another vehicle 𝐵 at time 𝑡 can accurately detect and track the relative coordinates 𝑐 (𝐵) 𝑡 −𝑐 (𝐴) 𝑡 = 𝑅(𝐴,𝐵) 𝑡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' The purple arrow between the green and blue vehicle in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' 1 represents this vehicle interaction type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' For the 𝑖-th car, the number of observable cars is 𝑚𝑖 𝑡 ∈ [0,𝐶].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Given all vehicle trajectories in a scene, our goal is to provide an anomaly score AS𝑡 ∈ R≥0 for each time 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='2 Architecture Figure 2 contains the complete architecture diagram of SABeR-VAE, which we discuss in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='1 Vehicle-Vehicle Self-Attention Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Our goal is to learn a representation of spatial interactions among vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Rather than using convolutional methods like those in prior works [25, 39], we encode the positions of vehicles on the road at each time with scaled dot-product multi-head self-attention, which allows each head to learn different features of the data [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Figure 1: Map discretization and interaction edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' We model the vehicle AD problem as an interaction graph with vehicle and lane nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' A continuous map of the road is discretized into blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Directed lane edges between lane nodes encode permissible routes for vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' The red vehicle has a directed edge toward the lane nodes in front and to its left because the driver can legally continue forward or merge left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Conversely, the green vehicle has no edge connecting to a left lane node since it cannot cross the road divider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Vehicle edges, shown in purple, exist for vehicles that are close enough to interact with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' We embed the coordinates of each car 𝑋𝑡 with a multi-layer perceptron (MLP) 𝑓 𝑉𝑉 𝑄 to obtain queries 𝑄𝑉𝑉 𝑡 : 𝑓 𝑉𝑉 𝑄 (𝑋𝑡) = 𝑄𝑉𝑉 𝑡 ∈ R1×𝐷, (1) where 𝐷 is the attention size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Let R𝑡 = � 𝑅(𝑖,1) 𝑡 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=', 𝑅(𝑖,𝑚) 𝑡 �⊺ be the displacements of all neighboring cars for the 𝑖-th car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' We use two other MLPs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' 𝑓 𝑉𝑉 𝐾 and 𝑓 𝑉𝑉 𝑉 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' to embed R𝑡 to obtain keys 𝐾𝑉𝑉 𝑡 and values 𝑉𝑉𝑉 𝑡 respectively: 𝑓 𝑉𝑉 𝐾 (R𝑡) = 𝐾𝑉𝑉 𝑡 ∈ R𝑚×𝐷 𝑓 𝑉𝑉 𝑉 (R𝑡) = 𝑉𝑉𝑉 𝑡 ∈ R𝑚×𝐷 (2) The final encoding of each vehicle position from this self-attention layer for time 𝑡 is calculated as: softmax ��� � 𝑄𝑉𝑉 𝑡 � 𝐾𝑉𝑉 𝑡 �⊺ √ 𝐷 ��� � 𝑉𝑉𝑉 𝑡 = 𝑝𝑉𝑉 𝑡 ∈ R1×𝐷 (3) Nonexistent or unobserved vehicles further than a distance 𝑑 cannot be allowed to contribute to the attention score of other vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Thus, we use a mask to set the score contributed from unobserved vehicles to −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='2 Lane-Vehicle Attention Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' We use available map and lane information in a separate lane-vehicle attention layer to model legal maneuvers in structured environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Similar to vehicle-vehicle attention, the query 𝑄𝐿𝑉 𝑡 is an embedding of 𝑋𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Lane information of each vehicle 𝐿𝑡 is used to produce keys 𝐾𝐿𝑉 𝑡 and values 𝑉 𝐿𝑉 𝑡 : 𝑓 𝐿𝑉 𝑄 (𝑋𝑡) = 𝑄𝐿𝑉 𝑡 ∈ R1×𝐷 𝑓 𝐿𝑉 𝐾 (𝐿𝑡) = 𝐾𝐿𝑉 𝑡 ∈ R3×𝐷 𝑓 𝐿𝑉 𝑉 (𝐿𝑡) = 𝑉 𝐿𝑉 𝑡 ∈ R3×𝐷 (4) The lane-conditioned vehicle embeddings are calculated as: softmax ��� � 𝑄𝐿𝑉 𝑡 � 𝐾𝐿𝑉 𝑡 �⊺ √ 𝐷 ��� � 𝑉 𝐿𝑉 𝑡 = 𝑝𝐿𝑉 𝑡 ∈ R1×𝐷 (5) Note that all three lane nodes may not always be permissible to a vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' For example, a car in the left-most lane of a road is unable to legally turn left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' As such, we mask out impermissible lane nodes like in the self-attention layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='3 Recurrent Encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' A gated recurrent unit (GRU) network encodes the sequence of self-attention features for each vehicle � 𝑝𝑉𝑉 0 , 𝑝𝑉𝑉 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=', 𝑝𝑉𝑉 𝑇−1 � into a sequence of Gaussian distributions in the latent space with temporal correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Thus, the latent space captures the stochastic nature of human behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Specifically, af- ter embedding the vehicle-vehicle attention feature with a network 𝑓𝑒, we pass the embedding through the GRU to get the hidden state of each vehicle for the current timestep: ℎ𝑡 = GRU � ℎ𝑡−1, 𝑓𝑒 � 𝑝𝑉𝑉 𝑡 �� (6) Mean and variance neural networks 𝑓𝜇 and 𝑓𝜎 produce parame- ters for a latent normal distribution of dimension 𝑗 conditioned on a vehicle’s hidden state at any time: 𝜇𝑉𝑉 𝑡 = 𝑓𝜇 (ℎ𝑡) , 𝜎𝑉𝑉 𝑡 = 𝑓𝜎 (ℎ𝑡) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' (7) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='4 Latent Propagation with Koopman Operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' While the GRU encoder encodes vehicle behaviors into the latent space solely conditioned on past and current vehicle interactions, we need a method to propagate the latent distributions in time to predict the future states of vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' To this end, we learn a stochastic Koopman operator conditioned on the lane-vehicle embeddings to perform this task, like Balakrishnan and Upadhyay [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' In Koopman operator theory, a discrete time system evolves according to potentially nonlinear dynamics𝑥𝑡+1 = 𝐹 (𝑥𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' However, a function 𝑔 maps the state 𝑥𝑡 into a space where dynamics evolve linearly with the Koopman operator K [4]: K𝑔 (𝑥𝑡) = 𝑔 (𝐹 (𝑥𝑡)) = 𝑔 (𝑥𝑡+1) (8) Similarly, the inverse of function 𝑔 translates an observable of 𝑥 back into the original dynamics space [4]: 𝑔−1 (K𝑔 (𝑥𝑡)) = 𝑥𝑡+1 (9) In our case, function 𝑔 is represented by the GRU encoder and neural networks 𝑓𝜇 and 𝑓𝜎, which altogether, produce a latent dis- tribution N (𝜇𝑡, 𝜎𝑡) conditioned on inter-vehicle embeddings 𝑝𝑉𝑉 𝑡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Like the Stochastic Adversarial Koopman (SAK) model [4], we use auxiliary neural networks 𝑓 𝜇 aux and 𝑓 𝜎 aux to predict tridiagonal Koopman matrices 𝐾𝜇,𝑡 and 𝐾𝜎,𝑡, rather than solving for their closed form solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' The outputs of 𝑓 𝜇 aux and 𝑓 𝜎 aux are conditioned on the current latent distributions N � 𝜇𝑉𝑉 𝑡 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' 𝜎𝑉𝑉 𝑡 � and the lane features 𝑝𝐿𝑉 𝑡 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' so that the Koopman operators capture legal route maneuvers in the latent space propagation: 𝐾𝜇,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='𝑡 = 𝑓 𝜇 aux � 𝜇𝑉𝑉 𝑡 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' 𝑝𝐿𝑉 𝑡 � 𝐾𝜎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='𝑡 = 𝑓 𝜎 aux � 𝜎𝑉𝑉 𝑡 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' 𝑝𝐿𝑉 𝑡 � (10) The predicted Koopman matrices are applied to the inter-vehicle distributions to linearly propagate the mean and variance of the latent distributions forward in time: 𝜇𝐿𝑉 𝑡+1 = 𝐾𝜇,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='𝑡 𝜇𝑉𝑉 𝑡 + 𝜇𝑉𝑉 𝑡 (11a) 𝜎𝐿𝑉 𝑡+1 = 𝐾𝜎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='𝑡 𝜎𝑉𝑉 𝑡 + 𝜎𝑉𝑉 𝑡 (11b) Intuitively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' we can interpret the GRU encoder as predicting a distribution of vehicle behaviors from their current trajectories,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' and the Koopman operator propagates to a one-step future distribution of behaviors based on lane information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='5 The Decoder Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' At this point, we have two sets of distributions in the 𝑗-dimensional latent space for the current states and future predictions of vehicles at each time: N � 𝜇𝑉𝑉 𝑡 , 𝜎𝑉𝑉 𝑡 � and N � 𝜇𝐿𝑉 𝑡+1, 𝜎𝐿𝑉 𝑡+1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' We utilize the reparameterization trick to sample a point from each of the distributions: 𝜖𝑉𝑉 𝑡 ∼ N (0, 1) 𝑧𝑉𝑉 𝑡 = 𝜇𝑉𝑉 𝑡 + 𝜖𝑉𝑉 𝑡 𝜎𝑉𝑉 𝑡 𝜖𝐿𝑉 𝑡+1 ∼ N (0, 1) 𝑧𝐿𝑉 𝑡+1 = 𝜇𝐿𝑉 𝑡+1 + 𝜖𝐿𝑉 𝑡+1𝜎𝐿𝑉 𝑡+1 (12) A multi-layer perceptron 𝑓dec is used as a decoder network, sim- ilar to 𝑔−1 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' 9, to predict a vehicle coordinate change from the sampled latent points: ˆ𝑋𝑡 = 𝑓dec � 𝑧𝑉𝑉 𝑡 � , ˆ𝑋𝑡+1 = 𝑓dec � 𝑧𝐿𝑉 𝑡+1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' (13) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='3 Training and Evaluation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='1 End-to-End Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' To fairly compare our method with prior convolutional approaches, we utilize a similar sliding window training approach performed by Wiederer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Specifically, whole trajectories of length 𝑇 are divided into small overlapping segments, or windows, of constant length 𝑇 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' In our training objective, we minimize the current reconstruc- tion loss and one-step future prediction loss of the model by split- ting our input ground truth trajectories into current states 𝑋 − = 𝑋0:𝑇 ′−2 and one-step future states 𝑋 + = 𝑋1:𝑇 ′−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' We also regularize the current distributions N � 𝜇𝑉𝑉 𝑡 , 𝜎𝑉𝑉 𝑡 � and propagated distribu- tions N � 𝜇𝐿𝑉 𝑡+1, 𝜎𝐿𝑉 𝑡+1 � to follow a standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Let 𝐷KL(𝜇, 𝜎) be the KL divergence between any Gaussian distribution N (𝜇, 𝜎) and the standard normal distribution N (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Then the regularized prediction and reconstruction losses are: Lpred = 𝛽1 · 𝐷KL � 𝜇𝐿𝑉 , 𝜎𝐿𝑉 � + ∥𝑋 + − ˆ𝑋 +∥2 Lrecon = 𝛽2 · 𝐷KL � 𝜇𝑉𝑉 , 𝜎𝑉𝑉 � + ∥𝑋 − − ˆ𝑋 −∥2, (14) where 𝛽1 and 𝛽2 are tunable weights applied to the regularization of the latent distributions similar to beta-VAE [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' The overall objective we optimize is: L = Lpred + Lrecon (15) GRU Encoder Decoder Attention Modules Koopman Propagation Vehicle Self-Attention GRU Lane-Vehicle Attention Figure 2: SABeR-VAE architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' The SABeR-VAE architecture attempts to predict the one-step future states of vehicles conditioned on current vehicle positions and structural lane information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Vehicle interactions are modeled by the self-attention module while permissible routes are encoded by the lane-vehicle attention module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' A GRU encoder processes the self-attention embeddings through time to produce a latent distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Then the Koopman operator conditioned on the lane embeddings propagates the latent distributions forward, which finally get decoded to predict next states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' The 𝑓dec network shares parameters for reconstruction and prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' We again mask out coordinates of unobserved vehicles so they do not contribute to the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' While SAK [4] applies maximum mean discrepancy (MMD) to synchronize the current and propagated distributions of their Koop- man model to any general distribution, we explicitly encourage the latent space distributions to follow the standard gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' We leave experimentation of various Koopman synchronization methods for the anomaly detection task as a future direction of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='2 Anomaly Detection Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' At test time, we follow the same sliding window practice as performed in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' First, we calculate the one-step future prediction loss Lpred,𝑡+1 = ∥𝑋 (𝑖) 𝑡+1 − ˆ𝑋 (𝑖) 𝑡+1∥2 for every vehicle at each timestep, within every window of a complete trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Then, we average the prediction loss of overlapping timesteps among all windows in the sequence, for each vehicle separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Suppose W (𝑡,𝑖) is the set of all windows in the complete trajec- tory containing time 𝑡 where vehicle 𝑖 is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' The averaged prediction error for car 𝑖 at 𝑡 is: ¯L(𝑖) pred,𝑡 = � 𝑤∈W (𝑡,𝑖) Lpred,𝑤(𝑖) 𝑡 |W (𝑡,𝑖) | , (16) where Lpred,𝑤 (𝑖) 𝑡 is the prediction error of time 𝑡 for vehicle 𝑖 in window 𝑤 of the set W (𝑡,𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Finally, we choose the anomaly score AS to be the maximum averaged prediction loss over all vehicles at a given timestep 𝑡: AS𝑡 = max 𝑖=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=',𝑛𝑡 ¯L(𝑖) pred,𝑡 (17) 4 EXPERIMENTAL SETUP AND RESULTS In this section, we first describe the MAAD dataset on which we performed experiments and detail baselines and ablations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' We also present our quantitative results and latent space interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='1 MAAD Dataset and Augmentation The MAAD dataset [39] consists of 2𝐷 trajectories of two vehicles on a straight two-lane highway with a divider separating the two Figure 3: Trajectories and SABeR-VAE anomaly scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' (top row) Examples of a normal overtaking (a,) abnormal off-road driv- ing (b,) and wrong-way driving (c) scenarios in the MAAD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' White arrows point toward direction of normal traffic flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' (bot- tom row) Predicted anomaly score curves for each scenario above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Colors of lines within the curves show the ground-truth labels of normal (green,) ignored (yellow,) & abnormal (red) timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' possible directions, as visualized in the top row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' There are 80 training and 66 test-split trajectories ranging from a length of 25 to 127 timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' These dynamic length trajectories are subsampled to produce approximately 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='3K training and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='1K testing windows of constant length𝑇 ′ = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' As the original dataset sequences did not come with map or lane details, we augmented the data to include this information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Specifically, we discretized the highway in the 𝑥-coordinate direction into blocks of length five meters as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' 1, and stored the 2𝐷 coordinates of the front, left, and right blocks for each vehicle at every timestep in all trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' All the training sequences consist of normal vehicle behaviors like driving side-by-side, overtaking, following, and driving in opposite directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' In contrast, the test-split contains both normal and 11 anomalous behavior classes like aggressive overtaking, pushing aside, tailgating, off-road, and wrong-way driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='2 Baseline Methods We compare against baselines implemented by Wiederer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' that depend on reconstruction loss rather than future prediction er- ror [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' (1) The Constant Velocity Model (CVM) is a standard baseline that predicts the next states of vehicles assuming each 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='0 : 10 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='8- 30 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='6- 201 :1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='0- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='0 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='4 - : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='5 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='5 / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='0 1 i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='0 20 40 60 80 10 15 20 25 5 10 15 20 25 0 Timestep Timestep Timestep (a) (b) (c)Table 1: Accuracy results of baselines, ablations, and SABeR methods over ten runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Method Detection Type AUROC ↑ AUPR-Abnormal ↑ AUPR-Normal ↑ FPR @ 95%-TPR ↓ CVM Reconstruction Loss 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='0 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='0 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='0 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='0 RAE-Recon* Reconstruction Loss 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='7 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='0 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='1 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='3 STGAE* Reconstruction Loss 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='8 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='1 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='8 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='2 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='3 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='8 ± 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='8 STGAE-KDE* One Class 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='7 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='2 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='7 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='5 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='0 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='9 RAE-Pred Prediction Loss 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='5 ± 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='3 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='5 ± 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='4 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='9 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='4 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='8 ± 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='6 VV-RAE Prediction Loss 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='2 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='9 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='9 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='5 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='4 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='1 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='3 SABeR-AE Prediction Loss 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='4 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='5 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='2 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='1 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='0 SABeR-VAE Prediction Loss 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='5 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='5 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='9 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='5 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='7 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='6 These results are presented in [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' vehicle travels at the same velocity as the last timestep, without modeling any inter-vehicle relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' (2) Recurrent Autoencoder (RAE-Recon) uses an LSTM network to encode and decode a se- quence of coordinates from an unregularized latent space, attempt- ing to minimize reconstruction loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' (3) Spatio-Temporal Graph Autoencoder (STGAE) is a convolutional method that models inter- vehicle behaviors, and outputs parameters for a bi-variate distri- bution describing the estimated state of the reconstructed pose of vehicles, and is trained to maximize the log-likelihood of the estimated probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Finally, (4) the STGAE-KDE baseline fits a Kernel Density Estimator (KDE) to the unregularized latent space of a trained STGAE model to predict the one-class probability of a set of points originating from a normal behavior window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Unlike the STGAE-KDE, our SABeR-VAE does not require an expensive KDE fitting procedure since our anomaly score solely relies on prediction error, and we still model inter-vehicle relations unlike CVM and RAE-Recon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' We additionally train ablation models with future prediction loss to identify the impact of different components in our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' We train (5) an unregularized Recurrent Autoencoder (RAE-Pred,) using a standard deterministic MLP to propagate latent points for- ward in time, without explicitly modeling any inter-vehicle be- haviors, like the RAE-Recon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' (6) A Recurrent Autoencoder with a vehicle-vehicle Self-Attention module (VV-RAE) minimizes predic- tion error while modeling inter-vehicle relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' We also train (7) a deterministic variant of SABeR-VAE without a regularized latent space, SABeR-AE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' SABeR-AE utilizes both vehicle self-attention and lane-vehicle attention like SABeR-VAE, but encodes trajectories into an unregularized (uninterpretable) latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='3 Quantitative Evaluation Metrics We quantitatively evaluate the effectiveness of models on the MAAD dataset using four metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' (1) Area Under Receiver-Operating Characteristic curve (AUROC) is calculated by plotting the False- Positive Rate (FPR) and True-Positive Rate (TPR) of a model over several decision thresholds, and computing the area under the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' A model with greater AUROC performs better, and a perfect classifier has an AUROC of 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Though, AUROC is skewed in datasets where there are very few positive labels, like in the field of outlier identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' As such, FPR may be misleadingly low, producing an optimistic AUROC value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' We compute (2) the Area Under Precision-Recall Curve (AUPR) with the anomalous points being the positive class (AUPR-Abnormal) and (3) with normal points being positive (AUPR-Normal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' The AUPR metric adjusts for skewed dataset distributions, and we evaluate model effective- ness of classifying anomalies, and not mis-classifying normal points with AUPR-Abnormal and AUPR-Normal respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Finally, we use (4) FPR @ 95%-TPR to check the rate of mis-labeling normal points when TPR is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='4 Training Several runs of SABeR-VAE were trained with learning rates rang- ing from 5𝑒 − 5 to 1𝑒 − 3, batch sizes from 32 to 128, KL-Divergence 𝛽1 = 𝛽2 weighting from 1𝑒 − 6 to 1𝑒 − 3, latent dimension size from 2 to 64, attention embedding sizes 32 and 64, and an inter-vehicle distance 𝑑 of 45 meters for the vehicle-vehicle attention mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' The RAE-Pred, VV-RAE, and SABeR-AE models were similarly trained, but without 𝛽, attention embedding sizes, and 𝑑 where irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Multi-head attention modules were instantiated with 8 heads each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Every model was trained for 500 epochs on the training split with a Tesla V100 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Like Wiederer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' , we calculate metrics for each hyperparameter choice on a 20% validation split of the whole test data, and choose to evaluate the best set of hyperparameters for each respective method on the complete test split [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='5 Accuracy Results Table 1 holds accuracy results of baselines, ablations, and the SABeR methods on the test split of the MAAD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Each method, be- sides CVM, was trained ten separate times with the same hyperpa- rameters, and we report the average and standard deviation of each methods’ results over the ten runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Figure 4 plots the ROC curves for each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Amongst baselines, the simple CVM model already performs well as an anomaly detector since its AUROC is only 3% less than that of the STGAE-KDE method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' CVM also has no variation of results since it is a deterministic model that is not trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' The LSTM-based RAE-Recon model is unable to effectively distinguish between anomalies and normal scenarios using reconstruction loss, since it does not model vehicle or lane information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' While recurrent models encode current timestep features based solely on previous timesteps, temporal convolution methods extract information from the whole trajectory, which helps to predict a more accurate re- construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Thus, the convolutional STGAE method drastically improves AUROC and AUPR scores over RAE-Recon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' However, once we incorporate a latent propagation network and predict future timesteps, the RAE-Pred ablation increases AUROC over RAE-Recon by 29% and even outperforms STGAE in the AUPR- Abnormal metric, without even modeling inter-vehicle behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' This result hints to the idea that recurrent networks learn to model normal behaviors more accurately with future prediction error, than reconstruction error of observed timesteps, which assists in AD performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Furthermore, recurrent methods are capable of reaching the same performance as convolutional methods, while relying on fewer trajectory data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Still, RAE-Pred is shown to be unstable as it produces a high variance in results over the ten trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' This variance was caused by two of the ten runs achieving only 45% AUROC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' STGAE also has the highest variance in results among baselines since it is a stochastic method reconstruct- ing a distribution over states, rather than the deterministic CVM and RAE-Recon approaches, but is more stable than RAE-Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Surprisingly, we see that adding a vehicle-vehicle self-attention layer to RAE-Pred to produce the VV-RAE model actually hinders performance, and gives results similar to RAE-Recon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Effectively, the vehicle-vehicle self-attention layer did not learn useful features for the future prediction task, and confused the model generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' This outcome could be a result of a low complexity neural network or a potentially poor choice for masking distance 𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' The one-class prediction model STGAE-KDE fits a KDE to the latent space of the STGAE under the assumption that normal be- havior latent points will be clustered together in the autoencoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' As such, this one-class classification approach improves detection rates and training stability over the STGAE such that it outperforms other baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' However, the fitting process of a KDE to a large dimensional space is a computationally complex and constrictive part the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' With gaussian regularization of the latent space, our SABeR-VAE clusters similar behaviors together and learns a latent distribution without fitting a KDE, which we discuss in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Finally, SABeR-AE and SABeR-VAE incorporate a lane-vehicle attention module to capture the effect of the structure of the envi- ronment on normal behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' We see that SABeR-AE outperforms all methods in AUROC and AUPR-Abnormal with low variance, showcasing the importance of modeling environment structure in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' SABeR-VAE performs slightly better than STGAE-KDE in AUROC, and significantly increases the AUPR-Abnormal score by 18%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' However, the stochasticity of the SABeR-VAE method hinders its reproducibility, and AUROC scores ranged from 84% to 89% over the ten training runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' SABeR-VAE further decreases the average FPR @ 95%-TPR of SABeR-AE by 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' STGAE-KDE and the two SABeR approaches have similar AUPR-Normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' We present examples of SABeR-VAE scoring anomalous timesteps in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' There, a normal overtaking maneuver was scored very low during the whole trajectory, whereas going off-road or driv- ing in the wrong direction were scored high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' We can also see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='b that timesteps where vehicles are acting normally prior to erratic behavior are still correctly scored low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Table 2 holds AU- ROC by anomaly type for CVM, STGAE-KDE, and SABeR methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Figure 4: ROC curves of tested methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Table 2: AUROC (↑) of methods by anomaly type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Anomaly Type CVM STGAE-KDE* SABeR-AE SABeR-VAE Reeving 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='7 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='6 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='9 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='5 Pushing Aside 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='5 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='4 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='6 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='3 Right Spreading 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='7 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='2 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='7 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='2 Left Spreading 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='9 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='6 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='2 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='9 Off-Road 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='7 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='2 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='2 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='7 Skidding 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='9 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='7 ∼100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='0 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='8 Wrong-way 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='2 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='2 ∼100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='0 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='3 These results are presented in [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' SABeR-VAE improves wrong-way driving AD by 35% over STGAE- KDE, while performing comparably in other metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' The complete version of Table 2 is provided in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='6 Latent Space Interpretation SABeR-VAE is a variational model with a continuous latent space such that observations with similar learned characteristics are clus- tered closer together in the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='a, we plot the test-split latent space of a SABeR-VAE model trained with a latent dimension size of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Points are clustered into three distinct regions in the latent space, which we will refer to as “bottom,” “middle,” and “top” clusters respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' We see from sampled trajectories that the bottom and middle clusters encode vehicles that travel toward the −𝑥 (left) direction in either of the top two lanes of the highway, while the top cluster encodes vehicles traveling to the right in the bottom two lanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' For example, points 1P, 2P, 3B, 4B, and 5B corre- spond to blue (B) and pink (P) cars that travel to the left in the top lanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Similarly, the trajectory of the pink car driving to the right in window 4 is encoded to point 4P in the top cluster of the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Vehicles that are also physically close and interacting with each other are encoded closely in the latent space, as shown with Receiver Operating Characteristic Curves 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='8 True Positive Rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='4 CVM RAE-Recon STGAE STGAE+KDE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='2 RAE-Pred VV-RAE SABeR-AE SABeR-VAE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='0 False Positive RateFigure 5: Koopman propagated latent space and corresponding trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' (a) We encode every window trajectory in the test-split of the MAAD dataset and plot the 2𝐷 sampled latent positions of the final timestep of the windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Blue points correspond to ground truth normal windows while orange are abnormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' (1-5) Five scenario windows are encoded into the latent space, and are explicitly annotated in (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=') Each of the five windows has two latent points for the pink and blue vehicles respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=', annotations “1B” & “1P” are the latent points of the blue and pink vehicles in road trajectory window 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=') Within the five trajectory windows, solid lines are the ground truth trajectory of the vehicle while open circles are predicted by SABeR-VAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' White arrows denote direction of traffic flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' (b & d) The prediction error curves for trajectories 4 and 5 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' (c & e) The trajectory of latent points for vehicle windows 4 and 5 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' A heat map of the original latent space is plotted in orange in the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Blue and pink circles are the latent trajectories of the blue and pink car through time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' The largest circles encode the initial timestep of the window, and they decrease in size as the window progresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' latent points corresponding to windows 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' The middle cluster embeds anomalous scenarios from the top lanes where vehicles are close enough to interact with each other, like window 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Furthermore, anomalous, non-interactive trajectories that are poorly predicted are encoded to the outskirts of the primary cluster distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' For example, the pink cars in trajectories 3 and 5 are driving in the wrong direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' These trajectories have high prediction error as visualized by the little overlap between the pre- dicted open circle positions and ground truth trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' As such, those poorly predicted points are encoded in the spaces between the bottom and middle, and middle and top clusters respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' In contrast, trajectories 1 and 4 have low loss and are encoded to positions within the primary clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Thus the latent space has learned a correspondence between permissible lane routings and expected vehicle behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Finally, we visualize the transformation of the latent space over time within one trajectory window to show the interpretable im- pact of the learned Koopman operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Figures 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='c and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='e show the transformation of the latent space as time progresses in trajectory windows 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' We can see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='c that the blue and pink latent trajectories stay in the bottom and top clusters respectively, since the vehicles follow the correct direction on the road throughout window 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Conversely, we see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='e that the pink latent trajec- tory begins in the top cluster since the pink vehicle in trajectory 5 is in one of the bottom two lanes on the road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' But, at timestep 6, the pink car crosses the road divider into the wrong direction lane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Thus, we see a jump in the pink car’s latent trajectory in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='e from the top cluster to the bottom and middle clusters that correspond to the top two lanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' At the same time, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='d has a spike in the prediction loss of the pink vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' For the remainder of the trajectory window, the pink car oscillates drastically in the latent space around the middle cluster, since the model expects the vehicle to be traveling to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Note, even though the pink vehicle in trajectory 5 is acting abnormally, this does not effect the latent trajectory of the blue vehicle in the same window, since the vehicles are not close enough to impact each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Overall, the Koopman operator explicitly models this transition from normal to anomalous states in the latent space, in an interpretable manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' 5 CONCLUSIONS AND FUTURE WORK In this paper, we propose a novel framework for anomaly detection with an unsupervised recurrent VAE network conditioned on struc- tured environment information and vehicle interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' We show that modeling this structured information is imperative to having high accuracy over a wide range of anomaly types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Furthermore, we have identified that our stochastic Koopman operator enforces interpretable propagation of the latent space using the lane embed- dings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' Several areas for possible future study include: (1) applying SABeR methods to real-world trajectory datasets containing anom- alies, (2) combining the anomaly detector with a vehicle controller, (3) using raw sensor data rather than processed ground truth po- sitions of vehicles for anomaly detection, and (4) performing user studies on preferences for notification and handling of anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' 1 Annotated Propagated Latent Space normal abnormal 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='0 40 3P 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='8 30/ 2P 2B右 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='8* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='4 5P 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='5 口 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content='0 2 10 12 10 12 Timestep Timestep H 5B 1B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} +page_content=' t=o C 1P t=6 t= 12 3B 3 4B (c) (e)ACKNOWLEDGMENTS We thank Julian Wiederer for providing access to the MAAD dataset and assisting with baseline code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfEQa1/content/2301.03634v1.pdf'} 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index 0000000000000000000000000000000000000000..70b0ee8574da219d6ce9c505d4fbb7d77305b414 --- /dev/null +++ b/cdAyT4oBgHgl3EQfwfli/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8f7ed87f99906e366f3d94d548f6ed440689bc75dbfec85cda27db832b9b2533 +size 54381 diff --git a/cdE2T4oBgHgl3EQfagdb/content/tmp_files/2301.03875v1.pdf.txt b/cdE2T4oBgHgl3EQfagdb/content/tmp_files/2301.03875v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..569973040373870d0e0c034bd29dc6fd40b8cf58 --- /dev/null +++ b/cdE2T4oBgHgl3EQfagdb/content/tmp_files/2301.03875v1.pdf.txt @@ -0,0 +1,1607 @@ +arXiv:2301.03875v1 [math.PR] 10 Jan 2023 +The number of ends in the uniform spanning tree +for recurrent unimodular random graphs. +Diederik van Engelenburg +Tom Hutchcroft +January 11, 2023 +Abstract +We prove that if a unimodular random rooted graph is recurrent, the number of ends +of its uniform spanning tree is almost surely equal to the number of ends of the graph. +Together with previous results in the transient case, this completely resolves the problem +of the number of ends of wired uniform spanning forest components in unimodular random +rooted graphs and confirms a conjecture of Aldous and Lyons (2006). +1 +Introduction +The uniform spanning tree of a finite connected graph G is defined by picking uniformly +at random a connected subgraph of G containing all vertices but no cycles. +To go from +finite to infinite graphs, it is possible to exhaust G by finite subgraphs and take weak limits +with appropriate boundary conditions. For two natural such choices of boundary conditions, +known as free and wired boundary conditions, Pemantle [24] proved that these infinite- +volume limits are always well-defined independently of the choice of exhaustion, and that the +choice of boundary conditions also does not affect the limit obtained when G = Zd. Since +connectivity of a subgraph is not a closed condition, these weak limits might be supported on +configurations that are forests rather than trees, and indeed Pemantle proved for Zd that the +limit is connected if and only if d ≤ 4. For a general infinite, connected, locally finite graph +the infinite-volume limit of the UST with free boundary conditions is called the free uniform +spanning forest (FUSF) and the infinite volume limit with wired boundary conditions is +called the wired uniform spanning forest (WUSF); when the two limits are the same we +refer to them simply as the uniform spanning forest (USF). In their highly influential work [6], +Benjamini, Lyons, Peres and Schramm resolved the connectivity question for the WUSF in +large generality: the wired uniform spanning tree is a single tree if and only if two random +walks intersect infinitely often. The connectivity of the FUSF appears to be a much more +subtle question and, outside of the case that the two forests are the same, is understood only +in a few examples [3,18,25,27]. For recurrent graphs, which are the main topic of this paper, +1 + +the infinite-volume limit of the UST is always defined independently of boundary conditions +and a.s. connected [6, Proposition 5.6], so that we can unambiguously refer to the uniform +spanning tree (UST) of an infinite, connected, locally finite, recurrent graph G. +After connectivity, the next most basic topological property of the USF is the number of +ends its components have. Here, we say that a graph has at least m ends whenever there exists +some finite set of vertices W such that G\W has at least m infinite connected components. The +graph is said to be m-ended if at has at least m but not m+1 ends. Understanding the number +of ends of the USF turns out to be rather more difficult than connectivity, with a significant +literature now devoted to the problem. For Cayley graphs, it follows from abstract principles +[3, Section 3.4] that every component has 1, 2, or infinitely many ends almost surely, and for +amenable Cayley graphs such as Zd (for which the WUSF and FUSF always coincide) is follows +by a Burton-Keane [10] type argument that every component has either one or two ends almost +surely; see [22, Chapter 10] for detailed background. For the wired uniform spanning forest on +transitive graphs, a complete solution to the problem was given by Benjamini, Lyons, Peres, +and Schramm [6] and Lyons, Morris, and Schramm [21], who proved that every component +of the WUSF of an infinite transitive graph is one-ended almost surely unless the graph in +question is rough-isometric to Z. Before going forward, let us emphasize that the recurrent +case of this result [6, Theorem 10.6] is established using a completely different argument to the +transient case, with the tools available for handling the two cases being largely disjoint. +Beyond the transitive setting, various works have established mild conditions under which +every component of the WUSF is one-ended almost surely, applying in particular to planar +graphs with bounded face degrees [18] and graphs satisfying isoperimetric conditions only +very slightly stronger than transience [15,21]. These proofs are quantitative, and recent works +studying critical exponents for the USF of Zd with d ≥ 3 [2,16,19] and Galton-Watson trees [20] +can be thought of as a direct continuation of the same line of research. +In parallel to this deterministic theory, Aldous and Lyons [1] observed that the methods of +[6] also apply to prove that the WUSF has one-ended components on any transient unimodular +random rooted graph of bounded degree, and the second author later gave new proofs of this +result with different methods that removed the bounded degree assumption [14,15]. It is also +proven in [17, 28] that every component of the free uniform spanning forest of a unimodular +random rooted graph is infinitely ended a.s. whenever the free and wired forests are different. +Here, unimodular random rooted graphs comprise a very large class of random graph models +including Benjamini-Schramm limits of finite graphs [7], Cayley graphs, and (suitable versions +of) Galton-Watson trees, as well as e.g. percolation clusters on such graphs; See Section 3.1 +for definitions and e.g. [1,11] for detailed background. +The aforementioned works [1,14,15,17,28] completely resolved the problem of the number +of ends of the WUSF and FUSF for transient unimodular random rooted graphs, but the +recurrent case remained open. Besides the fact that the transient methods do not apply, a +2 + +further complication of the recurrent case is that it is possible for the UST to be either one- +ended or two-ended according to the geometry of the graph: indeed, the UST of Z2 is one-ended +while the UST of Z is two-ended. +Aldous and Lyons conjectured [1, p. 1485] that the dependence of the number of ends +of the UST on the geometry of the graph is as simple as possible: The UST of a recurrent +unimodular random rooted graph is one-ended if and only if the graph is. +The fact that +two-ended unimodular random rooted graphs have two-ended USTs is trivial; the content of +the conjecture is that one-ended unimodular random rooted graphs have one-ended USTs. +Previously, the conjecture was resolved under the assumption of planarity in [3], while in [8] it +was proved (without using the planarity assumption) that the UST of a recurrent unimodular +random rooted graph is one-ended precisely when the “harmonic measure from infinity” is +uniquely defined. In this paper we resolve the conjecture. +Theorem 1. Let (G, o) be a recurrent unimodular random rooted graph and let T be the +uniform spanning tree of G. Then T has the same number of ends as G a.s. +To see that the theorem is not true without unimodularity, consider taking the line graph +Z and adding a path of length 2n connecting −n connecting to n for each n, making the +graph one-ended. Kirchoff’s effective resistance formula implies that the probability that the +additional path connecting −n to n is included in the UST is at most n/(2n +n), and a simple +Borel-Cantelli argument implies that the UST is two-ended almost surely. Similar examples +show that Theorem 1 does not apply to unimodular random rooted networks, since we can use +edges of very low conductance to make the network one-ended while having very little effect +on the geometry of the UST. +About the proof. We stress again that the tools used in the transient case do not apply +at all to the recurrent case, and we are forced to use completely different methods that are +specific to the recurrent case. We build on [8] which proved that the “harmonic measures +from infinity” are uniquely defined if and only if the uniform spanning tree is one-ended; A +self-contained treatment of (a slight generalization of) the results of [8] that we will need is +given in Appendix A. The set of harmonic measures from infinity can be thought of as a +“boundary at infinity” for the graph, analogously to the way the Martin boundary is used in +transient graphs. It is implicit in [8] that these measures correspond to the ways in which a +random walk “conditioned to never return to the root” can escape to infinity. We develop these +ideas further in Section 2, in which we make this connection precise. We then apply these ideas +inside an ergodic-theoretic framework to prove that if the UST has two ends, then the effective +resistance must grow linearly along the unique bi-infinite path in the tree, which implies in +particular that graph distances must also grow linearly. To conclude, we argue that this can +only happen when the graph has linear volume growth, which is known to be equivalent to +two-endedness for unimodular random rooted graphs [5,9]. +3 + +Acknowledgments +DvE wishes to thank Nathana¨el Berestycki for many useful discussions +and comments. Most of the research was carried out while DvE visited TH at Caltech and we +are grateful for the hospitality of the institute. DvE is supported by the FWF grant P33083, +“Scaling limits in random conformal geometry”. +2 +Boundary theory of recurrent graphs +In this section we develop the theory of harmonic measures from infinity on recurrent graphs, +their associated potential kernels and Doob transforms, and how this relates to the spanning +tree. Much of the theory we develop here is a direct analogue for recurrent graphs of the theory +of Martin boundaries of transient graphs [12, 30]. This theory is interesting in its own right, +and we were surprised to find how little attention has been paid to these notions outside of +some key motivating examples such as Z2 [13,26]. +All of the results in this section will concern deterministic infinite, connected, recurrent, +locally finite graphs; applications of the theory to unimodular random rooted graphs will be +given in Section 3. +2.1 +Harmonic measures from infinity +Let G = (V, E) be an infinite, connected, locally finite, recurrent graph. For each v ∈ V we +write Pv for the law of the simple random walk on G started at v, and for each set A ⊆ V +write TA and T + +A for the first visit time of the random walk to A and first positive visit time +of the random walk to A respectively. Given a probability measure µ on V , we also write Pµ +for the law of the random walk started at a µ-distributed vertex. +A harmonic measure from infinity h = (hB : B ⊂ V finite) on G is a collection of +probability measures on V indexed by the finite subsets B of V with the following properties: +1. hB is supported on ∂B for each B ⊂ V , where ∂B is the set of elements of B that are +adjacent to an element of V \ B. +2. For each pair of finite sets B ⊆ B′, hB and hB′ satisfy the consistency condition +hB(u) = +� +v∈B′ +hB′(v)Pv(XTB = u) +(1) +for every u ∈ B. +We denote the space of harmonic measures from infinity by H, which (identifying the measures +hB with their probability mass functions) is a compact convex subset of the space of functions +{finite subsets of V } → RV when equipped with the product topology. As mentioned above, +the space H plays a role for recurrent graphs analogous to that played by the Martin boundary +for transient graphs; the analogy will become clearer once we introduce potential kernels in +4 + +the next subsection. We say that the harmonic measure from infinity is uniquely defined +when H is a singleton. +If µn is a sequence of probability measures on V converging vaguely to the zero measure +in the sense that µn(v) → 0 as n → ∞ for each fixed v ∈ V then any subsequential limit of +the collections (Pµn(XTB = ·) : B ⊂ V finite) belongs to H, with these collections themselves +satisfying every property of a harmonic measure from infinity other than the condition that hB +is supported on ∂B for every finite B. (Indeed, the consistency condition (1) follows from the +strong Markov property of the random walk.) In fact every harmonic measure from infinity +can be written as such a limit. +Lemma 2. If h ∈ H is a harmonic measure from infinity then there exists a sequence of finitely +supported probability measures (µn)n≥1 on V such that µn(v) → 0 for every v ∈ V and +hB(·) = lim +n→∞ Pµn(XTB = · ) +for every B ⊂ V finite. +(2) +Proof. Fix h ∈ H. Let V1 ⊂ V2 ⊂ V3 · · · be an increasing sequence of finite subsets of V with +� +i Vi = V , and for each n ≥ 1 let µn = hVn. It follows from the consistency condition (1) that +hB(·) = Pµn(XTB = · ) +for every B ⊂ Vn, +and the claim follows since every finite set is eventually contained in Vn. +Since H is a weakly compact subspace of the set of functions from finite subsets of V to +RV , which is a locally convex topological vector space, it is a Choquet-simplex: Every element +can be written as a convex combination of the extremal points. In particular, if H has more +than one point then it must have more than one extremal point. This will be useful to us +because extremal points of H are always limits of harmonic measures from sequences of single +vertices. +Indeed, identifying each vertex v ∈ V with the collection of harmonic measures +(Pv(XTB = ·) : B ⊂ V finite) allows us to think of V ∪H as a compact Polish space containing +V (in which V might not be dense), and we say that a sequence of vertices (vn)n≥0 converges +to a point h ∈ H if hB(·) = limn→∞ Pvn(XTB = · ) for every B ⊂ V finite. +Lemma 3. If h ∈ H is extremal, there exists a sequence of vertices (vn)n≥0 such that vn +converges to h as n → ∞. +Proof. Let I be the set of functions h : {B ⊂ V finite} → RV of the form +hB(·) = Pµ(XTB = ·) +for every B ⊂ V finite +for some finitely supported measure µ on V . Lemma 2 implies that I = I ∪ H is a compact +convex subset of the space of all functions {B ⊂ V finite} → RV equipped with the product +topology, which is a locally convex topological vector space. By the Krein-Milman theorem, +a subset W of I ∪ H has closure containing the set of extremal points of I ∪ H if and only +5 + +if I ∪ H is contained in the closed convex hull of W. Thus, if we define Iext to be the set of +functions h : {B ⊂ V finite} → RV of the form +hB(·) = Pz(XTB = ·) +for every B ⊂ V finite +for some z ∈ V then I is clearly contained in the convex hull of Iext, so that I ∪H is contained +in the closed convex hull of Iext and, by the Krein-Milman theorem, the set of extremal points +of I ∪ H is contained in the closure of Iext. +Now, observe that for any non-trivial convex combination of an element of I and an element +of H, there must exist a finite set of vertices B and a point z in the interior of B (i.e., in B +and not adjacent to any element of V \ B) such that hB(z) ̸= 0; indeed, if the element of I +corresponds to some finitely supported measure µ, then any B containing the support of µ in +its interior and any z in the support of µ will do. Since no element of H can have this property, +it follows that non-trivial convex combinations of elements of I and H cannot belong to H and +hence that extremal points of H are also extremal in I ∪ H. It follows that the set of extremal +points of H is contained in the closure of Iext, which is equivalent to the claim. +Remark 1. The converse to this lemma is not true: A limit of a sequence of Dirac measures +need not be extremal. For example, if we construct a graph from Z by attaching a very long +path between −n and n for each n ≥ 1 and take zn to be a point in the middle of this path for +each n, the sequence (zn)n≥1 will converge to a non-extremal element of H that is the convex +combination of the limits of (n)n≥1 and (−n)n≥1. +2.2 +Potential kernels and Doob transforms +The arguments in [8] heavily rely on a correspondence between the harmonic measure from +infinity and its potential kernel. One important feature of the potential kernel is that, given +a vertex o ∈ V and a point h ∈ H, it provides a sensible way to “condition the random +walk to converge to h before returning to o”. We begin by discussing how conditioning the +random walk to hit a particular vertex before returning to o can be described in terms of Doob +transforms before developing the analogous limit theory. +Doob transforms and non-singular conditioning. Suppose that we are given two +distinct vertices o and z in an infinite, connected, locally finite recurrent graph G. Letting +Gz(x, y) be the expected number of times a random walk started at x visits y before hitting +z, we can compute that the function +a(x) = Gz(o, o) +deg(o) − Gz(x, o) +deg(o) += Px(To > Tz)Gz(o, o) +deg(o) +is harmonic at every vertex other than o and z, and has +∆a(o) = 0 − deg(o)Eo[a(X1)] = −Po(T + +o > Tz)Gz(o, o) = −1, +6 + +where ∆ denotes the graph Laplacian ∆f(x) = deg(x)f(x) − � +y∼x f(y) = deg(x)Ex[f(X0) − +f(X1)] (terms in this sum are counted with appropriate multiplicity if there is more than one +edge between x and y). Moreover, the quantity a(x) is strictly positive at every vertex x that +is neither equal to o nor disconnected from z by o in the sense that every path from x to z +must pass through o. Observe that the trivial identity +Po((X0, . . . , Xn) = (x0, . . . , xn)) = +n +� +i=1 +p(xi−1, xi) += +1 +a(xn)a(x1)p(o, x1) +n +� +i=2 +a(xi) +a(xi−1)p(xi−1, xi) +(3) +holds for every sequence of vertices x0, . . . , xn with x0 = o and a(xi) > 0 for every i > 0. Since +a(z) = Gz(o, o) = Po(Tz < T + +o )−1 it follows that +Po((X0, . . . , Xn) = (x0, . . . , xn) | Tz < T + +o ) = a(x1)p(o, x1) +n +� +i=2 +a(xi) +a(xi−1)p(xi−1, xi) +(4) +for every sequence of vertices x0, . . . , xn with x0 = o, xn = z, and xi /∈ {o, z} for every 0 < i < n +(which implies that a(xi) > 0 for every 1 ≤ i ≤ n). Now, the fact that a is harmonic off of +{o, z} and has ∆a(o) = −1 implies that we can define a stochastic matrix with state space +{x ∈ V : x = o or a(x) > 0} by +�pa(x, y) = + + + + + + + + + +a(y) +a(x)p(x, y) +x /∈ {o, z} +a(y) +x = 0 +1(y = z) +x = z, +and if we define the Doob transformed walk � +Xa to be the Markov chain with this transition +matrix started from o then it follows from (4) that ( � +Xa +n)Tz +n=0 has law equal to the conditional +law of the simple random walk (Xn)Tz +n=0 started at o and conditioned to hit z before returning +to o. Moreover, letting �Pa +o denote the law of � +Xa, it follows from the definition of � +Xa that +�Pa +o +� +( � +Xa +0 , . . . , � +Xa +n) = (x0, . . . , xn) +� += +n +� +i=1 +�pa(xi−1, xi) = a(x1) +n +� +i=2 +a(xi) +a(xi−1)p(xi−1, xi) += a(xn) +n +� +i=2 +p(xi−1, xi) = deg(o)a(xn)Po ((X0, . . . , Xn) = (x0, . . . , xn)) +(5) +for every sequence x0, . . . , xn with x0 = o and xi /∈ {o, z} for every 0 < i < n. +Defining the potential kernel. +We now define the potential kernel ah associated to a +point h ∈ H via the formula +ah(x, y) = hx,y(x)Reff(x ↔ y) +(6) +7 + +where we write hx,y = h{x,y}, so that ah(x, x) = 0 for each x ∈ V . The fact that this is a +sensible definition owes largely to the following lemma. +Lemma 4. For each h ∈ H, the potential kernel ah(x, y) = hx,y(x)Reff(x ↔ y) satisfies +∆ah( · , y) = −1( · = y), +(7) +so that the potential kernel ah(·, y) is harmonic away from y and subharmonic at y. +Proof. Since the map h �→ ah is affine and the equality (7) is linear, it suffices to prove the +lemma in the case that h is extremal. By Lemma 3, there exists a sequence of vertices (vn)n≥1 +such that vn converges to h. For each n ≥ 1 we define +an(x, y) = Gvn(y, y) +deg(y) +− Gvn(x, y) +deg(y) . +and claim that +ah(x, y) = lim +n→∞ an(x, y) +(8) +for every x, y ∈ V . (Note that this limit formula is often taken as the definition of the potential +kernel.) We will prove (8) with the aid of three standard identities for the Greens function: +1. By the strong Markov property, Gz(x, y) is equal to Px(Ty < Tz) Gz(y, y) for every three +distinct vertices x, y, and z. +2. By the strong Markov property, Gx(y, y) is equal to Px(Ty < T + +x )−1 for every pair of +distinct vertices x and y. It follows in particular that deg(y)−1 Gx(y, y) = Reff(x ↔ y) +and, since the effective resistance is symmetric in x and y, that deg(y)−1 Gx(y, y) = +deg(x)−1 Gy(x, x). +3. By time-reversal, deg(x) Gz(x, y) is equal to deg(y) Gz(y, x) for every three distinct ver- +tices x, y, and z. +Applying these three identities in order yields that +an(x, y) = Gvn(y, y) +deg(y) Px(Ty > Tvn) = Gy(vn, vn) +deg(vn) Px(Ty > Tvn) += Gy(x, vn) +deg(vn) += Gy(vn, x) +deg(x) +whenever x, y, and vn are distinct. Applying the first and second identities a second time then +yields that +an(x, y) = Pvn(Tx < Ty)Gy(x, x) +deg(x) += Pvn(Tx < Ty)Reff(x ↔ y) +(9) +whenever x, y, and vn are distinct. This is easily seen to imply the claimed limit formula (8). +8 + +In light of this lemma, we define Po to be the space of non-negative functions a : V → [0, ∞) +with a(o) = 0 and ∆a(x) = −1(x = o), so that ah( · , o) belongs to Po for each o ∈ V and h ∈ H +by Lemma 4. We will later show that the map h �→ ah is an affine isopmorphism between the +two convex spaces H and Po. We first describe how elements of Po can be used to define Doob +transformed walks. +Doob transforms and singular conditioning. +We now define the Doob transform asso- +ciated to an element of the space Po. Given a ∈ Po, we define � +Xa to be the Doob a-transform +of the simple random walk X on G, so that � +Xa has state space {x ∈ V : x = o or a(x) > 0} +and transition probabilities given by +�p a(x, y) := + + + +a(y) +a(x)p(x, y) +if x ̸= o +a(y) +if x = o, y ∼ o +where p is the transition kernel for the simple random walk. Similarly, given h ∈ H, we write +� +Xh = � +Xah(·,o) where ah is the potential kernel associated to h. Informally, we think of � +Xh +as the walk that is “conditioned to go to h before returning to o”. (In particular, when the +harmonic measure from infinity is unique and H and Po are singleton sets, we think of the +associated Doob transform as the random walk conditioned to never return to o.) We write +�Pa +o or �Ph +o for the law of � +Xa or � +Xh as appropriate. +As before, it follows from this definition that if a ∈ Po and we write X[0, m] for the initial +segment consisting of the first m steps of the random walk X then +�Pa +o( � +Xa[0, m] = γ) = +m +� +i=1 +�p a(γi−1, γi) = a(γ1) +m +� +i=2 +a(γi) +a(γi−1)p(γi−1, γi) += a(γm) +m +� +i=2 +p(γi−1, γi) = deg(o)a(γm)Po(X[0, m] = γ) +(10) +for every finite path γ = (γ0, . . . , γm) with γ0 = o and γi ̸= o for every i > 0. Summing over all +paths γ that begin at o, end at some point x ̸= o, and do not visit o or x at any intermediate +point yields in particular that if h ∈ H then +�Ph +o( � +X hits x) = deg(o)ah(x, o)Po(Tx < T + +o ) = ho,x(x), +(11) +where the last equality follows from (6) and the definition of the effective resistance. +Lemma 5. Let G = (V, E) be a recurrent graph and let a ∈ Po. Then the associated Doob- +transformed walk � +Xa is transient. +Proof. One can easily verify from the definitions that the sequence of reciprocals (a( � +Xa +n)−1)n≥1 +is a non-negative martingale with respect to its natural filtration, and hence converges almost +9 + +surely to some limiting random variable, which it suffices to prove is zero almost surely. It +follows from the identity (10) that +�Pa +o(a( � +Xa +n) ≤ M) = +� +v +1(a(v) ≤ M) deg(o)a(v)Po(Xn = v, T + +o > n) ≤ M deg(o)Po(T + +o > n), +for every n, M ≥ 1. +Since G is recurrent, the right hand side tends to zero as n → ∞ +for each fixed M. +It follows that lim supn→∞ a( � +Xa +n) = ∞ almost surely, and hence that +limn→∞ a( � +Xa +n) = ∞ almost surely since the limit is well-defined almost surely. This implies +that � +Xa is transient. +2.3 +An affine isomorphism +Let G = (V, E) be recurrent, fix o ∈ V , and let Po denote the set of positive functions +a : V → [0, ∞) with a(o) = 0 that satisfy ∆a(·) = −1( · = o). As we have seen, for each h ∈ H +the potential kernel ah(·, o) defines an element of Po. Moreover, the map sending h �→ ah( · , o) +is affine in the sense that if h = θh1 + (1 − θ)h2 then ah( · , o) = θah1( · , o) + (1 − θ)ah2( · , o). +We wish to show that this map defines an affine isomorphism between H and Po in the sense +that it is bijective (in which case its inverse is automatically affine). We begin by constructing +the inverse map from Po to H. +Lemma 6. Let G = (V, E) be a infinite, connected, locally finite recurrent graph and let o ∈ V . +For each a ∈ Po there exists a unique h ∈ H satisfying +hB(u) = �Pa +o( � +Xa visits B for the last time at u) +for every finite set B containing o. Moreover, this h satisfies ah(x, o) = a(x) for every x ∈ V . +Proof of Lemma 6. Fix a ∈ Po. We define a the family of probability measures h = (hB : B ⊂ +V finite) by +hB(u) = �Pa +o( � +Xa visits B for the last time at u) +for every u ∈ B if o ∈ B and +hB(u) = �Pa +o( � +Xa visits B ∪ {o} for the last time at u) ++ �Pa +o( � +Xa visits B ∪ {o} for the last time at o)Po(XTB = u) +for every u ∈ B if o /∈ B, so that if o /∈ B then +hB(u) = +� +v∈B∪{o} +hB∪{o}(v)Pv(XTB = u) +for every u ∈ V . We claim that this defines an element of H. It is clear that hB is a probability +measure that is supported on ∂B for each finite set B ⊂ V ; we need to verify that it satisfies +10 + +the consistency property (1). Once it is verified that h ∈ H, the fact that a = ah(·, o) follows +easily from the definition of ah together with the identity (10), which together yield that +ah(v, o) = hv,o(v)Reff(v ↔ o) = +�Pa +o( � +Xa visits {o, v} for the last time at v) +deg(o)Po(Tv < T + +o ) += +�Pa +o( � +Xa hits v) +deg(o)Po(Tv < T + +o ) = deg(o)a(v)Po(Tv < T + +o ) +deg(o)Po(Tv < T + +o ) += a(v) +for each v ∈ V . +We now prove that h satisfies the consistency property (1). We will prove the required +identity in the case o ∈ B, the remaining case o /∈ B following from this case and the definition. +Let B ⊆ B′ be finite sets with o ∈ B and let (Vn)n≥1 be an exhaustion of V by finite sets such +that B′ ⊆ Vn for every n ≥ 1. Writing V c +n = V \ Vn for each n ≥ 1 and τn for the first time +the walk visits V c +n, we have that +hB(u) = lim +n→∞ +�Pa +o( � +X[0, τn] last visits B at u) += lim +n→∞ +� +b∈V c +n +�Pa +o( � +X[0, τn] last visits B at u, � +Xτn = b) +and hence by (10) and time-reversal that +hB(u) = lim +n→∞ +� +b∈V c +n +deg(o)a(b)Po(X[0, τn] last visits B at u, Xτn = b) += lim +n→∞ +� +b∈V c +n +deg(b)a(b)Pb(XTB = u, To < T + +V c +n ). +(12) +It follows from this together with the strong Markov property that +hB(u) = lim +n→∞ +� +v∈B′ +� +b∈V c +n +deg(b)a(b)Pb(XT ′ +B = v, XTB = u, To < T + +V c +n ) += lim +n→∞ +� +v∈B′ +� +b∈V c +n +deg(b)a(b)Pb(XT ′ +B = v, TB′ < T + +V c +n )Pv(XTB = u, To < T + +V c +n ). +Now, we have by the strong Markov property that for each b ∈ V c +n and v ∈ B′ +Pb(XTB′ = v, To < T + +V c +n ) = Pb(XTB′ = v, TB′ < T + +V c +n )Pv(TV c +n > To). +and by recurrence that limn→∞ Pv(To < T + +V c +n ) = 1, so that +hB(u) = lim +n→∞ +� +v∈B′ +� +b∈V c +n +deg(b)a(b)Pb(XT ′ +B = v, To < T + +V c +n )Pv(XTB = u). +The claimed identity (1) follows from this together with the identity (12) applied to the larger +set B′. +11 + +Theorem 7. Let G be an infinite, recurrent, locally finite graph, and let o ∈ V . The map +h �→ ah(·, o) is an affine isomorphism H → Po. In particular, this map identifies extremal +elements of H with extremal elements of Po. +Proof. It remains only to prove that h �→ ah is injective. To prove this it suffices by definition +of ah to prove that hB is determined by (hx,o(x) : x ∈ ∂B) for each finite set B ⊂ V containing +the vertex o. Fix one such set B. We have by definition of H that +hx,o(x) = +� +y∈∂B +hB(y)Py(Tx < To) = +� +y∈∂B +A(x, y)hB(y) +for each x ∈ ∂B where A(x, y) := Py(Tx < To) for each x, y ∈ ∂B, so that it suffices to prove +that the matrix A (which is indexed by ∂B) is invertible. Define a matrix Q indexed by ∂B +by +Q(x, y) = Py(T + +∂B < To, XT + +∂B = x). +Then we have by the strong Markov property that +A(x, y) − 1(x = y)Px(T + +x ≥ To) = Py(T + +x < To) += +� +z∈∂B +Pz(Tx < To)Q(z, y) = Px(T + +x ≥ To)Q(x, y) + +� +z∈∂B +Pz(T + +x < To)Q(z, y) +and hence inductively that +Py(T + +x < To) = Px(T + +x ≥ To) +n +� +i=1 +Qn(x, y) + +� +z∈∂B +Pz(T + +x < To)Qn(z, y) +for every n ≥ 1. Since Q is irreducible and substochastic, we can take the limit as n → ∞ to +obtain that +A(x, y) = 1(x = y)Px(T + +x ≥ To) + Py(T + +x < To) = Px(T + +x ≥ To) +∞ +� +i=0 +Qn(x, y) +for every x, y ∈ ∂B. It follows by a standard argument that the matrix A is invertible with +inverse A−1 = Px(T + +x ≥ To)−1(1 − Q) as required. +2.4 +The Liouville property for extremal Doob transforms +In this section we prove a kind of tail-triviality property of the Doob-transformed walk cor- +responding to an extremal point h ∈ H. Letting G = (V, E) be a graph, we recall that an +event A ⊆ V N is said to be invariant if (x0, x1, . . .) ∈ A implies that (x1, x2, . . .) ∈ A for every +(x0, x1, . . .) ∈ V N. +Theorem 8. Let G = (V, E) be an infinite, connected, recurrent, locally finite graph and let +o ∈ V . If h ∈ H is extremal then the Doob transformed random walk ˆXh does not have any +non-trivial invariant events: If A ⊆ V N is an invariant event then �Ph +o(A) ∈ {0, 1}. +12 + +Proof. It suffices to prove the corresponding statement for � +Xa when a is an extremal element +of Po. Suppose not, so that A is a non-trivial invariant event. We have by Levy’s 0-1 law that +Po( � +Xa ∈ A | � +Xa +1 , . . . , � +Xa +n) → 1( � +Xa ∈ A) almost surely as n → ∞. +(13) +Moreover, we also have by invariance that +�Pa +x( � +Xa ∈ A) = +� +y∈V +a(y) +a(x)p(x, y)�Pa +y( � +Xa ∈ A) +and that +�Pa +o( � +Xa ∈ A) = +� +y∈V +a(y)�Pa +y( � +Xa ∈ A). +Since similar inequalities hold when we replace A by Ac it follows that we can write a as a +non-trivial convex combination of two elements of Po +a(x) = �Pa +o( � +Xa ∈ A) · a(x)�Pa +x( � +Xa ∈ A) +�Pao( � +Xa ∈ A) ++ �Pa +o( � +Xa /∈ A) · a(x)�Pa +x( � +Xa /∈ A) +�Pao( � +Xa /∈ A) +, +these two factors being different by (13), contradicting extremality of a. +Remark 2. Underlying this proposition is the fact that once we fix a ∈ Po, we can identify Po +with the Martin boundary of the conditioned walk � +Xa. Theorem 8 is the recurrent version +of the fact that Doob transforming by an extremal element of the Martin boundary yields a +process with trivial invariant sigma-algebra. +For our purposes, the most important output of the Liouville property is the following +proposition, which lets us easily tell apart the trajectories of two different Doob transformed +walks � +Xh and � +Xh′ by looking at any infinite subset of their traces (and, in particular, from +their loop-erasures). +Proposition 9. Let h, h′ be distinct extremal elements of H and let � +Xh be the Doob-transformed +simple random walk corresponding to h. Then +ah′( � +Xh +n, o) +ah( � +Xhn, o) +→ 0 +almost surely as n → ∞. +Proof. We prove the corresponding statement in which a, a′ are distinct extremal elements of +Po. Let � +X and � +X′ have laws �Pa +o and �Pa′ +o respectively. One can easily verify from the definitions +that +(Zn)n≥1 = +� +a′( � +Xn) +a( � +Xn) +� +n≥1 +and +(Z′ +n)n≥1 = +� +a( � +X′ +n) +a′( � +X′n) +� +n≥1 +are both non-negative martingales with respect to their natural filtrations, and hence converge +almost surely to some limiting random variables Z and Z′. Since Z and Z′ are measurable with +13 + +respect to the invariant σ-algebras of � +X and � +X′ respectively and a and a′ are both extremal, +there must exist constants α and α′ such that Z = α and Z′ = α′ almost surely. We also have +that EZn = EZ′ +n = 1 for every n ≥ 1 and hence that α, α′ ≤ 1. We wish to prove that α = 0. +It follows from (10) that the conditional distributions of the initial segments � +X[0, m] and +� +X′[0, m] are the same if we condition on � +Xm = � +X′ +m = v for any v ∈ V for any v ∈ V and +m ≥ 1 and that +Pa +o( � +Xm = v) +Pa′ +o ( � +Xa′ +m = v) += a(v) +a′(v) +for every m ≥ 1 and v ∈ V . If α > 0 then for every ε > 0 there exists M such that the +distribution of � +Xm puts mass at least 1 − ε on the set of vertices with a′(v)/a(v) ≥ (1 − ε)α +for every m ≥ M, and it follows that for each m ≥ M there is a coupling of the two walks � +X′ +and � +X so that their initial segments of length m coincide with probability at least (1 − ε)2α. +Taking a weak limit as m → ∞ and ε → 0, it follows that there exists a coupling of the two +walks � +X′ and � +X such that the two walks coincide forever with probabilty at least α > 0. If we +couple the walks in this way then on this event we must have that Z′ = 1/Z, which can occur +with positive probability only if α′ = 1/α. Since α, α′ ≤ 1 we must have that α = α′ = 1 and +that we can couple the two walks to be exactly the same almost surely. This is clearly only +possible if a = a′, and since a ̸= a′ by assumption we must have that α = 0. +2.5 +Potential kernels and the uniform spanning tree +We now use Lemma 11 to show that the UST of a recurrent graph can always be sampled using +a variant of Wilson’s algorithm [6,29] in which we ‘root at a point in H’, where again we are +thinking intuitively of H as a kind of boundary at infinity of the graph. Fix h ∈ ex(H) and let +� +Xh be the conditioned walk of the previous section. Fix some enumeration V = {v1, v2, . . .} +of V with v1 = o. Set E0 = LE( � +Xh[0, ∞)) (which is well defined because � +Xh is transient) and +for each i ≥ 1 define Ei given Ei−1 recursively as follows: +• if vi ∈ Ei−1, set Ei = Ei−1 +• otherwise, set Ei = Ei−1 ∪ LE(Y [0, τ)) where Y is the simple random walk started at vi +and stopped at τ, the hitting time of Ei−1. +Last, define T = �∞ +i=0 Ei. We refer to this procedure as Wilson’s algorithm rooted at h. +The random tree T generated by Wilson’s algorithm rooted at h is clearly a spanning tree of +G; the next lemma shows that it is distributed as the UST of G. +Lemma 10 (Wilson meets Doob). Let G = (V, E) be an infinite, connected, locally finite, +recurrent graph and let h ∈ ext(H). The tree T generated by Wilson’s algorithm rooted at h +is distributed as the uniform spanning tree of G. In particular, the law of T is independent of +the chosen enumeration of V and the choice of h ∈ ext(H). +14 + +Remark 3. It follows by taking convex combinations that the same statement also holds when +h is not extremal. +We will deduce Lemma 10 from the following lemma, which allows us to think of the Doob- +transformed walk � +Xh as a limit of conditioned simple random walks on G. For the purposes of +this lemma we think of our walks as belonging to the space of sequences in V equipped with +the product topology. +Lemma 11 (Local convergence). Let G = (V, E) be an infinite, connected, locally finite, +recurrent graph and suppose that zn is a sequence of vertices of G such that zn converges +to h ∈ H. +If X denotes the random walk on G started at o and � +Xh denotes the Doob- +transformation of X as above, then the conditional law of X given that it hits zn before first +returning to o converges weakly to the law of � +Xh. +Proof of Lemma 11. This is a classical result concerning Doob transforms, and can also be +deduced from the limit formula (8). We give a brief proof. Let Tzn be the first time the walk +hits zn, let T + +o be the first positive time the walk hits o, and let ϕ = (o, ϕ1, . . . , ϕm) be a path +of length m for some m ≥ 1 with ϕi ̸= o for every i > 0. By the Markov property for the +simple random walk, +Po(X[0, m] = ϕ, Tzn < T + +o ) = Po(X[0, m] = ϕ)Pϕm(Tzn < To), +and it follows from (10) that +Po(X[0, m] = ϕ, Tzn < T + +o ) = +1 +deg(o)ah(ϕm, o)Po( � +Xh[0, m] = ϕ)Pϕm(Tzn < To). +The result follows once multiplying both sides by the effective resistance between o and zn and +using the representation (6) for the potential kernel. +Proof of Lemma 10. The standard implementation of Wilson’s algorithm rooted at zn allows +us to sample the uniform spanning tree of G in a manner exactly analogous to above, except +that we start with a walk run from o until it first hits zn. Now, it is a combinatorial fact that +the loop erasure of the walk run from o until it first hits zn does not change its distribution +if we condition the walk to hit zn before returning to o: Indeed, the loop-erasure of the entire +unconditioned walk is equal to the loop-erasure of the final segment of the walk between its +last visit to o and its first visit to zn, and this final segment is distributed as the conditioned +walk. +Thus, in the standard implementation of Wilson’s algorithm, we do not change the +distribution of the obtained tree if we condition the first walk to hit zn before returning to o. +The claim then follows by taking the limit as zn → ∞ and using Lemma 11. +This leads to the following connection between the ends of the UST and the extremal points +of the set of harmonic measures from infinity H. +15 + +Proposition 12. Let G = (V, E) be an infinite, connected, locally finite, recurrent graph, let +T be the uniform spanning tree of T, and let H be a countable subset of ext(H). Almost surely, +for each h ∈ H there exists an infinite simple path Γ = (Γ1, Γ2, . . .) in T such that +ah′(Γi, x) +ah(Γi, x) → 0 +as i → ∞ for each h′ ∈ H \ {h}. +In particular, T almost surely has at least as many ends as there are extremal points of H. +(In the last sentence of this proposition we are not distinguishing between different infinite +cardinalities, but merely claiming that if H has infinitely many extremal points then T has +infinitely many ends almost surely.) +Proof. This is an immediate consequence of Proposition 9 and Lemma 10. +Orientations. Let G = (V, E) be an infinite, connected, locally finite, recurrent graph and +let h ∈ ext(H). When we generate the UST T of G using Wilson’s algorithm rooted at h, the +algorithm also provides a natural orientation of T, where each edge is oriented in the direction +that it is crossed by the loop-erased random walk that contributed that edge to the tree. When +T almost surely has the same number of ends as there are extremal points in H, and both +numbers are finite (which will always be the case in the unimodular setting by the results of +[8]), it follows from Proposition 12 that this orientation is a.s. determined by the (unoriented +tree): Almost surely, for each h ∈ H and v ∈ V there is a unique infinite ray (Γ1, Γ2, . . .) +starting from v such that +ah′(Γi, v) +ah(Γi, v) → 0 +as i → ∞ for each h′ ∈ ext(H) \ {h}, +and if we orient the tree in the direction of this ray we must recover the same orientation as if +we had generated the oriented tree using Wilson’s algorithm rooted at h. This fact will play a +key role in the proof of our main theorem. +3 +Proof of the main theorem +3.1 +Reversible and unimodular graphs +We now give a very brief introduction to unimodular random rooted graphs, referring the +reader to [1,11] for detailed introductions. Let us just recall that G•,• is the separable metric +space of doubly rooted graphs (G, x, y) (modulo graph isomorphisms), equipped with the local +topology, also known as Benjamini-Schramm topology. +Similarly defined is the space G• of +rooted graphs (G, o). A mass transport is a measurable function f : G•,• → [0, ∞]. A measure +P on G• is called unimodular whenever the mass transport principle +�E +�� +x∈V +f(G, o, x) +� += �E +�� +x∈V +f(G, x, o) +� +16 + +holds for all mass transports f. A probability measure P on G• is called reversible if (G, o, X1) d= +(G, X1, o) where X1 is the first step of the simple random walk. The law P is called station- +ary if (G, o) d= (G, X1) and clearly any reversible graph is stationary. For recurrent graphs, +stationarity and reversibility are equivalent [4]. +If P is the law of a unimodular random graph, with finite expected degree, then biasing it +by deg(o) gives a reversible random graph and whenever P is the law of a reversible random +graph, then biasing by deg(o)−1 gives a unimodular random graph; see for example [4]. +A set A ⊆ G• is said to be rerooting invariant if ((g, v) ∈ A) ⇒ ((g, u) ∈ A) for every +rooted graph (g, v) ∈ G• and every u in the vertex set of g. A unimodular random rooted +graph (G, o) is said to be ergodic if it has probability 0 or 1 to belong to any given re-rooting +invariant event in G•. As explain in [1, Section 4], this is equivalent to the law of (G, o) being +extremal in the weakly compact convex set of unimodular probability measures on G•. As +such, it follows by Choquet theory that every unimodular measure on G• may be written as a +mixture of ergodic unimodular measures. For our purposes, the upshot of this is that we may +assume without loss of generality that (G, o) is ergodic when proving Theorem 1. +We will also rely on the following characterization of two-ended unimodular random rooted +graphs due to Bowen, Kun, and Sabok [9], which builds on work of Benjamini and the second +author [5]. Here, a graph G is said to have linear volume growth if for each vertex v of G +there exists a constant Cv such that |B(v, r)| ≤ Cvr for every r ≥ 1, where B(v, r) denotes the +graph distance ball of radius r around v. +Proposition 13 ([9], Proposition 2.1). Let (G, o) be an infinite unimodular random rooted +graph. Then G is two-ended almost surely if and only if it has linear volume growth almost +surely. +To prove Thorem 1, it will therefore suffice to prove that if (G, o) is a recurrent unimodular +random rooted graph whose UST is two-ended almost surely then G has linear volume growth +almost surely. +3.2 +The effective resistance is linear on the spine +Let P be the joint law of an ergodic recurrent unimodular random rooted graph (G, o) and its +uniform spanning tree T, which we think of as a triple (G, o, T). It follows by tail triviality of +the UST [6, Theorem 8.3] that the number of ends of T is deterministic conditional on (G, o), +and since (G, o) is ergodic that T has some constant number of ends almost surely. Moreover, +it follows from [1, Theorem 6.2 and Proposition 7.1] that this number of ends is either 1 or 2 +almost surely, so that T is either one-ended almost surely or two-ended almost surely. +We wish to prove that if T is two-ended almost surely then G is two-ended almost surely +also. We will rely on the following theorem of Berestycki and the first author. +17 + +Theorem 14 ([8], Theorem 1). Let (G, o) be a recurrent unimodular random rooted graph with +E deg(o) < ∞. Almost surely, the uniform spanning tree of G is one-ended if and only if the +harmonic measure from infinity is uniquely defined. +To avoid the unnecessary assumption that E deg(o) < ∞, we will use the following mild +generalization of this theorem, whose proof is given in Appendix A. +Theorem 15. Let (G, o) be a recurrent unimodular random rooted graph. Almost surely, the +uniform spanning tree of G is one-ended if and only if the harmonic measure from infinity is +uniquely defined. +It follows from this theorem together with Proposition 12 that if T is two-ended almost +surely then |ext(H)| = 2 almost surely. +Suppose that T is two-ended almost surely and let S be the spine of T, i.e., the unique +double-infinite simple path contained in T. We give T an orientation by choosing uniformly +at random one of the two ends of S and directing every edge towards that end, letting the +resulting oriented tree be denoted T → with oriented spine S→. +Since the law of T → is a +rerooting-equivariant function of the graph (G, o), the triple (G, T →, o) is unimodular. Since +“everything that can happen somewhere can happen at the root” [1, Lemma 2.3] we also have +that the origin belongs to S with positive probability and hence that we can define a law PS +on triplets (G, T →, o) (which we can view as a rooted network) by conditioning o to belong +to S. The law PS has the very useful property that it is stationary under shifts along the spine, +which we now define. Each vertex v ∈ S has a unique oriented edge emanating from it in S→, +and we will write σ(v) for the vertex on the other end of this edge. The map v �→ σ(v) can be +thought of as a shift, following the orientation along the spine, and there is also a well-defined +backwards shift σ−1 mapping each x ∈ S to the unique vertex v ∈ S with σ(v) = x. +Lemma 16. The law PS is invariant under the shift σ. +Proof. Let A be any Borel set of triples (g, t→, v) where (g, v) is a rooted graph and t→ is an +oriented spanning tree of g, and define the mass transport +f(g, t→, v, w) := 1 (t→ is two-ended, w is in the spine of t→, v = σ(w), and (g, t→, w) ∈ A) . +Note that there only exists one vertex v such that v = σ(w) and, vice-versa, for each v in the +spine of t→ there is only one v in the spine of t→ such that σ(v) = o and v ∈ S. Therefore, +� +v∈V +f(G, T →, v, o) = 1 (T → is two-ended, o is in the spine of T →, and (G, T →, o) ∈ A) +and +� +v∈V +f(G, T →, v, o) = 1 (T → is two-ended, o is in the spine of T →, and (G, T →, σ(o)) ∈ A) +18 + +Using the mass-transport principle we thus have that +P (T → is two-ended, o is in the spine of T →, and (G, T →, o) ∈ A) += P (T → is two-ended, o is in the spine of T →, and (G, T →, σ(o)) ∈ A) +which shows the result because P(o ∈ S) > 0 and T is two-ended a.s. by assumption. +The main goal of this section is to show that along the spine of the UST, the effective +resistances on the original graph must grow linearly under the assumption that the UST has +two ends (and thus a well-defined spine). Heuristically, this tells us that if a graph is unimodular +and the uniform spanning tree is two-ended, then the actual graph should in some sense be +“close” to the line Z. +Proposition 17. The limit limn→∞ 1 +nReff(o ↔ σn(o)) = limn→∞ 1 +nReff(o ↔ σ−n(o)) exists +and is positive PS-a.s. +Note that the existence part of this proposition is an immediate consequence of the subad- +ditive ergodic theorem; the content of the proposition is that the limit is positive. +As discussed above, it follows from Proposition 12 and Theorem 14 that, PS-almost surely, +there are exactly two extremal elements of H, which we call “ℓ” and “r”, which satisfy +ar(σn(o), v) +aℓ(σn(o), v) → + + + +∞ +as n → +∞ +0 +as n → −∞ +(14) +for every v ∈ V . (In particular, the random choice of orientation of T we made when defining +PS is equivalent to randomly choosing which of the two extremal elements of H to call “r”.) +Consider the function V → R defined by +Mo(x) := ar(x, o) − aℓ(x, o). +We will show that Mo(σn(o)) grows linearly in n and deduce from this that the effective +resistance does too. The latter fact can be seen using (6), from which it follows that +Mo(x) = (rx,o(x) − ℓx,o(x))Reff(o ↔ x). +In the remainder we will slightly abuse notation to write Mm(n) := Mσm(o)(σn(o)) for n, m ∈ Z. +The first main ingredient is that Mo(n) is an additive cocyle. +Lemma 18. Mo(n + m) = Mo(n) + Mn(n + m) for every n, m ∈ Z. +Proof. This is a direct consequence of Proposition 3.5 in [8], stating that +a#(x, o) − a#(y, o) = a#(x, y) − Gy(x, o) +deg(o) +19 + +for each # ∈ {ℓ, r} and all x, y ∈ V . Indeed, it follows from this identity that +Mo(n + m) − Mo(n) = ar(σn+m(o), o) − aℓ(σn+m(o), o) − ar(σn(o), o) + aℓ(σn(o), o) += +� +ar(σn+m(o), σn(o)) − Gσn(o)(σn+m(o), o) +deg(o) +� +− +� +aℓ(σn+m(o), σn(o)) − Gσn(o)(σn+m(o), o) +deg(o) +� += ar(σn+m(o), σn(o)) − aℓ(σn+m(o), σn(o)) = Mn(n + m) +for every n, m ∈ Z as claimed. +Let us also make note of the following key property of this additive cocycle. +Lemma 19. PS-almost surely, Mo(n) is positive for all sufficiently large positive n and negative +for all sufficiently large negative n. Moreover, +Mo(n) ∼ ar(σn(o), o) = rσn(o),o(σn(o))Reff(o ↔ σn(o)) +PS-almost surely as n → ∞. +Proof. This follows immediately from (14) and the definition of Mo(n). +We will deduce Proposition 17 from Lemma 19 together with the following general fact +about stationary sequences. +Proposition 20. Let (Zi)i∈Z be a stationary sequence of real-valued random variables and sup- +pose that �n +i=0 Z−i > 0 for all sufficiently large n almost surely. Then lim supn→∞ +1 +n +�n +i=0 Zi > +0 almost surely. +Proof. For each n ∈ Z let Rn = inf{m ≥ 0 : �n+m +i=n Zi > 0}, so that Rn = 0 whenever Zn > 0 +and (Rn)n∈Z is a stationary sequence of {0, 1, . . .}-valued random variables. It follows from +the definitions that if n ≤ m then either n + Rn < m or n + Rn ≥ m + Rm, so that the +intervals [n, n + Rn] and [m, m + Rm] are either disjoint or ordered by inclusion. On the other +hand, we have by stationarity and the hypotheses of the Proposition that for each n ∈ Z +there almost surely exists Nn < ∞ such that �n−1 +i=n−m Zi > 0 for every m ≥ Nn and hence +that Rn−m + (n − m) < n for every m ≥ Nn, so that each n ∈ Z is contained in at most +finitely many of the intervals [m, m + Rm] almost surely. Using the fact that these intervals +are either disjoint or ordered by inclusion, it follows that there is a unique decomposition of Z +into maximal intervals of this form +Z = +�� +[k, k + Rk] : k ∈ Z, [k, k + Rk] ⊈ [m, m + Rm] for every m ∈ Z \ {k} +� +. +Thus, if we define Yn by +Yn = + + + +�n +i=k Zi +n = k + Rk for some k ∈ Z such that [k, k + Rk] maximal +0 +otherwise +20 + +then (Yn)n∈Z is a stationary sequence of non-negative random variables such that Yn is positive +whenever n is the right endpoint of a maximal interval. Since Yn is non-negative and the set +of n such that Yn ̸= 0 is almost surely non-empty, it follows from the ergodic theorem applied +to (min{Yn, 1})n∈Z that +lim inf +n→∞ +1 +n +n +� +i=0 +Yn > 0 +almost surely. +The claim follows since if −m is the left endpoint of the maximal interval +containing 0 then +n +� +i=0 +Yn = +n +� +i=−m +Zi +for every n that is the right endpoint of some maximal interval. +Proof. It follows from Lemma 16 that (Mn(n + 1))n∈Z is a stationary sequence under PS and +from Lemma 18 that Mo(n) = �n−1 +i=0 Mi(i+1) for every n ≥ 0 and Mo(−n) = �−1 +i=−n Mi(i+1) +for every n ≤ 0. Thus, Lemma 19 implies that the stationary sequence (Mn(n+1))n∈Z satisfies +the hypotheses of Proposition 20 and hence that +lim sup +n→∞ +Mo(n) +n +> 0 +almost surely. +On the other hand, the subadditive ergodic theorem implies that the limit +limn→∞ 1 +nReff(o ↔ σn(o)) exists PS-a.s., and since +Mo(n) = +� +rσn(o),o(σn(o)) − ℓσn(o),o(σn(o)) +� +Reff(o ↔ σn(o)) ≤ Reff(o ↔ σn(o)) +we must have that +lim +n→∞ +1 +nReff(o ↔ σn(o)) > 0 +PS-a.s. as claimed. The fact that the negative-n limit limn→∞ 1 +nReff(o ↔ σ−n(o)) also exists +and is equal to the positive-n limit a.s. follows from the subadditive ergodic theorem. +3.3 +Completing the proof +We now complete the proof of the main theorem. +Proof of Theorem 1. It suffices by Proposition 13 to prove that if (G, o) is a recurrent unimod- +ular random rooted graph whose UST is two-ended almost surely then G has linear volume +growth almost surely. As before, we write S for the spine of the oriented UST T →, write PS +for the conditional law of (G, T →, o) given that o ∈ S, and write σ for the shift along the spine +as in Lemma 16. +For each x ∈ V let S(x) be an element of S of minimal graph distance to x, choosing one +of the finitely many possibilities uniformly and independently at random for each x where this +21 + +point is not unique. Letting S−1(v) = {x ∈ V : S(x) = v} for each v ∈ S, we have by the +mass-transport principle that +ES|S−1(o)| = E +� +|S−1(o)| | o ∈ S +� += P(o ∈ S)−1E +�� +x∈V +1(o = S(x)) +� += P(o ∈ S)−1E +�� +x∈V +1(x = S(o)) +� += P(o ∈ S)−1 < ∞. +We thus have a stationary sequence of random variables (|S−1(σi(o))|)i∈Z with uniformly finite +mean, and the ergodic theorem implies that +lim +i→∞ +1 +2n +n +� +i=−n +|S−1(σi(o))| < ∞ +(15) +almost surely. 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Wilson, Generating random spanning trees more quickly than the cover time, Proceedings of the +twenty-eighth annual acm symposium on theory of computing, 1996, pp. 296–303. +[30] W. Woess, Random walks on infinite graphs and groups, Cambridge university press, 2000. +23 + +A +Uniqueness of the potential kernel implies one-endedness of +the UST, without finite expected degree +In this appendix we prove Theorem 15, which generalizes the theorem of Berestycki and the +first author concerning the equivalence of the UST being one-ended and uniqueness of the +harmonic measure from infinity to the case that the unimodular random rooted graph does +not necessarily have finite expected degree. A secondary purpose of this appendix is to give +a brief and self-contained account of those results of [8] that are needed for our main results. +Since recurrent graphs whose USTs are one-ended always have unique harmonic measure from +infinity [6, Theorem 14.2], it suffices to prove that the converse holds under the additional +assumption of unimodularity. Moreover, it suffices as usual to consider the case that (G, o) is +ergodic. +Suppose that (G, o) is an ergodic recurrent unimodular random rooted graph for which H is +a singleton almost surely. We write h for the unique element of H and a for the corresponding +potential kernel. For each c > 0 consider the event Ac = {lim supx→∞ hx,o(x) ≥ c} = {for each +ε > 0 there exist infinitely many vertices x with hx,o(x) ≥ c − ε}. As explained in detail in +[8, Lemma 5.3] (which concerns deterministic recurrent graphs), we have that +hx,o(x) ∼ hx,w(x) +as x → ∞ for each fixed w ∈ V , +(17) +which implies that Ac is re-rooting invariant. Since (G, o) was assumed to be ergodic we deduce +the following. +Lemma 21. Let (G, o) be an ergodic unimodular random rooted graph. If G is almost surely +recurrent with a uniquely defined harmonic measure from infinity then the event Ac has prob- +ability 0 or 1 for each c ∈ (0, 1). +The next lemma is proven in [8] using an argument that relies on reversibility (and hence +on the assumption E deg(o) < ∞). We give an alternative proof using Følner sequences that +works without this assumption. +Lemma 22. Let (G, o) be an ergodic unimodular random rooted graph. If G is almost surely +recurrent with a uniquely defined harmonic measure from infinity then the event A1/2 holds +almost surely. +Proof. It suffices to prove that Ac holds with positive probability for every c < 1/2. Since +(G, o) is recurrent, it follows from the results of [1, §8] that (G, o) is hyperfinite, meaning that +there exists a sequence of random subsets (ωn)n≥1 of E such that +1. Every component of the subgraph spanned by ωn is finite almost surely for each n ≥ 1. +2. ωn ⊆ ωn+1 for each n ≥ 1 and � +n≥1 ωn = E. +24 + +3. The random rooted edge-labelled graph (G, o, (ωn)n≥1) is unimodular. +Let n ≥ 1 and let Kn be the component of o in ωn. Then we have by the mass-transport +principle that +E +� +1 +|Kn| +� +x∈Kn +1 +� +hx,o(x) ≥ 1 +2 +�� += E +� +1 +|Kn| +� +x∈Kn +1 +� +hx,o(o) ≥ 1 +2 +�� +, +and since the sum of the two sides is at least 1 it follows that +E +� +1 +|Kn| +� +x∈Kn +1 +� +hx,o(x) ≥ 1 +2 +�� +≥ 1 +2 +and hence by Markov’s inequality that +P +���{x ∈ Kn : hx,o(x) ≥ 1 +2} +�� ≥ 1 +4|Kn| +� +≥ 1 − 4 +3E +� +1 +|Kn| +� +x∈Kn +1 +� +hx,o(x) < 1 +2 +�� +≥ 1 +3. +Since |Kn| → ∞ almost surely as n → ∞, it follows from this and Fatou’s lemma that +P(A1/2) ≥ P +���{x ∈ Kn : hx,o(x) ≥ 1 +2} +�� ≥ 1 +4|Kn| for infinitely many n +� +≥ 1 +3 +and hence by ergodicity that P(A1/2) = 1 as claimed. +Lemma 23. Let G = (V, E) be an infinite, connected, locally finite recurrent graph with +uniquely defined harmonic measure from infinity h, let o ∈ V and let a be the associated poten- +tial kernel. If A is any infinite set of vertices with infx∈A hx,o(x) > 0, the Doob-transformed +walk � +X visits A infinitely often almost surely. +Proof. We have by (11) that �Po( � +X hits x) = ho,x(x) for every x ∈ V , and it follows by Fatou’s +lemma that �P(hit A infinitely often) ≥ infx∈A hx,o(x) > 0. On the other hand, we have by +Theorem 8 and the assumption that h is unique that � +X has trivial tail σ-algebra, so that �P(hit +A infinitely often) = 1 as claimed. +Proposition 24. Let G = (V, E) be an infinite, connected, locally finite recurrent graph with +uniquely defined harmonic measure from infinity h, let o ∈ V , let a be the associated potential +kernel, and suppose that lim infx→∞ hx,o(x) > 0. If � +X and �Y are independent copies of the +Doob-transformed walk started at some vertices x and y, then { � +Xn : n ≥ 0} ∩ {�Yn : n ≥ 0} is +infinite almost surely. +Proof. Let δ > 0 be such that A = {x ∈ V : hx,o(x) ≥ δ} is infinite. Applying Lemma 23 +yields that � +X ∩ A is infinite almost surely, and applying Lemma 23 a second time yields that +�Y ∩ � +X ∩ A is infinite almost surely. +25 + +Applying this proposition together with the results of [23], which imply that an independent +Markov process and loop-erased Markov process intersect infinitely almost surely whenever +the corresponding two independent Markov processes do, we deduce the following immediate +corollary. +Corollary 25. Let G = (V, E) be an infinite, connected, locally finite recurrent graph with +uniquely defined harmonic measure from infinity h, let o ∈ V , let a be the associated potential +kernel, and suppose that lim infx→∞ hx,o(x) > 0. If � +X and �Y are independent copies of the +Doob-transformed walk started at some vertices x and y, then { � +Xn : n ≥ 0}∩{LE(�Y )n : n ≥ 0} +is infinite almost surely. +Proposition 26. Let G = (V, E) be an infinite, connected, locally finite recurrent graph with +uniquely defined harmonic measure from infinity h, let o ∈ V , let a be the associated potential +kernel, and suppose that lim infx→∞ hx,o(x) > 0. For each x ∈ V , let X be a random walk +started at x and let �Y be a Doob-transformed walk started at o. Then +lim +x→∞ P +� +{Xn : 0 ≤ n ≤ To} ∩ {LE(�Y )m : m ≥ 0} = {o} +� += 0. +Proof. As x → ∞, the law of the time-reversed final segment (XTo, XTo−1, . . . , XTo−k) con- +verges to that of ( � +X0, . . . , � +Xk) for each k ≥ 1, and the claim follows from Corollary 25. +Proof of Theorem 15. The fact that G has a unique harmonic measure from infinity means +that we can endow the uniform spanning tree of G with an orientation in a canonical way: +Suppose that we exhuast V by finite sets V = � Vn and let G∗ +n be defined by contracting V \Vn +into a single boundary vertex ∂n, so that the UST of G can be expressed as the weak limit of +the USTs of the graphs G∗ +n. If for each n ≥ 1 we orient the UST of G∗ +n towards the boundary +vertex ∂n to obtain an oriented tree T → +n , then the uniqueness of the harmonic measure from +infinity on G implies that the law of T → +n +converges weakly to the law of an oriented spanning +tree T → of G, which can be thought of as a canonical (but potentially random) orientation of +the UST of G. Indeed, if we fix an enumeration v1, v2, . . . of V with v1 = o we can sample +T → +n +using Wilson’s algorithm rooted at ∂n, starting with the vertices in the order they appear +in the enumeration of V , and orienting the edges of the tree in the direction they are crossed +by the loop-erased walk that contributed them to the tree. In the infinite-volume limit (since +only the part of the first walk after its final visit to o contributes to its loop erasure), this +corresponds to doing Wilson’s algorithm where the first walk started at o is Doob-transformed +and the remaining walks are ordinary simply random walks. +An important consequence of this discussion is that if we sample the oriented uniform +spanning tree using Wilson’s algorithm rooted at infinity, where the first random walk is a +Doob-transformed walk started at o and the remaining walks are ordinary simple random +walks, the distribution of the resulting oriented tree T → does not depend on the choice of the +root vertex o, since it is given by the limit of the USTs of G∗ +n oriented towards ∂n independently +26 + +of the choice of exhaustion. Given the oriented tree T →, we say that a vertex u is in the future +of a vertex v if the unique infinite oriented path emanating from v passes through v, and say +that u is in the past of v if v is in the future of u. +Let (ωn)n≥1 be a sequence witnessing the fact that (G, o) is hyperfinite as in the proof of +Lemma 22 and let Kn be the cluster of o in ωn for each n ≥ 1. We have by the mass-transport +principle that +E +� +1 +|Kn| +� +x∈Kn +1 (x in past of o) +� += E +� +1 +|Kn| +� +x∈Kn +1 (x in future of o) +� +. +On the other hand, letting S be the set of vertices belonging to a doubly infinite path in T, +we also have that +E +� +1 +|Kn| +� +x∈Kn +1 (x in past or future of o) +� +≥ E +� +1 +|Kn| +� +x∈Kn +1(o, x ∈ S) +� +and we can use the mass-transport principle again to bound +E +� +1 +|Kn| +� +x∈Kn +1(o, x ∈ S) +� += E + + +1 +|Kn|2 +� +x,y∈Kn +1(o, x ∈ S) + + = E + + +1 +|Kn|2 +� +x,y∈Kn +1(x, y ∈ S) + + += E +��|Kn ∩ S| +|Kn| +�2� +≥ E +�|Kn ∩ S| +|Kn| +�2 += P(o ∈ S)2. +Putting these two estimates together, it follows that +E +� +1 +|Kn| +� +x∈Kn +1 (x in past of o) +� +≥ 1 +2P(o ∈ S)2. +(18) +On the other hand, if we sample T → using Wilson’s algorithm rooted at infinity, starting with +a Doob-transformed �Y started at o followed by an ordinary random walk X started at x, the +vertex x belongs to the past of o if and only if the walk X first hits the loop-erasure of �Y at the +vertex o. Proposition 26 implies that this probability tends to zero as x → ∞,and it follows +by bounded convergence that +E +� +1 +|Kn| +� +x∈Kn +1 (x in past of o) +� +→ 0 +(19) +as n → ∞. Putting together (18) and (19) yields that P(o ∈ S) = 0. Since “everything that +can happen somewhere can happen at the root” [1, Lemma 2.3], it follows that S = ∅ almost +surely and hence that T is one-ended almost surely as claimed. +27 + diff --git a/cdE2T4oBgHgl3EQfagdb/content/tmp_files/load_file.txt b/cdE2T4oBgHgl3EQfagdb/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..92a0962cec8b9bf5071b3a2e4d8d49f941d7383b --- /dev/null +++ b/cdE2T4oBgHgl3EQfagdb/content/tmp_files/load_file.txt @@ -0,0 +1,695 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf,len=694 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content='03875v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content='PR] 10 Jan 2023 The number of ends in the uniform spanning tree for recurrent unimodular random graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Diederik van Engelenburg Tom Hutchcroft January 11, 2023 Abstract We prove that if a unimodular random rooted graph is recurrent, the number of ends of its uniform spanning tree is almost surely equal to the number of ends of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Together with previous results in the transient case, this completely resolves the problem of the number of ends of wired uniform spanning forest components in unimodular random rooted graphs and confirms a conjecture of Aldous and Lyons (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' 1 Introduction The uniform spanning tree of a finite connected graph G is defined by picking uniformly at random a connected subgraph of G containing all vertices but no cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' To go from finite to infinite graphs, it is possible to exhaust G by finite subgraphs and take weak limits with appropriate boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' For two natural such choices of boundary conditions, known as free and wired boundary conditions, Pemantle [24] proved that these infinite- volume limits are always well-defined independently of the choice of exhaustion, and that the choice of boundary conditions also does not affect the limit obtained when G = Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Since connectivity of a subgraph is not a closed condition, these weak limits might be supported on configurations that are forests rather than trees, and indeed Pemantle proved for Zd that the limit is connected if and only if d ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' For a general infinite, connected, locally finite graph the infinite-volume limit of the UST with free boundary conditions is called the free uniform spanning forest (FUSF) and the infinite volume limit with wired boundary conditions is called the wired uniform spanning forest (WUSF);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' when the two limits are the same we refer to them simply as the uniform spanning forest (USF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' In their highly influential work [6], Benjamini, Lyons, Peres and Schramm resolved the connectivity question for the WUSF in large generality: the wired uniform spanning tree is a single tree if and only if two random walks intersect infinitely often.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' The connectivity of the FUSF appears to be a much more subtle question and, outside of the case that the two forests are the same, is understood only in a few examples [3,18,25,27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' For recurrent graphs, which are the main topic of this paper, 1 the infinite-volume limit of the UST is always defined independently of boundary conditions and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' connected [6, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content='6], so that we can unambiguously refer to the uniform spanning tree (UST) of an infinite, connected, locally finite, recurrent graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' After connectivity, the next most basic topological property of the USF is the number of ends its components have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Here, we say that a graph has at least m ends whenever there exists some finite set of vertices W such that G\\W has at least m infinite connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' The graph is said to be m-ended if at has at least m but not m+1 ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Understanding the number of ends of the USF turns out to be rather more difficult than connectivity, with a significant literature now devoted to the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' For Cayley graphs, it follows from abstract principles [3, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content='4] that every component has 1, 2, or infinitely many ends almost surely, and for amenable Cayley graphs such as Zd (for which the WUSF and FUSF always coincide) is follows by a Burton-Keane [10] type argument that every component has either one or two ends almost surely;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' see [22, Chapter 10] for detailed background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' For the wired uniform spanning forest on transitive graphs, a complete solution to the problem was given by Benjamini, Lyons, Peres, and Schramm [6] and Lyons, Morris, and Schramm [21], who proved that every component of the WUSF of an infinite transitive graph is one-ended almost surely unless the graph in question is rough-isometric to Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Before going forward, let us emphasize that the recurrent case of this result [6, Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content='6] is established using a completely different argument to the transient case, with the tools available for handling the two cases being largely disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Beyond the transitive setting, various works have established mild conditions under which every component of the WUSF is one-ended almost surely, applying in particular to planar graphs with bounded face degrees [18] and graphs satisfying isoperimetric conditions only very slightly stronger than transience [15,21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' These proofs are quantitative, and recent works studying critical exponents for the USF of Zd with d ≥ 3 [2,16,19] and Galton-Watson trees [20] can be thought of as a direct continuation of the same line of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' In parallel to this deterministic theory, Aldous and Lyons [1] observed that the methods of [6] also apply to prove that the WUSF has one-ended components on any transient unimodular random rooted graph of bounded degree, and the second author later gave new proofs of this result with different methods that removed the bounded degree assumption [14,15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' It is also proven in [17, 28] that every component of the free uniform spanning forest of a unimodular random rooted graph is infinitely ended a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' whenever the free and wired forests are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Here, unimodular random rooted graphs comprise a very large class of random graph models including Benjamini-Schramm limits of finite graphs [7], Cayley graphs, and (suitable versions of) Galton-Watson trees, as well as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' percolation clusters on such graphs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' See Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content='1 for definitions and e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' [1,11] for detailed background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' The aforementioned works [1,14,15,17,28] completely resolved the problem of the number of ends of the WUSF and FUSF for transient unimodular random rooted graphs, but the recurrent case remained open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Besides the fact that the transient methods do not apply, a 2 further complication of the recurrent case is that it is possible for the UST to be either one- ended or two-ended according to the geometry of the graph: indeed, the UST of Z2 is one-ended while the UST of Z is two-ended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Aldous and Lyons conjectured [1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' 1485] that the dependence of the number of ends of the UST on the geometry of the graph is as simple as possible: The UST of a recurrent unimodular random rooted graph is one-ended if and only if the graph is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' The fact that two-ended unimodular random rooted graphs have two-ended USTs is trivial;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' the content of the conjecture is that one-ended unimodular random rooted graphs have one-ended USTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Previously, the conjecture was resolved under the assumption of planarity in [3], while in [8] it was proved (without using the planarity assumption) that the UST of a recurrent unimodular random rooted graph is one-ended precisely when the “harmonic measure from infinity” is uniquely defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' In this paper we resolve the conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Let (G, o) be a recurrent unimodular random rooted graph and let T be the uniform spanning tree of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Then T has the same number of ends as G a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' To see that the theorem is not true without unimodularity, consider taking the line graph Z and adding a path of length 2n connecting −n connecting to n for each n, making the graph one-ended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Kirchoff’s effective resistance formula implies that the probability that the additional path connecting −n to n is included in the UST is at most n/(2n +n), and a simple Borel-Cantelli argument implies that the UST is two-ended almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Similar examples show that Theorem 1 does not apply to unimodular random rooted networks, since we can use edges of very low conductance to make the network one-ended while having very little effect on the geometry of the UST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' About the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' We stress again that the tools used in the transient case do not apply at all to the recurrent case, and we are forced to use completely different methods that are specific to the recurrent case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' We build on [8] which proved that the “harmonic measures from infinity” are uniquely defined if and only if the uniform spanning tree is one-ended;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' A self-contained treatment of (a slight generalization of) the results of [8] that we will need is given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' The set of harmonic measures from infinity can be thought of as a “boundary at infinity” for the graph, analogously to the way the Martin boundary is used in transient graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' It is implicit in [8] that these measures correspond to the ways in which a random walk “conditioned to never return to the root” can escape to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' We develop these ideas further in Section 2, in which we make this connection precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' We then apply these ideas inside an ergodic-theoretic framework to prove that if the UST has two ends, then the effective resistance must grow linearly along the unique bi-infinite path in the tree, which implies in particular that graph distances must also grow linearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' To conclude, we argue that this can only happen when the graph has linear volume growth, which is known to be equivalent to two-endedness for unimodular random rooted graphs [5,9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' 3 Acknowledgments DvE wishes to thank Nathana¨el Berestycki for many useful discussions and comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Most of the research was carried out while DvE visited TH at Caltech and we are grateful for the hospitality of the institute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' DvE is supported by the FWF grant P33083, “Scaling limits in random conformal geometry”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' 2 Boundary theory of recurrent graphs In this section we develop the theory of harmonic measures from infinity on recurrent graphs, their associated potential kernels and Doob transforms, and how this relates to the spanning tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Much of the theory we develop here is a direct analogue for recurrent graphs of the theory of Martin boundaries of transient graphs [12, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' This theory is interesting in its own right, and we were surprised to find how little attention has been paid to these notions outside of some key motivating examples such as Z2 [13,26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' All of the results in this section will concern deterministic infinite, connected, recurrent, locally finite graphs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' applications of the theory to unimodular random rooted graphs will be given in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content='1 Harmonic measures from infinity Let G = (V, E) be an infinite, connected, locally finite, recurrent graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' For each v ∈ V we write Pv for the law of the simple random walk on G started at v, and for each set A ⊆ V write TA and T + A for the first visit time of the random walk to A and first positive visit time of the random walk to A respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Given a probability measure µ on V , we also write Pµ for the law of the random walk started at a µ-distributed vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' A harmonic measure from infinity h = (hB : B ⊂ V finite) on G is a collection of probability measures on V indexed by the finite subsets B of V with the following properties: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' hB is supported on ∂B for each B ⊂ V , where ∂B is the set of elements of B that are adjacent to an element of V \\ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' For each pair of finite sets B ⊆ B′, hB and hB′ satisfy the consistency condition hB(u) = � v∈B′ hB′(v)Pv(XTB = u) (1) for every u ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' We denote the space of harmonic measures from infinity by H, which (identifying the measures hB with their probability mass functions) is a compact convex subset of the space of functions {finite subsets of V } → RV when equipped with the product topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' As mentioned above, the space H plays a role for recurrent graphs analogous to that played by the Martin boundary for transient graphs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' the analogy will become clearer once we introduce potential kernels in 4 the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' We say that the harmonic measure from infinity is uniquely defined when H is a singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' If µn is a sequence of probability measures on V converging vaguely to the zero measure in the sense that µn(v) → 0 as n → ∞ for each fixed v ∈ V then any subsequential limit of the collections (Pµn(XTB = ·) : B ⊂ V finite) belongs to H, with these collections themselves satisfying every property of a harmonic measure from infinity other than the condition that hB is supported on ∂B for every finite B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' (Indeed, the consistency condition (1) follows from the strong Markov property of the random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=') In fact every harmonic measure from infinity can be written as such a limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' If h ∈ H is a harmonic measure from infinity then there exists a sequence of finitely supported probability measures (µn)n≥1 on V such that µn(v) → 0 for every v ∈ V and hB(·) = lim n→∞ Pµn(XTB = · ) for every B ⊂ V finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' (2) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Fix h ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Let V1 ⊂ V2 ⊂ V3 · · · be an increasing sequence of finite subsets of V with � i Vi = V , and for each n ≥ 1 let µn = hVn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' It follows from the consistency condition (1) that hB(·) = Pµn(XTB = · ) for every B ⊂ Vn, and the claim follows since every finite set is eventually contained in Vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Since H is a weakly compact subspace of the set of functions from finite subsets of V to RV , which is a locally convex topological vector space, it is a Choquet-simplex: Every element can be written as a convex combination of the extremal points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' In particular, if H has more than one point then it must have more than one extremal point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' This will be useful to us because extremal points of H are always limits of harmonic measures from sequences of single vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Indeed, identifying each vertex v ∈ V with the collection of harmonic measures (Pv(XTB = ·) : B ⊂ V finite) allows us to think of V ∪H as a compact Polish space containing V (in which V might not be dense), and we say that a sequence of vertices (vn)n≥0 converges to a point h ∈ H if hB(·) = limn→∞ Pvn(XTB = · ) for every B ⊂ V finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' If h ∈ H is extremal, there exists a sequence of vertices (vn)n≥0 such that vn converges to h as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Let I be the set of functions h : {B ⊂ V finite} → RV of the form hB(·) = Pµ(XTB = ·) for every B ⊂ V finite for some finitely supported measure µ on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Lemma 2 implies that I = I ∪ H is a compact convex subset of the space of all functions {B ⊂ V finite} → RV equipped with the product topology, which is a locally convex topological vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' By the Krein-Milman theorem, a subset W of I ∪ H has closure containing the set of extremal points of I ∪ H if and only 5 if I ∪ H is contained in the closed convex hull of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Thus, if we define Iext to be the set of functions h : {B ⊂ V finite} → RV of the form hB(·) = Pz(XTB = ·) for every B ⊂ V finite for some z ∈ V then I is clearly contained in the convex hull of Iext, so that I ∪H is contained in the closed convex hull of Iext and, by the Krein-Milman theorem, the set of extremal points of I ∪ H is contained in the closure of Iext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Now, observe that for any non-trivial convex combination of an element of I and an element of H, there must exist a finite set of vertices B and a point z in the interior of B (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=', in B and not adjacent to any element of V \\ B) such that hB(z) ̸= 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' indeed, if the element of I corresponds to some finitely supported measure µ, then any B containing the support of µ in its interior and any z in the support of µ will do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Since no element of H can have this property, it follows that non-trivial convex combinations of elements of I and H cannot belong to H and hence that extremal points of H are also extremal in I ∪ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' It follows that the set of extremal points of H is contained in the closure of Iext, which is equivalent to the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' The converse to this lemma is not true: A limit of a sequence of Dirac measures need not be extremal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' For example, if we construct a graph from Z by attaching a very long path between −n and n for each n ≥ 1 and take zn to be a point in the middle of this path for each n, the sequence (zn)n≥1 will converge to a non-extremal element of H that is the convex combination of the limits of (n)n≥1 and (−n)n≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content='2 Potential kernels and Doob transforms The arguments in [8] heavily rely on a correspondence between the harmonic measure from infinity and its potential kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' One important feature of the potential kernel is that, given a vertex o ∈ V and a point h ∈ H, it provides a sensible way to “condition the random walk to converge to h before returning to o”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' We begin by discussing how conditioning the random walk to hit a particular vertex before returning to o can be described in terms of Doob transforms before developing the analogous limit theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Doob transforms and non-singular conditioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Suppose that we are given two distinct vertices o and z in an infinite, connected, locally finite recurrent graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Letting Gz(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' y) be the expected number of times a random walk started at x visits y before hitting z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' we can compute that the function a(x) = Gz(o,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' o) deg(o) − Gz(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' o) deg(o) = Px(To > Tz)Gz(o,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' o) deg(o) is harmonic at every vertex other than o and z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' and has ∆a(o) = 0 − deg(o)Eo[a(X1)] = −Po(T + o > Tz)Gz(o,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' o) = −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' 6 where ∆ denotes the graph Laplacian ∆f(x) = deg(x)f(x) − � y∼x f(y) = deg(x)Ex[f(X0) − f(X1)] (terms in this sum are counted with appropriate multiplicity if there is more than one edge between x and y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Moreover, the quantity a(x) is strictly positive at every vertex x that is neither equal to o nor disconnected from z by o in the sense that every path from x to z must pass through o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Observe that the trivial identity Po((X0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' , Xn) = (x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' , xn)) = n � i=1 p(xi−1, xi) = 1 a(xn)a(x1)p(o, x1) n � i=2 a(xi) a(xi−1)p(xi−1, xi) (3) holds for every sequence of vertices x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' , xn with x0 = o and a(xi) > 0 for every i > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Since a(z) = Gz(o, o) = Po(Tz < T + o )−1 it follows that Po((X0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' , Xn) = (x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' , xn) | Tz < T + o ) = a(x1)p(o, x1) n � i=2 a(xi) a(xi−1)p(xi−1, xi) (4) for every sequence of vertices x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' , xn with x0 = o, xn = z, and xi /∈ {o, z} for every 0 < i < n (which implies that a(xi) > 0 for every 1 ≤ i ≤ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' the fact that a is harmonic off of {o,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' z} and has ∆a(o) = −1 implies that we can define a stochastic matrix with state space {x ∈ V : x = o or a(x) > 0} by �pa(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' y) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 a(y) a(x)p(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' y) x /∈ {o,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' z} a(y) x = 0 1(y = z) x = z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' and if we define the Doob transformed walk � Xa to be the Markov chain with this transition matrix started from o then it follows from (4) that ( � Xa n)Tz n=0 has law equal to the conditional law of the simple random walk (Xn)Tz n=0 started at o and conditioned to hit z before returning to o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Moreover, letting �Pa o denote the law of � Xa, it follows from the definition of � Xa that �Pa o � ( � Xa 0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' , � Xa n) = (x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' , xn) � = n � i=1 �pa(xi−1, xi) = a(x1) n � i=2 a(xi) a(xi−1)p(xi−1, xi) = a(xn) n � i=2 p(xi−1, xi) = deg(o)a(xn)Po ((X0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' , Xn) = (x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' , xn)) (5) for every sequence x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' , xn with x0 = o and xi /∈ {o, z} for every 0 < i < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Defining the potential kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' We now define the potential kernel ah associated to a point h ∈ H via the formula ah(x, y) = hx,y(x)Reff(x ↔ y) (6) 7 where we write hx,y = h{x,y}, so that ah(x, x) = 0 for each x ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' The fact that this is a sensible definition owes largely to the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' For each h ∈ H, the potential kernel ah(x, y) = hx,y(x)Reff(x ↔ y) satisfies ∆ah( · , y) = −1( · = y), (7) so that the potential kernel ah(·, y) is harmonic away from y and subharmonic at y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Since the map h �→ ah is affine and the equality (7) is linear, it suffices to prove the lemma in the case that h is extremal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' By Lemma 3, there exists a sequence of vertices (vn)n≥1 such that vn converges to h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' For each n ≥ 1 we define an(x, y) = Gvn(y, y) deg(y) − Gvn(x, y) deg(y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' and claim that ah(x, y) = lim n→∞ an(x, y) (8) for every x, y ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' (Note that this limit formula is often taken as the definition of the potential kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=') We will prove (8) with the aid of three standard identities for the Greens function: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' By the strong Markov property, Gz(x, y) is equal to Px(Ty < Tz) Gz(y, y) for every three distinct vertices x, y, and z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' By the strong Markov property, Gx(y, y) is equal to Px(Ty < T + x )−1 for every pair of distinct vertices x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' It follows in particular that deg(y)−1 Gx(y, y) = Reff(x ↔ y) and, since the effective resistance is symmetric in x and y, that deg(y)−1 Gx(y, y) = deg(x)−1 Gy(x, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' By time-reversal, deg(x) Gz(x, y) is equal to deg(y) Gz(y, x) for every three distinct ver- tices x, y, and z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Applying these three identities in order yields that an(x, y) = Gvn(y, y) deg(y) Px(Ty > Tvn) = Gy(vn, vn) deg(vn) Px(Ty > Tvn) = Gy(x, vn) deg(vn) = Gy(vn, x) deg(x) whenever x, y, and vn are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Applying the first and second identities a second time then yields that an(x, y) = Pvn(Tx < Ty)Gy(x, x) deg(x) = Pvn(Tx < Ty)Reff(x ↔ y) (9) whenever x, y, and vn are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' This is easily seen to imply the claimed limit formula (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' 8 In light of this lemma, we define Po to be the space of non-negative functions a : V → [0, ∞) with a(o) = 0 and ∆a(x) = −1(x = o), so that ah( · , o) belongs to Po for each o ∈ V and h ∈ H by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' We will later show that the map h �→ ah is an affine isopmorphism between the two convex spaces H and Po.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' We first describe how elements of Po can be used to define Doob transformed walks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Doob transforms and singular conditioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' We now define the Doob transform asso- ciated to an element of the space Po.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Given a ∈ Po, we define � Xa to be the Doob a-transform of the simple random walk X on G, so that � Xa has state space {x ∈ V : x = o or a(x) > 0} and transition probabilities given by �p a(x, y) := \uf8f1 \uf8f2 \uf8f3 a(y) a(x)p(x, y) if x ̸= o a(y) if x = o, y ∼ o where p is the transition kernel for the simple random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Similarly, given h ∈ H, we write � Xh = � Xah(·,o) where ah is the potential kernel associated to h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Informally, we think of � Xh as the walk that is “conditioned to go to h before returning to o”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' (In particular, when the harmonic measure from infinity is unique and H and Po are singleton sets, we think of the associated Doob transform as the random walk conditioned to never return to o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=') We write �Pa o or �Ph o for the law of � Xa or � Xh as appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' As before, it follows from this definition that if a ∈ Po and we write X[0, m] for the initial segment consisting of the first m steps of the random walk X then �Pa o( � Xa[0, m] = γ) = m � i=1 �p a(γi−1, γi) = a(γ1) m � i=2 a(γi) a(γi−1)p(γi−1, γi) = a(γm) m � i=2 p(γi−1, γi) = deg(o)a(γm)Po(X[0, m] = γ) (10) for every finite path γ = (γ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' , γm) with γ0 = o and γi ̸= o for every i > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Summing over all paths γ that begin at o, end at some point x ̸= o, and do not visit o or x at any intermediate point yields in particular that if h ∈ H then �Ph o( � X hits x) = deg(o)ah(x, o)Po(Tx < T + o ) = ho,x(x), (11) where the last equality follows from (6) and the definition of the effective resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Let G = (V, E) be a recurrent graph and let a ∈ Po.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Then the associated Doob- transformed walk � Xa is transient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' One can easily verify from the definitions that the sequence of reciprocals (a( � Xa n)−1)n≥1 is a non-negative martingale with respect to its natural filtration, and hence converges almost 9 surely to some limiting random variable, which it suffices to prove is zero almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' It follows from the identity (10) that �Pa o(a( � Xa n) ≤ M) = � v 1(a(v) ≤ M) deg(o)a(v)Po(Xn = v, T + o > n) ≤ M deg(o)Po(T + o > n), for every n, M ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Since G is recurrent, the right hand side tends to zero as n → ∞ for each fixed M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' It follows that lim supn→∞ a( � Xa n) = ∞ almost surely, and hence that limn→∞ a( � Xa n) = ∞ almost surely since the limit is well-defined almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' This implies that � Xa is transient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content='3 An affine isomorphism Let G = (V, E) be recurrent, fix o ∈ V , and let Po denote the set of positive functions a : V → [0, ∞) with a(o) = 0 that satisfy ∆a(·) = −1( · = o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' As we have seen, for each h ∈ H the potential kernel ah(·, o) defines an element of Po.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Moreover, the map sending h �→ ah( · , o) is affine in the sense that if h = θh1 + (1 − θ)h2 then ah( · , o) = θah1( · , o) + (1 − θ)ah2( · , o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' We wish to show that this map defines an affine isomorphism between H and Po in the sense that it is bijective (in which case its inverse is automatically affine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' We begin by constructing the inverse map from Po to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Let G = (V, E) be a infinite, connected, locally finite recurrent graph and let o ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' For each a ∈ Po there exists a unique h ∈ H satisfying hB(u) = �Pa o( � Xa visits B for the last time at u) for every finite set B containing o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Moreover, this h satisfies ah(x, o) = a(x) for every x ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Fix a ∈ Po.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' We define a the family of probability measures h = (hB : B ⊂ V finite) by hB(u) = �Pa o( � Xa visits B for the last time at u) for every u ∈ B if o ∈ B and hB(u) = �Pa o( � Xa visits B ∪ {o} for the last time at u) + �Pa o( � Xa visits B ∪ {o} for the last time at o)Po(XTB = u) for every u ∈ B if o /∈ B, so that if o /∈ B then hB(u) = � v∈B∪{o} hB∪{o}(v)Pv(XTB = u) for every u ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' We claim that this defines an element of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' It is clear that hB is a probability measure that is supported on ∂B for each finite set B ⊂ V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' we need to verify that it satisfies 10 the consistency property (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Once it is verified that h ∈ H, the fact that a = ah(·, o) follows easily from the definition of ah together with the identity (10), which together yield that ah(v, o) = hv,o(v)Reff(v ↔ o) = �Pa o( � Xa visits {o, v} for the last time at v) deg(o)Po(Tv < T + o ) = �Pa o( � Xa hits v) deg(o)Po(Tv < T + o ) = deg(o)a(v)Po(Tv < T + o ) deg(o)Po(Tv < T + o ) = a(v) for each v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' We now prove that h satisfies the consistency property (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' We will prove the required identity in the case o ∈ B, the remaining case o /∈ B following from this case and the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Let B ⊆ B′ be finite sets with o ∈ B and let (Vn)n≥1 be an exhaustion of V by finite sets such that B′ ⊆ Vn for every n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Writing V c n = V \\ Vn for each n ≥ 1 and τn for the first time the walk visits V c n, we have that hB(u) = lim n→∞ �Pa o( � X[0, τn] last visits B at u) = lim n→∞ � b∈V c n �Pa o( � X[0, τn] last visits B at u, � Xτn = b) and hence by (10) and time-reversal that hB(u) = lim n→∞ � b∈V c n deg(o)a(b)Po(X[0, τn] last visits B at u, Xτn = b) = lim n→∞ � b∈V c n deg(b)a(b)Pb(XTB = u, To < T + V c n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' (12) It follows from this together with the strong Markov property that hB(u) = lim n→∞ � v∈B′ � b∈V c n deg(b)a(b)Pb(XT ′ B = v, XTB = u, To < T + V c n ) = lim n→∞ � v∈B′ � b∈V c n deg(b)a(b)Pb(XT ′ B = v, TB′ < T + V c n )Pv(XTB = u, To < T + V c n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Now, we have by the strong Markov property that for each b ∈ V c n and v ∈ B′ Pb(XTB′ = v, To < T + V c n ) = Pb(XTB′ = v, TB′ < T + V c n )Pv(TV c n > To).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' and by recurrence that limn→∞ Pv(To < T + V c n ) = 1, so that hB(u) = lim n→∞ � v∈B′ � b∈V c n deg(b)a(b)Pb(XT ′ B = v, To < T + V c n )Pv(XTB = u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' The claimed identity (1) follows from this together with the identity (12) applied to the larger set B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' 11 Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Let G be an infinite, recurrent, locally finite graph, and let o ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' The map h �→ ah(·, o) is an affine isomorphism H → Po.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' In particular, this map identifies extremal elements of H with extremal elements of Po.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' It remains only to prove that h �→ ah is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' To prove this it suffices by definition of ah to prove that hB is determined by (hx,o(x) : x ∈ ∂B) for each finite set B ⊂ V containing the vertex o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Fix one such set B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' We have by definition of H that hx,o(x) = � y∈∂B hB(y)Py(Tx < To) = � y∈∂B A(x, y)hB(y) for each x ∈ ∂B where A(x, y) := Py(Tx < To) for each x, y ∈ ∂B, so that it suffices to prove that the matrix A (which is indexed by ∂B) is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Define a matrix Q indexed by ∂B by Q(x, y) = Py(T + ∂B < To, XT + ∂B = x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Then we have by the strong Markov property that A(x, y) − 1(x = y)Px(T + x ≥ To) = Py(T + x < To) = � z∈∂B Pz(Tx < To)Q(z, y) = Px(T + x ≥ To)Q(x, y) + � z∈∂B Pz(T + x < To)Q(z, y) and hence inductively that Py(T + x < To) = Px(T + x ≥ To) n � i=1 Qn(x, y) + � z∈∂B Pz(T + x < To)Qn(z, y) for every n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Since Q is irreducible and substochastic, we can take the limit as n → ∞ to obtain that A(x, y) = 1(x = y)Px(T + x ≥ To) + Py(T + x < To) = Px(T + x ≥ To) ∞ � i=0 Qn(x, y) for every x, y ∈ ∂B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' It follows by a standard argument that the matrix A is invertible with inverse A−1 = Px(T + x ≥ To)−1(1 − Q) as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content='4 The Liouville property for extremal Doob transforms In this section we prove a kind of tail-triviality property of the Doob-transformed walk cor- responding to an extremal point h ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Letting G = (V, E) be a graph, we recall that an event A ⊆ V N is said to be invariant if (x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=') ∈ A implies that (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=') ∈ A for every (x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=') ∈ V N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Let G = (V, E) be an infinite, connected, recurrent, locally finite graph and let o ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' If h ∈ H is extremal then the Doob transformed random walk ˆXh does not have any non-trivial invariant events: If A ⊆ V N is an invariant event then �Ph o(A) ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' 12 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' It suffices to prove the corresponding statement for � Xa when a is an extremal element of Po.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Suppose not, so that A is a non-trivial invariant event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' We have by Levy’s 0-1 law that Po( � Xa ∈ A | � Xa 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' , � Xa n) → 1( � Xa ∈ A) almost surely as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' (13) Moreover, we also have by invariance that �Pa x( � Xa ∈ A) = � y∈V a(y) a(x)p(x, y)�Pa y( � Xa ∈ A) and that �Pa o( � Xa ∈ A) = � y∈V a(y)�Pa y( � Xa ∈ A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Since similar inequalities hold when we replace A by Ac it follows that we can write a as a non-trivial convex combination of two elements of Po a(x) = �Pa o( � Xa ∈ A) · a(x)�Pa x( � Xa ∈ A) �Pao( � Xa ∈ A) + �Pa o( � Xa /∈ A) · a(x)�Pa x( � Xa /∈ A) �Pao( � Xa /∈ A) , these two factors being different by (13), contradicting extremality of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Underlying this proposition is the fact that once we fix a ∈ Po, we can identify Po with the Martin boundary of the conditioned walk � Xa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Theorem 8 is the recurrent version of the fact that Doob transforming by an extremal element of the Martin boundary yields a process with trivial invariant sigma-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' For our purposes, the most important output of the Liouville property is the following proposition, which lets us easily tell apart the trajectories of two different Doob transformed walks � Xh and � Xh′ by looking at any infinite subset of their traces (and, in particular, from their loop-erasures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Let h, h′ be distinct extremal elements of H and let � Xh be the Doob-transformed simple random walk corresponding to h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Then ah′( � Xh n, o) ah( � Xhn, o) → 0 almost surely as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' We prove the corresponding statement in which a, a′ are distinct extremal elements of Po.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Let � X and � X′ have laws �Pa o and �Pa′ o respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' One can easily verify from the definitions that (Zn)n≥1 = � a′( � Xn) a( � Xn) � n≥1 and (Z′ n)n≥1 = � a( � X′ n) a′( � X′n) � n≥1 are both non-negative martingales with respect to their natural filtrations, and hence converge almost surely to some limiting random variables Z and Z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Since Z and Z′ are measurable with 13 respect to the invariant σ-algebras of � X and � X′ respectively and a and a′ are both extremal, there must exist constants α and α′ such that Z = α and Z′ = α′ almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' We also have that EZn = EZ′ n = 1 for every n ≥ 1 and hence that α, α′ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' We wish to prove that α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' It follows from (10) that the conditional distributions of the initial segments � X[0, m] and � X′[0, m] are the same if we condition on � Xm = � X′ m = v for any v ∈ V for any v ∈ V and m ≥ 1 and that Pa o( � Xm = v) Pa′ o ( � Xa′ m = v) = a(v) a′(v) for every m ≥ 1 and v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' If α > 0 then for every ε > 0 there exists M such that the distribution of � Xm puts mass at least 1 − ε on the set of vertices with a′(v)/a(v) ≥ (1 − ε)α for every m ≥ M, and it follows that for each m ≥ M there is a coupling of the two walks � X′ and � X so that their initial segments of length m coincide with probability at least (1 − ε)2α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Taking a weak limit as m → ∞ and ε → 0, it follows that there exists a coupling of the two walks � X′ and � X such that the two walks coincide forever with probabilty at least α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' If we couple the walks in this way then on this event we must have that Z′ = 1/Z, which can occur with positive probability only if α′ = 1/α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Since α, α′ ≤ 1 we must have that α = α′ = 1 and that we can couple the two walks to be exactly the same almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' This is clearly only possible if a = a′, and since a ̸= a′ by assumption we must have that α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content='5 Potential kernels and the uniform spanning tree We now use Lemma 11 to show that the UST of a recurrent graph can always be sampled using a variant of Wilson’s algorithm [6,29] in which we ‘root at a point in H’, where again we are thinking intuitively of H as a kind of boundary at infinity of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Fix h ∈ ex(H) and let � Xh be the conditioned walk of the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Fix some enumeration V = {v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content='} of V with v1 = o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Set E0 = LE( � Xh[0, ∞)) (which is well defined because � Xh is transient) and for each i ≥ 1 define Ei given Ei−1 recursively as follows: if vi ∈ Ei−1, set Ei = Ei−1 otherwise, set Ei = Ei−1 ∪ LE(Y [0, τ)) where Y is the simple random walk started at vi and stopped at τ, the hitting time of Ei−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Last, define T = �∞ i=0 Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' We refer to this procedure as Wilson’s algorithm rooted at h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' The random tree T generated by Wilson’s algorithm rooted at h is clearly a spanning tree of G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' the next lemma shows that it is distributed as the UST of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Lemma 10 (Wilson meets Doob).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Let G = (V, E) be an infinite, connected, locally finite, recurrent graph and let h ∈ ext(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' The tree T generated by Wilson’s algorithm rooted at h is distributed as the uniform spanning tree of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' In particular, the law of T is independent of the chosen enumeration of V and the choice of h ∈ ext(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' 14 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' It follows by taking convex combinations that the same statement also holds when h is not extremal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' We will deduce Lemma 10 from the following lemma, which allows us to think of the Doob- transformed walk � Xh as a limit of conditioned simple random walks on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' For the purposes of this lemma we think of our walks as belonging to the space of sequences in V equipped with the product topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Lemma 11 (Local convergence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Let G = (V, E) be an infinite, connected, locally finite, recurrent graph and suppose that zn is a sequence of vertices of G such that zn converges to h ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' If X denotes the random walk on G started at o and � Xh denotes the Doob- transformation of X as above, then the conditional law of X given that it hits zn before first returning to o converges weakly to the law of � Xh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Proof of Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' This is a classical result concerning Doob transforms, and can also be deduced from the limit formula (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' We give a brief proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Let Tzn be the first time the walk hits zn, let T + o be the first positive time the walk hits o, and let ϕ = (o, ϕ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' , ϕm) be a path of length m for some m ≥ 1 with ϕi ̸= o for every i > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' By the Markov property for the simple random walk, Po(X[0, m] = ϕ, Tzn < T + o ) = Po(X[0, m] = ϕ)Pϕm(Tzn < To), and it follows from (10) that Po(X[0, m] = ϕ, Tzn < T + o ) = 1 deg(o)ah(ϕm, o)Po( � Xh[0, m] = ϕ)Pϕm(Tzn < To).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' The result follows once multiplying both sides by the effective resistance between o and zn and using the representation (6) for the potential kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Proof of Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' The standard implementation of Wilson’s algorithm rooted at zn allows us to sample the uniform spanning tree of G in a manner exactly analogous to above, except that we start with a walk run from o until it first hits zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Now, it is a combinatorial fact that the loop erasure of the walk run from o until it first hits zn does not change its distribution if we condition the walk to hit zn before returning to o: Indeed, the loop-erasure of the entire unconditioned walk is equal to the loop-erasure of the final segment of the walk between its last visit to o and its first visit to zn, and this final segment is distributed as the conditioned walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Thus, in the standard implementation of Wilson’s algorithm, we do not change the distribution of the obtained tree if we condition the first walk to hit zn before returning to o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' The claim then follows by taking the limit as zn → ∞ and using Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' This leads to the following connection between the ends of the UST and the extremal points of the set of harmonic measures from infinity H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' 15 Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Let G = (V, E) be an infinite, connected, locally finite, recurrent graph, let T be the uniform spanning tree of T, and let H be a countable subset of ext(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Almost surely, for each h ∈ H there exists an infinite simple path Γ = (Γ1, Γ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=') in T such that ah′(Γi, x) ah(Γi, x) → 0 as i → ∞ for each h′ ∈ H \\ {h}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' In particular, T almost surely has at least as many ends as there are extremal points of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' (In the last sentence of this proposition we are not distinguishing between different infinite cardinalities, but merely claiming that if H has infinitely many extremal points then T has infinitely many ends almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=') Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' This is an immediate consequence of Proposition 9 and Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Let G = (V, E) be an infinite, connected, locally finite, recurrent graph and let h ∈ ext(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' When we generate the UST T of G using Wilson’s algorithm rooted at h, the algorithm also provides a natural orientation of T, where each edge is oriented in the direction that it is crossed by the loop-erased random walk that contributed that edge to the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' When T almost surely has the same number of ends as there are extremal points in H, and both numbers are finite (which will always be the case in the unimodular setting by the results of [8]), it follows from Proposition 12 that this orientation is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' determined by the (unoriented tree): Almost surely, for each h ∈ H and v ∈ V there is a unique infinite ray (Γ1, Γ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=') starting from v such that ah′(Γi, v) ah(Γi, v) → 0 as i → ∞ for each h′ ∈ ext(H) \\ {h}, and if we orient the tree in the direction of this ray we must recover the same orientation as if we had generated the oriented tree using Wilson’s algorithm rooted at h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' This fact will play a key role in the proof of our main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' 3 Proof of the main theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content='1 Reversible and unimodular graphs We now give a very brief introduction to unimodular random rooted graphs, referring the reader to [1,11] for detailed introductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Let us just recall that G•,• is the separable metric space of doubly rooted graphs (G, x, y) (modulo graph isomorphisms), equipped with the local topology, also known as Benjamini-Schramm topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Similarly defined is the space G• of rooted graphs (G, o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' A mass transport is a measurable function f : G•,• → [0, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' A measure P on G• is called unimodular whenever the mass transport principle �E �� x∈V f(G, o, x) � = �E �� x∈V f(G, x, o) � 16 holds for all mass transports f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' A probability measure P on G• is called reversible if (G, o, X1) d= (G, X1, o) where X1 is the first step of the simple random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' The law P is called station- ary if (G, o) d= (G, X1) and clearly any reversible graph is stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' For recurrent graphs, stationarity and reversibility are equivalent [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' If P is the law of a unimodular random graph, with finite expected degree, then biasing it by deg(o) gives a reversible random graph and whenever P is the law of a reversible random graph, then biasing by deg(o)−1 gives a unimodular random graph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' see for example [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' A set A ⊆ G• is said to be rerooting invariant if ((g, v) ∈ A) ⇒ ((g, u) ∈ A) for every rooted graph (g, v) ∈ G• and every u in the vertex set of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' A unimodular random rooted graph (G, o) is said to be ergodic if it has probability 0 or 1 to belong to any given re-rooting invariant event in G•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' As explain in [1, Section 4], this is equivalent to the law of (G, o) being extremal in the weakly compact convex set of unimodular probability measures on G•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' As such, it follows by Choquet theory that every unimodular measure on G• may be written as a mixture of ergodic unimodular measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' For our purposes, the upshot of this is that we may assume without loss of generality that (G, o) is ergodic when proving Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' We will also rely on the following characterization of two-ended unimodular random rooted graphs due to Bowen, Kun, and Sabok [9], which builds on work of Benjamini and the second author [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Here, a graph G is said to have linear volume growth if for each vertex v of G there exists a constant Cv such that |B(v, r)| ≤ Cvr for every r ≥ 1, where B(v, r) denotes the graph distance ball of radius r around v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Proposition 13 ([9], Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Let (G, o) be an infinite unimodular random rooted graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Then G is two-ended almost surely if and only if it has linear volume growth almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' To prove Thorem 1, it will therefore suffice to prove that if (G, o) is a recurrent unimodular random rooted graph whose UST is two-ended almost surely then G has linear volume growth almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content='2 The effective resistance is linear on the spine Let P be the joint law of an ergodic recurrent unimodular random rooted graph (G, o) and its uniform spanning tree T, which we think of as a triple (G, o, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' It follows by tail triviality of the UST [6, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content='3] that the number of ends of T is deterministic conditional on (G, o), and since (G, o) is ergodic that T has some constant number of ends almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Moreover, it follows from [1, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content='2 and Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content='1] that this number of ends is either 1 or 2 almost surely, so that T is either one-ended almost surely or two-ended almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' We wish to prove that if T is two-ended almost surely then G is two-ended almost surely also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' We will rely on the following theorem of Berestycki and the first author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' 17 Theorem 14 ([8], Theorem 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Let (G, o) be a recurrent unimodular random rooted graph with E deg(o) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Almost surely, the uniform spanning tree of G is one-ended if and only if the harmonic measure from infinity is uniquely defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' To avoid the unnecessary assumption that E deg(o) < ∞, we will use the following mild generalization of this theorem, whose proof is given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Theorem 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Let (G, o) be a recurrent unimodular random rooted graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Almost surely, the uniform spanning tree of G is one-ended if and only if the harmonic measure from infinity is uniquely defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' It follows from this theorem together with Proposition 12 that if T is two-ended almost surely then |ext(H)| = 2 almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Suppose that T is two-ended almost surely and let S be the spine of T, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=', the unique double-infinite simple path contained in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' We give T an orientation by choosing uniformly at random one of the two ends of S and directing every edge towards that end, letting the resulting oriented tree be denoted T → with oriented spine S→.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Since the law of T → is a rerooting-equivariant function of the graph (G, o), the triple (G, T →, o) is unimodular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Since “everything that can happen somewhere can happen at the root” [1, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content='3] we also have that the origin belongs to S with positive probability and hence that we can define a law PS on triplets (G, T →, o) (which we can view as a rooted network) by conditioning o to belong to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' The law PS has the very useful property that it is stationary under shifts along the spine, which we now define.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Each vertex v ∈ S has a unique oriented edge emanating from it in S→, and we will write σ(v) for the vertex on the other end of this edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' The map v �→ σ(v) can be thought of as a shift, following the orientation along the spine, and there is also a well-defined backwards shift σ−1 mapping each x ∈ S to the unique vertex v ∈ S with σ(v) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Lemma 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' The law PS is invariant under the shift σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Let A be any Borel set of triples (g, t→, v) where (g, v) is a rooted graph and t→ is an oriented spanning tree of g, and define the mass transport f(g, t→, v, w) := 1 (t→ is two-ended, w is in the spine of t→, v = σ(w), and (g, t→, w) ∈ A) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Note that there only exists one vertex v such that v = σ(w) and, vice-versa, for each v in the spine of t→ there is only one v in the spine of t→ such that σ(v) = o and v ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Therefore, � v∈V f(G, T →, v, o) = 1 (T → is two-ended, o is in the spine of T →, and (G, T →, o) ∈ A) and � v∈V f(G, T →, v, o) = 1 (T → is two-ended, o is in the spine of T →, and (G, T →, σ(o)) ∈ A) 18 Using the mass-transport principle we thus have that P (T → is two-ended, o is in the spine of T →, and (G, T →, o) ∈ A) = P (T → is two-ended, o is in the spine of T →, and (G, T →, σ(o)) ∈ A) which shows the result because P(o ∈ S) > 0 and T is two-ended a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' The main goal of this section is to show that along the spine of the UST, the effective resistances on the original graph must grow linearly under the assumption that the UST has two ends (and thus a well-defined spine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Heuristically, this tells us that if a graph is unimodular and the uniform spanning tree is two-ended, then the actual graph should in some sense be “close” to the line Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Proposition 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' The limit limn→∞ 1 nReff(o ↔ σn(o)) = limn→∞ 1 nReff(o ↔ σ−n(o)) exists and is positive PS-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Note that the existence part of this proposition is an immediate consequence of the subad- ditive ergodic theorem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' the content of the proposition is that the limit is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' As discussed above, it follows from Proposition 12 and Theorem 14 that, PS-almost surely, there are exactly two extremal elements of H, which we call “ℓ” and “r”, which satisfy ar(σn(o), v) aℓ(σn(o), v) → \uf8f1 \uf8f2 \uf8f3 ∞ as n → +∞ 0 as n → −∞ (14) for every v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' (In particular, the random choice of orientation of T we made when defining PS is equivalent to randomly choosing which of the two extremal elements of H to call “r”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=') Consider the function V → R defined by Mo(x) := ar(x, o) − aℓ(x, o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' We will show that Mo(σn(o)) grows linearly in n and deduce from this that the effective resistance does too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' The latter fact can be seen using (6), from which it follows that Mo(x) = (rx,o(x) − ℓx,o(x))Reff(o ↔ x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' In the remainder we will slightly abuse notation to write Mm(n) := Mσm(o)(σn(o)) for n, m ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' The first main ingredient is that Mo(n) is an additive cocyle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Lemma 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Mo(n + m) = Mo(n) + Mn(n + m) for every n, m ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' This is a direct consequence of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content='5 in [8], stating that a#(x, o) − a#(y, o) = a#(x, y) − Gy(x, o) deg(o) 19 for each # ∈ {ℓ, r} and all x, y ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Indeed, it follows from this identity that Mo(n + m) − Mo(n) = ar(σn+m(o), o) − aℓ(σn+m(o), o) − ar(σn(o), o) + aℓ(σn(o), o) = � ar(σn+m(o), σn(o)) − Gσn(o)(σn+m(o), o) deg(o) � − � aℓ(σn+m(o), σn(o)) − Gσn(o)(σn+m(o), o) deg(o) � = ar(σn+m(o), σn(o)) − aℓ(σn+m(o), σn(o)) = Mn(n + m) for every n, m ∈ Z as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Let us also make note of the following key property of this additive cocycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Lemma 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' PS-almost surely, Mo(n) is positive for all sufficiently large positive n and negative for all sufficiently large negative n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Moreover, Mo(n) ∼ ar(σn(o), o) = rσn(o),o(σn(o))Reff(o ↔ σn(o)) PS-almost surely as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' This follows immediately from (14) and the definition of Mo(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' We will deduce Proposition 17 from Lemma 19 together with the following general fact about stationary sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Proposition 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Let (Zi)i∈Z be a stationary sequence of real-valued random variables and sup- pose that �n i=0 Z−i > 0 for all sufficiently large n almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Then lim supn→∞ 1 n �n i=0 Zi > 0 almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' For each n ∈ Z let Rn = inf{m ≥ 0 : �n+m i=n Zi > 0}, so that Rn = 0 whenever Zn > 0 and (Rn)n∈Z is a stationary sequence of {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' }-valued random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' It follows from the definitions that if n ≤ m then either n + Rn < m or n + Rn ≥ m + Rm, so that the intervals [n, n + Rn] and [m, m + Rm] are either disjoint or ordered by inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' On the other hand, we have by stationarity and the hypotheses of the Proposition that for each n ∈ Z there almost surely exists Nn < ∞ such that �n−1 i=n−m Zi > 0 for every m ≥ Nn and hence that Rn−m + (n − m) < n for every m ≥ Nn, so that each n ∈ Z is contained in at most finitely many of the intervals [m, m + Rm] almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Using the fact that these intervals are either disjoint or ordered by inclusion, it follows that there is a unique decomposition of Z into maximal intervals of this form Z = �� [k, k + Rk] : k ∈ Z, [k, k + Rk] ⊈ [m, m + Rm] for every m ∈ Z \\ {k} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Thus, if we define Yn by Yn = \uf8f1 \uf8f2 \uf8f3 �n i=k Zi n = k + Rk for some k ∈ Z such that [k, k + Rk] maximal 0 otherwise 20 then (Yn)n∈Z is a stationary sequence of non-negative random variables such that Yn is positive whenever n is the right endpoint of a maximal interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Since Yn is non-negative and the set of n such that Yn ̸= 0 is almost surely non-empty, it follows from the ergodic theorem applied to (min{Yn, 1})n∈Z that lim inf n→∞ 1 n n � i=0 Yn > 0 almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' The claim follows since if −m is the left endpoint of the maximal interval containing 0 then n � i=0 Yn = n � i=−m Zi for every n that is the right endpoint of some maximal interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' It follows from Lemma 16 that (Mn(n + 1))n∈Z is a stationary sequence under PS and from Lemma 18 that Mo(n) = �n−1 i=0 Mi(i+1) for every n ≥ 0 and Mo(−n) = �−1 i=−n Mi(i+1) for every n ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Thus, Lemma 19 implies that the stationary sequence (Mn(n+1))n∈Z satisfies the hypotheses of Proposition 20 and hence that lim sup n→∞ Mo(n) n > 0 almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' On the other hand, the subadditive ergodic theorem implies that the limit limn→∞ 1 nReff(o ↔ σn(o)) exists PS-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=', and since Mo(n) = � rσn(o),o(σn(o)) − ℓσn(o),o(σn(o)) � Reff(o ↔ σn(o)) ≤ Reff(o ↔ σn(o)) we must have that lim n→∞ 1 nReff(o ↔ σn(o)) > 0 PS-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' The fact that the negative-n limit limn→∞ 1 nReff(o ↔ σ−n(o)) also exists and is equal to the positive-n limit a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' follows from the subadditive ergodic theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content='3 Completing the proof We now complete the proof of the main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' It suffices by Proposition 13 to prove that if (G, o) is a recurrent unimod- ular random rooted graph whose UST is two-ended almost surely then G has linear volume growth almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' As before, we write S for the spine of the oriented UST T →, write PS for the conditional law of (G, T →, o) given that o ∈ S, and write σ for the shift along the spine as in Lemma 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' For each x ∈ V let S(x) be an element of S of minimal graph distance to x, choosing one of the finitely many possibilities uniformly and independently at random for each x where this 21 point is not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Letting S−1(v) = {x ∈ V : S(x) = v} for each v ∈ S, we have by the mass-transport principle that ES|S−1(o)| = E � |S−1(o)| | o ∈ S � = P(o ∈ S)−1E �� x∈V 1(o = S(x)) � = P(o ∈ S)−1E �� x∈V 1(x = S(o)) � = P(o ∈ S)−1 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' We thus have a stationary sequence of random variables (|S−1(σi(o))|)i∈Z with uniformly finite mean, and the ergodic theorem implies that lim i→∞ 1 2n n � i=−n |S−1(σi(o))| < ∞ (15) almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' On the other hand, letting B(o, r) be the graph distance ball of radius r around o for each r ≥ 1, we have by definition of S−1 that B(o, r) ⊆ � � S−1 (σn(o)) : n ∈ Z, d(o, σn(o)) ≤ 2r � (16) for each r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Proposition 17 together with the trivial inequality Reff(x ↔ y) ≤ d(x, y) imply that there exists a positive constant c > 0 such that d(o, σn(o)) ≥ c|n| for all sufficiently large (positive or negative) n almost surely, and together with (15) and (16) this implies that lim supr→∞ 1 r|B(o, r)| < ∞ almost surely.' metadata={'source': 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(2017), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' 1-2, 113–152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' MR3651050 [18] , Uniform spanning forests of planar graphs, Forum Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Sigma 7 (2019), Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' e29, 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' MR4010561 [19] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Hutchcroft and P.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Tim´ar, Indistinguishability of the components of random spanning forests, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' 46 (2018), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' 4, 2221–2242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' MR3813990 [29] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Wilson, Generating random spanning trees more quickly than the cover time, Proceedings of the twenty-eighth annual acm symposium on theory of computing, 1996, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' 296–303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' [30] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Woess, Random walks on infinite graphs and groups, Cambridge university press, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' 23 A Uniqueness of the potential kernel implies one-endedness of the UST, without finite expected degree In this appendix we prove Theorem 15, which generalizes the theorem of Berestycki and the first author concerning the equivalence of the UST being one-ended and uniqueness of the harmonic measure from infinity to the case that the unimodular random rooted graph does not necessarily have finite expected degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' A secondary purpose of this appendix is to give a brief and self-contained account of those results of [8] that are needed for our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Since recurrent graphs whose USTs are one-ended always have unique harmonic measure from infinity [6, Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content='2], it suffices to prove that the converse holds under the additional assumption of unimodularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Moreover, it suffices as usual to consider the case that (G, o) is ergodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Suppose that (G, o) is an ergodic recurrent unimodular random rooted graph for which H is a singleton almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' We write h for the unique element of H and a for the corresponding potential kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' For each c > 0 consider the event Ac = {lim supx→∞ hx,o(x) ≥ c} = {for each ε > 0 there exist infinitely many vertices x with hx,o(x) ≥ c − ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' As explained in detail in [8, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content='3] (which concerns deterministic recurrent graphs), we have that hx,o(x) ∼ hx,w(x) as x → ∞ for each fixed w ∈ V , (17) which implies that Ac is re-rooting invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Since (G, o) was assumed to be ergodic we deduce the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Let (G, o) be an ergodic unimodular random rooted graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' If G is almost surely recurrent with a uniquely defined harmonic measure from infinity then the event Ac has prob- ability 0 or 1 for each c ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' The next lemma is proven in [8] using an argument that relies on reversibility (and hence on the assumption E deg(o) < ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' We give an alternative proof using Følner sequences that works without this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Lemma 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Let (G, o) be an ergodic unimodular random rooted graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' If G is almost surely recurrent with a uniquely defined harmonic measure from infinity then the event A1/2 holds almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' It suffices to prove that Ac holds with positive probability for every c < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Since (G, o) is recurrent, it follows from the results of [1, §8] that (G, o) is hyperfinite, meaning that there exists a sequence of random subsets (ωn)n≥1 of E such that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Every component of the subgraph spanned by ωn is finite almost surely for each n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' ωn ⊆ ωn+1 for each n ≥ 1 and � n≥1 ωn = E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' 24 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' The random rooted edge-labelled graph (G, o, (ωn)n≥1) is unimodular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Let n ≥ 1 and let Kn be the component of o in ωn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Then we have by the mass-transport principle that E � 1 |Kn| � x∈Kn 1 � hx,o(x) ≥ 1 2 �� = E � 1 |Kn| � x∈Kn 1 � hx,o(o) ≥ 1 2 �� , and since the sum of the two sides is at least 1 it follows that E � 1 |Kn| � x∈Kn 1 � hx,o(x) ≥ 1 2 �� ≥ 1 2 and hence by Markov’s inequality that P ���{x ∈ Kn : hx,o(x) ≥ 1 2} �� ≥ 1 4|Kn| � ≥ 1 − 4 3E � 1 |Kn| � x∈Kn 1 � hx,o(x) < 1 2 �� ≥ 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Since |Kn| → ∞ almost surely as n → ∞, it follows from this and Fatou’s lemma that P(A1/2) ≥ P ���{x ∈ Kn : hx,o(x) ≥ 1 2} �� ≥ 1 4|Kn| for infinitely many n � ≥ 1 3 and hence by ergodicity that P(A1/2) = 1 as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Lemma 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Let G = (V, E) be an infinite, connected, locally finite recurrent graph with uniquely defined harmonic measure from infinity h, let o ∈ V and let a be the associated poten- tial kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' If A is any infinite set of vertices with infx∈A hx,o(x) > 0, the Doob-transformed walk � X visits A infinitely often almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' We have by (11) that �Po( � X hits x) = ho,x(x) for every x ∈ V , and it follows by Fatou’s lemma that �P(hit A infinitely often) ≥ infx∈A hx,o(x) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' On the other hand, we have by Theorem 8 and the assumption that h is unique that � X has trivial tail σ-algebra, so that �P(hit A infinitely often) = 1 as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Proposition 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Let G = (V, E) be an infinite, connected, locally finite recurrent graph with uniquely defined harmonic measure from infinity h, let o ∈ V , let a be the associated potential kernel, and suppose that lim infx→∞ hx,o(x) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' If � X and �Y are independent copies of the Doob-transformed walk started at some vertices x and y, then { � Xn : n ≥ 0} ∩ {�Yn : n ≥ 0} is infinite almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Let δ > 0 be such that A = {x ∈ V : hx,o(x) ≥ δ} is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Applying Lemma 23 yields that � X ∩ A is infinite almost surely, and applying Lemma 23 a second time yields that �Y ∩ � X ∩ A is infinite almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' 25 Applying this proposition together with the results of [23], which imply that an independent Markov process and loop-erased Markov process intersect infinitely almost surely whenever the corresponding two independent Markov processes do, we deduce the following immediate corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Corollary 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Let G = (V, E) be an infinite, connected, locally finite recurrent graph with uniquely defined harmonic measure from infinity h, let o ∈ V , let a be the associated potential kernel, and suppose that lim infx→∞ hx,o(x) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' If � X and �Y are independent copies of the Doob-transformed walk started at some vertices x and y, then { � Xn : n ≥ 0}∩{LE(�Y )n : n ≥ 0} is infinite almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Proposition 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Let G = (V, E) be an infinite, connected, locally finite recurrent graph with uniquely defined harmonic measure from infinity h, let o ∈ V , let a be the associated potential kernel, and suppose that lim infx→∞ hx,o(x) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' For each x ∈ V , let X be a random walk started at x and let �Y be a Doob-transformed walk started at o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Then lim x→∞ P � {Xn : 0 ≤ n ≤ To} ∩ {LE(�Y )m : m ≥ 0} = {o} � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' As x → ∞, the law of the time-reversed final segment (XTo, XTo−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' , XTo−k) con- verges to that of ( � X0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' , � Xk) for each k ≥ 1, and the claim follows from Corollary 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Proof of Theorem 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' The fact that G has a unique harmonic measure from infinity means that we can endow the uniform spanning tree of G with an orientation in a canonical way: Suppose that we exhuast V by finite sets V = � Vn and let G∗ n be defined by contracting V \\Vn into a single boundary vertex ∂n, so that the UST of G can be expressed as the weak limit of the USTs of the graphs G∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' If for each n ≥ 1 we orient the UST of G∗ n towards the boundary vertex ∂n to obtain an oriented tree T → n , then the uniqueness of the harmonic measure from infinity on G implies that the law of T → n converges weakly to the law of an oriented spanning tree T → of G, which can be thought of as a canonical (but potentially random) orientation of the UST of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Indeed, if we fix an enumeration v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' of V with v1 = o we can sample T → n using Wilson’s algorithm rooted at ∂n, starting with the vertices in the order they appear in the enumeration of V , and orienting the edges of the tree in the direction they are crossed by the loop-erased walk that contributed them to the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' In the infinite-volume limit (since only the part of the first walk after its final visit to o contributes to its loop erasure), this corresponds to doing Wilson’s algorithm where the first walk started at o is Doob-transformed and the remaining walks are ordinary simply random walks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' An important consequence of this discussion is that if we sample the oriented uniform spanning tree using Wilson’s algorithm rooted at infinity, where the first random walk is a Doob-transformed walk started at o and the remaining walks are ordinary simple random walks, the distribution of the resulting oriented tree T → does not depend on the choice of the root vertex o, since it is given by the limit of the USTs of G∗ n oriented towards ∂n independently 26 of the choice of exhaustion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Given the oriented tree T →, we say that a vertex u is in the future of a vertex v if the unique infinite oriented path emanating from v passes through v, and say that u is in the past of v if v is in the future of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Let (ωn)n≥1 be a sequence witnessing the fact that (G, o) is hyperfinite as in the proof of Lemma 22 and let Kn be the cluster of o in ωn for each n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' We have by the mass-transport principle that E � 1 |Kn| � x∈Kn 1 (x in past of o) � = E � 1 |Kn| � x∈Kn 1 (x in future of o) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' On the other hand, letting S be the set of vertices belonging to a doubly infinite path in T, we also have that E � 1 |Kn| � x∈Kn 1 (x in past or future of o) � ≥ E � 1 |Kn| � x∈Kn 1(o, x ∈ S) � and we can use the mass-transport principle again to bound E � 1 |Kn| � x∈Kn 1(o, x ∈ S) � = E \uf8ee \uf8f0 1 |Kn|2 � x,y∈Kn 1(o, x ∈ S) \uf8f9 \uf8fb = E \uf8ee \uf8f0 1 |Kn|2 � x,y∈Kn 1(x, y ∈ S) \uf8f9 \uf8fb = E ��|Kn ∩ S| |Kn| �2� ≥ E �|Kn ∩ S| |Kn| �2 = P(o ∈ S)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Putting these two estimates together, it follows that E � 1 |Kn| � x∈Kn 1 (x in past of o) � ≥ 1 2P(o ∈ S)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' (18) On the other hand, if we sample T → using Wilson’s algorithm rooted at infinity, starting with a Doob-transformed �Y started at o followed by an ordinary random walk X started at x, the vertex x belongs to the past of o if and only if the walk X first hits the loop-erasure of �Y at the vertex o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Proposition 26 implies that this probability tends to zero as x → ∞,and it follows by bounded convergence that E � 1 |Kn| � x∈Kn 1 (x in past of o) � → 0 (19) as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Putting together (18) and (19) yields that P(o ∈ S) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' Since “everything that can happen somewhere can happen at the root” [1, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content='3], it follows that S = ∅ almost surely and hence that T is one-ended almost surely as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE2T4oBgHgl3EQfagdb/content/2301.03875v1.pdf'} +page_content=' 27' metadata={'source': 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Matchev,1, † Katia Matcheva,1, ‡ +Alexander Roman,1, § Eyup Unlu,1, ¶ and Sarunas Verner1, ∗∗ +1Institute for Fundamental Theory, Physics Department, +University of Florida, Gainesville, FL 32611, USA +(Dated: January 12, 2023) +We design a deep-learning algorithm for the discovery and identification of the continuous group +of symmetries present in a labeled dataset. +We use fully connected neural networks to model +the symmetry transformations and the corresponding generators. We construct loss functions that +ensure that the applied transformations are symmetries and that the corresponding set of generators +forms a closed (sub)algebra. Our procedure is validated with several examples illustrating different +types of conserved quantities preserved by symmetry. +In the process of deriving the full set of +symmetries, we analyze the complete subgroup structure of the rotation groups SO(2), SO(3), and +SO(4), and of the Lorentz group SO(1, 3). Other examples include squeeze mapping, piecewise +discontinuous labels, and SO(10), demonstrating that our method is completely general, with many +possible applications in physics and data science. Our study also opens the door for using a machine +learning approach in the mathematical study of Lie groups and their properties. +CONTENTS +I. Introduction +1 +II. Setup and notations +3 +III. Deep Learning Approach +4 +A. Invariance +4 +B. Infinitesimality +4 +C. Orthogonality +5 +D. Closure +5 +IV. Linear Algebra Approach +5 +V. Length-preserving Transformations in Two +Dimensions +6 +VI. Length-preserving Transformations in Three +Dimensions +8 +VII. Length-preserving Transformations in Four +Dimensions +9 +A. Two generator subalgebras +9 +B. Three generator subalgebras +10 +C. Four generator subalgebras +10 +∗ roy.forestano@ufl.edu +† matchev@ufl.edu +‡ matcheva@ufl.edu +§ alexroman@ufl.edu +¶ eyup.unlu@ufl.edu +∗∗ verner.s@ufl.edu +D. Six generator algebras +12 +VIII. Lorentz Transformations in Four Dimensional +Minkowski Space +12 +A. Two generator subalgebras +13 +B. Three generator subalgebras +14 +C. Four generator subalgebras +16 +D. Six generator algebras +16 +IX. Squeeze Mapping in Two Dimensions +16 +X. Discontinuous Oracles +17 +A. Piecewise Linear Oracle +17 +B. Manhattan Distance Oracle +18 +XI. Conclusions +18 +Acknowledgments +19 +References +19 +I. +INTRODUCTION +Symmetries play a fundamental role in modern physics +[1]. Physical systems with continuous symmetries exhibit +conservation laws that are universally applicable and in- +dispensable in understanding the system’s behavior and +evolution. In particle physics, symmetries provide an or- +ganizing principle behind the observed particle zoo and +its interactions, and guide model-builders in the search +for viable extensions of the Standard Model (SM)[2]. At +the same time, the mathematical study of symmetries is +arXiv:2301.05638v1 [hep-ph] 13 Jan 2023 + +2 +interesting in its own right and has a rich history. +Over the last decade, there has been increased inter- +est in applications of machine learning (ML) to high- +dimensional physics data as a sensitive tool for event +simulation, data analysis, and statistical inference [3–5]. +More recently, ML is also being used to facilitate tasks +that traditionally have fallen within the domain of the- +orists, e.g., performing symbolic computations [6, 7] or +deriving analytical formulas by training a symbolic re- +gression on synthetic data [8–20]. +Applications of ML to the study of symmetries have +been pursued by a number of groups in different contexts. +One line of work investigates how a given symmetry is +reflected in a learned representation of the data [21, 22] +or in the ML architecture itself, e.g., in the embedding +layer of a neural network (NN) [23]. Several proposals +attempt to design special ML architectures (equivariant +NNs) which have a desired symmetry property built in +from the outset [24–30] and test their performance [31]. +Incorporating the symmetry directly into the ML model +makes it more economical (in terms of learned represen- +tations), interpretable and trainable. The approach can +be extended to cover discrete (permutation) symmetries +as well [32, 33]. Such efforts pave the way for data-driven +blind searches for new physics which stress-test the data +for violations of a well-established symmetry of the SM +[34–36]. +More recently, machine learning is also being applied to +address more formal theoretical questions. For example, +a good understanding of the symmetries present in the +problem can reveal conserved quantities [37, 38] or hint +at a more fundamental unified picture [39]. ML has been +used to discover the symmetry of a potential [23, 40, 41], +to decide whether a given pair of inputs is related by sym- +metry or not [42], to distinguish between scale-invariant +and conformal symmetries [43], and to explore the string +landscape [44–46]. Recent work made use of Generative +Adversarial Networks to learn transformations that pre- +serve probability distributions [47]. ML applications have +also found their way into group theory, which provides +the abstract mathematical language of symmetries. For +example, recent work used ML to compute tensor prod- +ucts and branching rules of irreducible representations of +Lie groups [48] and to obtain Lie group generators of a +symmetry present in the data [49, 50]. +The main goal of this paper is to design a deep- +learning method that mimics the traditional theorist’s +thinking and is capable of discovering and categorizing +the full set of (continuous) symmetries in a given dataset +from first principles, i.e., without any prior assumptions +or prejudice. +The only input to our procedure is a +labeled dataset {x; y} like the one in eq. (1) below. An +oracle ϕ(x) = y can then be learned from the dataset, +or alternatively, can be provided externally. With those +ingredients, we go through the following objectives: +• Discovery of symmetries. In the first step, de- +scribed in Section III A, we learn to generate a sym- +metry transformation, x +f→ x′, which preserves the +oracle values. The transformation f is encoded in a +neural network trained on the dataset. In general, +there will be many possible symmetry transforma- +tions f, and their study and categorization from a +group theory point of view is the main goal of this +paper. +• Discovery of symmetry generators. +Having +learned how to generate arbitrary (finite) symme- +try transformations f, we then focus on infinitesi- +mal transformations δf, which give us in turn the +symmetry generators J. +• Discovery of Lie subalgebras. By adding suit- +able terms to the loss function, our procedure re- +quires that the learned set of generators {Jα} forms +a closed algebra. This allows us to discover subal- +gebras of the symmetry group, and identify them +by their structure constants. Since the number of +generators Ng is a free input, in cases when the +training is unsuccessful (as quantified by the loss), +we can rule out the existence of Ng-dimensional +subalgebras. +• Discovery of the full Lie algebra. The maxi- +mum value of Ng which gives a vanishing loss, in- +dicates the dimension of the full Lie algebra. The +corresponding learned set of generators describes +the full symmetry group of the dataset. +• Identification of the symmetry group and its +subgroups. The learned sets of generators found +in the previous two steps are then used to obtain +the structure constants of the respective full alge- +bra and its subalgebras, and thus to identify the +corresponding symmetry group and its subgroups. +Our study complements and extends previous related +work in [23, 40, 41, 49, 50]. We note that our procedure +is completely general, and does not anticipate what sym- +metries might be present in the dataset — instead, the +symmetries are learned from scratch. In addition, our +method is basis-independent since we do not choose a +specific convenient basis for the learned transformations +and generators. Consequently, our results will not always +match the nice canonical forms of the generators found in +the group theory textbooks. Nevertheless, as the exam- +ples below explicitly demonstrate, all our learned trans- +formations and generators will satisfy the defining prop- +erties of the respective symmetry groups and subgroups. +The paper is organized as follows. In Section II we in- +troduce the setup for our analysis and the corresponding +notation and conventions. The main steps of our deep- +learning procedure are outlined in Section III, which also +explains and motivates the necessary ingredients for the + +3 +loss function. For readers who are not yet fully comfort- +able with a deep-learning approach, Section IV outlines +an analogous linear algebra approach that often (but not +always — see Section X for counterexamples) can accom- +plish similar objectives. Each of the remaining sections +contains a separate completely worked-out example illus- +trating our method. The examples are distinguished by +the choice of oracle and the dimensionality of the fea- +ture space. In Sections V-VII we choose an oracle that +returns the Euclidean distance in feature space, whose +dimensionality, in turn, is chosen to be n = 2 in Sec- +tion V, n = 3 in Section VI, and n = 4 in Section VII. +In Section VIII we focus on the Lorentz group, i.e., the +oracle computes the pseudo-Euclidean distance in four- +dimensional Minkowski space-time. +In Section IX we +consider the squeeze transformations in n = 2 dimen- +sions whereby the oracle returns the product of the two +features. To demonstrate the universal applicability of +our technique, in Section X we show two examples of +discontinuous oracles. +We summarize and conclude in +Section XI. +II. +SETUP AND NOTATIONS +Our starting point is a labeled dataset containing m +samples of n features and a target label y: +m samples +� +� +� +� +� +� +� +� +� +� +� +x(1) +1 , x(2) +1 , . . . , x(n) +1 ; y1 +x(1) +2 , x(2) +2 , . . . , x(n) +2 ; y2 +... +... +... +... +... +x(1) +m , x(2) +m , . . . , x(n) +m ; ym +. +(1) +In ML parlance, the dataset (1) is an m × n dataframe +with n features and m samples. In what follows, we use +the sample index a lot more often than the feature index, +thus we use an explicit subscript for the sample index and +hide the feature index in the boldface vector notation x: +x ≡ {x(1), x(2), . . . , x(n)}. +(2) +This allows us to write the input features in a compact +form as +{xi} ≡ {x1, x2, . . . , xm} . +(3) +In order to study the symmetries of the data (1), we +need to know the function y(x), which can be given ana- +lytically or numerically in terms of an oracle ϕ : Rn → R +capable of computing the corresponding output target +labels y1, y2, . . . , ym: +yi = ϕ(xi) , +i = 1, 2, . . . , m . +(4) +This leads to two basic scenarios: +• The function y(x) is known analytically, and that +same function has been used as in (4) to compute +the labels in (1) exactly. This case is of interest to +theorists, and this is the approach adopted in this +paper as well — for each of our examples below, we +specify the relevant analytic oracle ϕ(x) and pro- +ceed to study the resulting symmetries. Note, how- +ever, that we never take advantage of the knowl- +edge of the analytical form of the oracle, e.g., we +do not differentiate or symbolically manipulate in +any other way the function ϕ(x). For our purposes, +we only use the oracle numerically — our method +treats it simply as a black box, which, given the +values of x, can produce the numerical value of the +label y. +• The functional dependence y(x) is a priori unknown +and the dataset (1) is all that is available to us. +This is the typical case encountered by data scien- +tists. Now, one needs to go through a preliminary +step of first creating the oracle ϕ (usually in the +form of a neural network trained on the dataset +(1)), which is capable of computing and reporting +the values of y = ϕ(x) to us. This is a standard +regression task which is of no interest here since it +can be accomplished using one of the many estab- +lished ML regression techniques. We can therefore +safely assume that this preliminary step has already +been completed and we have such a numerical ora- +cle y = ϕ(x) already available. +Since we are only using the oracle numerically, from our +point of view there is no real difference between the above +two cases. In what follows the oracle ϕ(x) will be used +in the exact same way, regardless of its origin. +Given this general setup, our main task is to derive the +symmetry transformation f : Rn → Rn +x′ = f(x) , +(5) +which preserves the ϕ-induced labels of our dataset (1). +In other words, we want to find the function f(x) for +which +ϕ(x′ +i) ≡ ϕ(f(xi)) = ϕ(xi), +∀i = 1, 2, . . . , m . +(6) +The particular instantiation of the symmetry f(x) will +depend on the initialization of our parameters, so by re- +peating the procedure with different initializations, we +will in principle obtain a whole family of symmetry trans- +formations. +Next, we focus on infinitesimal symmetry transforma- +tions and proceed to study the corresponding set of gener- +ators {Jα}, with α = 1, 2, . . . , Ng, where we use lowercase +Greek letters to label the generators of symmetry trans- +formations. A given set of generators {Jα} forms a Lie +algebra if the closure condition is satisfied, i.e., if all Lie +brackets +� +. , . +� +can be represented as linear combinations +of the generators in the set: +� +Jα, Jβ +� += +Ng +� +γ=1 +a[αβ]γJγ. +(7) + +4 +The coefficients a[αβ]γ are the structure constants (anti- +symmetric in their first two indices, as implied by the +square brackets) whose values will reveal the symmetry +group present in our dataset. +In principle, the number of generators Ng is a hyperpa- +rameter that must be specified ahead of time (similarly +to the choice of the number of clusters in certain cluster- +ing algorithms like K-means). Therefore, when we find +a closed algebra at a given Ng value, we are only guar- +anteed that it is a subalgebra, and we must proceed to +try out higher values for Ng as well. The full algebra will +then correspond to the maximum value of Ng for which +a closed algebra of distinct generators is found to exist. +III. +DEEP LEARNING APPROACH +In our approach, we model the output function f with +a neural network (NN) FW with n neurons in the out- +put layer, corresponding to the n transformed features +of the data point x′. The trainable network parameters +(weights and biases) will be generically denoted with W. +During training, they will evolve and converge to the cor- +responding trained values � +W of the parameters of the +trained network F� +W, i.e., the hat symbol will denote the +result of the training. Once the parameters � +W are found, +we can find the structure constants using standard meth- +ods. We choose a suitable loss function that ensures the +desired properties of the trained network. The following +subsections discuss the individual contributions to the +loss function, which in our implementation are combined +and minimized simultaneously. +The neural network FW is implemented as a sequential +feed-forward neural network in PyTorch [51]. The ex- +amples in Sections V-VIII are simple enough to be done +with no hidden layers, no bias, and no activation func- +tion, i.e., with linear transformations (see Section IV). +For the examples in the later sections, we do add hidden +layers. Optimizations are performed with the Adam op- +timizer with a learning rate between 0.03 and 0.1. The +loss functions were designed to achieve a fast and efficient +training process without the need for extensive hyperpa- +rameter tuning. The training data (3) was typically on +the order of a few hundred points and was sampled from +a standard normal distribution. +An alternative approach to predicting the generators +from the model parameters, which utilizes an identi- +cal loss function, can be carried out where a NN, F : +{Gα, a} → { �Gα, ˆa}, takes a set of randomly initialized +n×n generators {Gα} and an Nb ×Ng structure constant +array {a}, flattens each individual generator and the +structure constant array, and proceeds to converge the +elements to the desired set of generators {� +Gα} and struc- +ture constant array {ˆa}. Here Nb = +�Ng +2 +� += +Ng(Ng−1) +2 +denotes the number of unique one-directional Lie brack- +ets for a given number of generators Ng. +This alternative approach can be implemented as a +module list of sequential neural network layers consist- +ing of two hidden layers for the generators and a single +sequential layer consisting of two hidden layers for the +structure constants. +Each layer consists of a bias and +the hidden layers use the Rectified Linear Unit (ReLU) +activation function. +The optimizer, average learning +rate, model hyperparameters, and training data gener- +ation were identical to the alternative implementation +described above. Whereas the original method feeds the +data directly into the network, the data in this approach +is fed into the loss function to be transformed by the +model’s predicted generators. +A. +Invariance +The invariance under the transformation (5) is en- +forced by requiring that the labels before and after the +transformation remain the same. For this purpose, we +include the following mean squared error (MSE) term in +the loss function L: +Linv(W, {xi}) = 1 +m +m +� +i=1 +[ϕ (FW(xi)) − ϕ(xi)]2 . +(8) +A NN trained with this loss function will find an ar- +bitrarily general (finite) symmetry transformation F� +W +parametrized by the values of the trained network pa- +rameters � +W. +In order to find multiple symmetries, the process can +be repeated. +Alternatively, several networks can be +trained concurrently, by modifying the loss function to +ensure that the resulting transformations are sufficiently +distinct (see Section III C below). +B. +Infinitesimality +In order to focus on the generators of the possible set +of symmetry transformations, we restrict ourselves to in- +finitesimal transformations δF in the vicinity of the iden- +tity transformation I: +δF ≡ I + ε GW , +(9) +where ε is an infinitesimal parameter and the parameters +W of the new neural network G will be forced to be finite, +which ensures that (9) is an infinitesimal transformation. +The loss function (8) can then be rewritten as +Linf(W, {xi}) = +1 +mε2 +m +� +i=1 +[ϕ(xi + εGW(xi)) − ϕ(xi)]2 , +(10) + +5 +where we have introduced an explicit factor of ε2 in the +denominator to account for the fact that generic transfor- +mations scale as ε [41]. Once we minimize the loss func- +tion, the trained NN G� +W will represent a corresponding +generator +J = G� +W , +(11) +where +� +W ≡ arg min +W +� +L(W, {xi}) +� +(12) +are the learned values of the NN parameters which min- +imize the loss function. The result for � +W, and therefore, +for J, will in principle depend on the starting values W0 +of the network parameters at initialization. If we now +repeat the training Ng times under different initial con- +ditions W0 (or with different values of the hyperparam- +eters), we will obtain a set of Ng (in general distinct) +generators {Jα}, α = 1, 2, . . . , Ng. +C. +Orthogonality +In order to make the set of generators {Jα}, α = +1, 2, . . . , Ng, found in the previous step maximally differ- +ent, we introduce the following additional orthogonality +term to the loss function +Lortho(W, {xi}) = +Ng +� +α<β +�� +p +W(p) +α W(p) +β +�2 +, +(13) +where the index p runs over the individual NN parame- +ters W(p) +α . +D. +Closure +In order to test whether a certain set of distinct gener- +ators {Jα}, α = 1, 2, . . . , Ng, found in the previous steps, +generates a group, we need to check the closure of the +algebra (7). This can be done in several ways. The most +principled would be to minimize +Lclosure(a[αβ]γ) = +� +α<β +Tr +� +CT +[αβ]C[αβ] +� +, +(14) +with respect to the candidate structure constant param- +eters a[αβ]γ, where the closure mismatch is defined by +C[αβ](a[αβ]γ) ≡ +� +Jα, Jβ +� +− +Ng +� +γ=1 +a[αβ]γJγ. +(15) +Since Lclosure is positive semi-definite, Lclosure = 0 would +indicate that the algebra is closed and we are thus dealing +with a genuine (sub)group. In practice,1 we perform the +minimization of (14) simultaneously with the previously +discussed contributions to the loss function, by replacing +Jα → GWα in (15). The advantage of this simultaneous +construction is that every set of generators that we obtain +at any given value of Ng is already forming an algebra +that is “as closed as possible”. +Then, the size of the +achieved total training loss is an indicator whether for +that value of Ng a closed algebra exists or not. +Once a closed set of valid generators has been found, +we can retrain the NN in a conveniently chosen canonical +basis and obtain the canonical form of the set of genera- +tors, whose structure constants in turn reveal the nature +of the group behind the found symmetry transformations +G� +Wα, α = 1, 2, . . . , Ng. +Sections V-X illustrate the steps above with several +examples of increasing complexity. +IV. +LINEAR ALGEBRA APPROACH +The universal approximation theorems [52] guarantee +that the deep-learning approach of the previous section +can handle almost any symmetry transformation, includ- +ing a highly non-linear one. At the same time, a large +class of interesting symmetries arising in physics are lin- +ear transformations for which the usual formalism of lin- +ear algebra would suffice. Furthermore, the analysis of +the symmetry generators involves infinitesimal transfor- +mations, which are represented with linear operators. For +those reasons and to captivate the readers who are not +yet at ease with the technical intricacies of machine learn- +ing, in this section, we reformulate our analysis from the +previous section in the language of linear algebra. We fol- +low the same notation and conventions, but replace the +calligraphic font symbols representing neural networks +with corresponding blackboard-bold symbols represent- +ing n × n matrices acting on the n-dimensional feature +space. +In this section we are interested in the linear subclass +of the transformations (5), which are encoded in a generic +matrix F +x′ = F x , +(16) +whose n2 components Fij are determined by minimizing +the loss function +Linv(F, {xi}) = 1 +m +m +� +i=1 +[ϕ (F xi) − ϕ(xi)]2 . +(17) +1 Another possible approach is to minimize the out-of-space com- +ponents of the commutators with respect to the space of gener- +ators, after flattening and Gram–Schmidt orthonormalization. + +6 +If the minimization is successful, then such a linear sym- +metry exists and is represented with the learned matrix +�F. +In analogy to (9), we can write the corresponding in- +finitesimal linear transformation as +δF(ε) = I + ε G , +(18) +where I is the unit n×n matrix and G is an n×n matrix +whose components Gij are yet to be determined through +the optimization. In order to obtain a single generator +matrix J, we can use a loss function analogous to (10) +LJ(G, {xi}) = hinv +mε2 +m +� +i +� +ϕ +� +xi + ε G xi +� +− ϕ(xi) +�2 ++ hnorm +� +Tr +� +GT G +� +− 2 +�2 +, +(19) +where the constants hinv and hnorm in (19) are hyperpa- +rameter weights determining the relative contribution of +the two terms in the loss function (19) enforcing the sym- +metry invariance and finite normalization2 conditions, re- +spectively. The actual generator J is then obtained by +minimizing (19): +J = arg min +G +� +LJ(G, {xi}) +� +. +(20) +By repeating this procedure several times with differ- +ent initial conditions, we obtain a different generator J +each time. Alternatively, we can produce all Ng genera- +tors in one go by adding together and minimizing simul- +taneously Ng copies of the loss function (19): +Ng +� +α=1 +LJ(Gα, {xi}) , +(21) +which will lead to a set of Ng generators Jα, α = +1, 2, . . . , Ng, and their respective infinitesimal transfor- +mations δFα ≡ I + ε Jα. +At this point the generators Jα are completely decou- +pled and independent of each other. We can force them +to be different by adding a loss term analogous to (13): +Lortho(G, {xi}) = +Ng +� +α<β +� +Tr(GT +αGβ) +�2 . +(22) +Finally, we can enforce the closure property by includ- +ing a loss term analogous to (14): +Lclosure(a[αβ]γ) = +� +α<β +Tr +� +CT +[αβ]C[αβ] +� +, +(23) +2 In order for (18) to be an infinitesimal transformation, the matrix +G needs to be finite. We choose to normalize our generators as +Tr(GT +αGβ) = 2δαβ, hence the factor of 2 in the second line of +(19). +FIG. 1. A representative symmetry transformation preserving +the oracle function (25), found by our procedure in the two- +dimensional exercise in Section V. In this and all subsequent +such figures, the arrows represent the displacements x′ − x +resulting from the symmetry transformation. The color map +illustrates the oracle function ϕ(x). +where +C[αβ](a[αβ]γ) ≡ +� +Gα, Gβ +� +− +Ng +� +γ=1 +a[αβ]γGγ , +(24) +and minimizing the total loss function with respect to +the parameters a[αβ]γ as well. +V. +LENGTH-PRESERVING +TRANSFORMATIONS IN TWO DIMENSIONS +In this and the next two sections, we focus on trans- +formations which preserve the Euclidean length of the +feature vector, i.e., our oracle ϕ will return +ϕ(x) ≡ |x| = +� n +� +i=1 +[x(i)]2 +�1/2 +. +(25) +In this case we expect our method to discover the sym- +metry of the orthogonal group O(n), whose generators +can be written in terms of Kronecker deltas as +(Omn)ij = δi +mδj +n − δj +mδi +n , +(26) +in which case the algebra is given by +[Omn, Opq] = δnpOmq + δmqOnp +− δmpOnq − δnqOmp. +(27) +Note that the generators Omn are labeled by two indices, +which indicate the plane of rotation. + +2 +2.7 +2.4 +1 +2.1 +1.8 +1.5 +(x)d +0 +X +1.2 +0.9 +-1 +0.6 +0.3 +-2 +0.0 +-2 +-1 +0 +1 +2 +×(1)7 +FIG. 2. +The evolution of the training loss with the num- +ber of epochs for the two-dimensional exercise in Section V. +Our algorithm finds one symmetry generator (the training +loss shown in blue steadily decreases) but not two different +symmetry generators (the training loss shown in orange stays +large). +In this section we start with the simplest case of two di- +mensions, n = 2, which should correspond to the single- +generator group O(2). +First we try to generate a sin- +gle generic (i.e., not necessarily infinitesimal) symmetry +transformation. For this purpose, we train our network +FW with the invariance loss (8). This exercise was suc- +cessful and, depending on the initialization, we found var- +ious symmetry transformations. They all involved a ro- +tation around the origin in the (x(1), x(2)) plane. One +representative symmetry transformation is depicted in +Figure 1. +Next, we turn to the algebra of symmetry generators. +We begin with a single generator, Ng = 1, in which case +we do not need to include the orthogonality and closure +terms in the loss function. The training is successful and +the loss function is driven to zero, as seen by the blue +solid line in Figure 2. The resulting generator in matrix +form is pictorially visualized in the top row of Figure 3, +and we immediately recognize the familiar matrix O12 +from (26) generating rotations in the 12-plane +O12 = +� +0 1 +−1 0 +� +. +(28) +Note that the generator has the expected antisymmetric +property. +Having found one symmetry generator, we next check +if there is a second distinct generator. For this purpose, +we add the orthogonality and closure terms in the loss +function and repeat the training. This time the training +is unsuccessful and the loss flattens after about 50 epochs, +as seen by the orange solid line in Figure 2. In the second +row of Figure 3 we show the result for the two candidate +generators found in this case. We note that while they do +FIG. 3. Top row: a successfully learned generator for n = 2 +and Ng = 1. Bottom row: unsuccessfully learned “genera- +tors” for n = 2 and Ng = 2. In this and all subsequent such +figures, each panel represents a learned generator Jα in ma- +trix form, where the values of the individual elements of the +matrix are indicated by the color bar. +have the expected form for a generator of rotations in two +dimensions, they are essentially the same transformation, +and differ only by an overall sign. This implies that they +fail the orthogonality condition — indeed, we find that +the dominant contribution to the large total loss in that +case is from the orthogonality loss (13). Since the total +loss is large and does not improve with further training +(see the orange line for Ng = 2 in Figure 2), these two +are not valid generators and should be discarded. +We +thus conclude that there is no Lie algebra with Ng = 2 +distinct generators in this scenario. +We note in passing that upon repeated training runs +in the Ng = 2 case, we sometimes find that the algo- +rithm chooses to create generators which are orthogonal +to each other, but are not genuine symmetry transfor- +mations, i.e., the orthogonality loss is driven to zero, but +only at the expense of a large invariance loss (10). In +principle, it is difficult to predict what the machine will +choose to do when presented with two mutually exclu- +sive requirements. For example, in some runs the two +candidate generators ended up being identical instead of +differing by a minus sign. Also note that the normaliza- +tion of the two candidate generators in the second row of +Figure 3 is off — this is because the program chose to vi- +olate the normalization condition as well, in order to help +the orthogonality loss, which is the dominant penalty in +that case. In any case, since the failed training examples +are of little theoretical value, in the following we shall +only show results for the successful training runs which +led to valid (sub)algebras. +Next, we check the cases with Ng > 2. Similarly, we + +n= 2 +10' +10 +Loss +10- +-5 +Ng = 1 +Ng = 2 +10 +-14 +0 +100 +200 +300 +400 +500 +600 +EpochsGenerator 1 +1.00 +0.75 +1 - +0.50 + 0.25 ++ 0.00 +0.25 +2 - +0.50 +0.75 +1.00 +1 +2Generator 1 +Generator 2 +1.00 +0.75 +1 +0.50 +0.25 + 0.00 +0.25 +2 +2 +0.50 +0.75 +1.00 +1 +2 +1 +28 +FIG. 4. The axes of the eigenvectors of the three generators +found at intermediate stages of the training: after 1 epoch +(top left), after 10 epochs (top right), after 100 epochs (bot- +tom left), and after 300 epochs (bottom right). For conve- +nience, at the top of each panel we list the angle (in degrees) +between each pair of axes. +find that the training ends up in a large loss and that +there is no consistent solution for a closed algebra. We +therefore conclude that in this n = 2 example there is +only a one-parameter symmetry group with a generator +given by (28). +Although the example of this section was rather trivial, +it did outline and validate the main steps of our method. +More complicated examples follow in the next sections. +VI. +LENGTH-PRESERVING +TRANSFORMATIONS IN THREE DIMENSIONS +In this section, we proceed to study symmetry trans- +formations which preserve the oracle (25) in n = 3 di- +mensions. +Once again, we find that training with the +invariance loss (8) alone always leads to a valid sym- +metry transformation. We also observe that the matrix +form F is antisymmetric, in agreement with the expecta- +tion for the orthogonal group O(3). Now that the data +is three-dimensional, however, it is difficult to visualize +the symmetry transformation directly as in Figure 1. In- +stead, we choose to plot the symmetry axis (in this case +the axis of rotation) defined by the real eigenvector. Fig- +ure 4 illustrates the transformations found at different +stages of the training for the case of Ng = 3 generators. +At the top of each panel we list the relative angle in de- +grees for each pair of axes. Note how the symmetry axes +FIG. 5. The same as Fig. 2, but for the n = 3 exercise consid- +ered in Section VI. The training results in valid closed algebras +with one or three generators, but not two or four. +start out oriented at random, but the orthogonality loss +term (13) gradually drives them to a mutually orthogonal +configuration. +In order to analyze the group structure of the found +symmetry transformations, we proceed to study the gen- +erators of infinitesimal transformations. First we try to +determine the dimensionality of the full algebra, i.e., the +maximal number of linearly independent generators re- +sulting in a closed algebra. Figure 5 shows loss curves for +several different values of Ng: 1, 2, 3 and 4 (we do not +show results for Ng ≥ 5 since they had large losses). We +observe that the training was successful only for the cases +of Ng = 1 and Ng = 3. This implies that the full algebra +has 3 generators, in agreement with the expectation of +n(n − 1)/2 generators for an orthogonal O(n) group. At +the same time, the successful training at Ng = 1 reveals +a single generator subgroup of O(3) which is nothing but +the O(2) discussed in the previous section. +The results from a typical training run at Ng = 3 are +shown in Figure 6, where the top row depicts the three +successfully learned generators Jα, α = 1, 2, 3, while the +bottom panel is a pictorial representation of the structure +constants in matrix form as follows. Each row (labeled +αβ = 12, 31, 23) represents one of the three possible com- +mutators, whereas the columns (labeled γ = 1, 2, 3) rep- +resent the found generators Jγ shown in the top panels +of the figure. Then, the entry in each cell represents the +structure constant a[αβ]γ from the defining equation (7). +A careful inspection of the top row in Figure 6 reveals +that the three generators found in our example are ap- +proximately +J1 ≈ +� +� +0 0 1 +0 0 0 +−1 0 0 +� +� = − O31, +(29a) + +Epoch: 0 Angles = 66.16°, 92.56°, 44.16°Epoch: 10 |Angles = 51.74°, 94.25°, 69.22°Ep0ch: 100 |Angles = 92.02°, 90.32°, 93.08°Ep0ch: 300 |Angles = 90.0°, 90.0°, 90.0°n= 3 [30(3)] +102 +102, +Ng= 1 +SS +Lo +10 +-0 +Ng = 2 +Ng = 3 +10 +-10 +10° +-14 +0 +100 +200 +300 +400 +500 +600 +Epochs9 +FIG. 6. Top row: the successfully learned generators for the +n = 3, Ng = 3 exercise considered in Section VI. Bottom +panel: the corresponding structure constants in matrix form +(see text). +J2 ≈ +1 +√ +2 +� +� +0 −1 +0 +1 +0 −1 +0 +1 +0 +� +� = − 1 +√ +2 (O12 + O23), (29b) +J3 ≈ +1 +√ +2 +� +� +0 −1 0 +1 +0 1 +0 −1 0 +� +� = − 1 +√ +2 (O12 − O23). +(29c) +The bottom panel shows that the algebra of the found +three generators Jα is given by +[Jα, Jβ] = − ϵαβγJγ, +(30) +in which we recognize the usual SO(3) algebra involving +the Levi-Civita permutation symbol ϵαβγ. From now on +we will be using the Einstein summation convention for +repeated generator-type indices α, β, γ, . . . . +VII. +LENGTH-PRESERVING +TRANSFORMATIONS IN FOUR DIMENSIONS +In this section, we generalize the discussion from the +previous two sections to the case of n = 4 dimensions. +The new twist here will be the existence of multiple non- +trivial subalgebras of different dimensionality. +The discovery of a single finite symmetry transfor- +mation with the invariance loss (8) is straightforward +and always succeeds in finding some finite rotation in +four dimensions, represented with an orthogonal matrix. +Therefore in this section, we focus only on the identi- +fication of the (sub)algebras. As before, we repeat the +training for various number of multiple distinct genera- +tors (Ng = 2, 3, 4, 5, 6, 7) and with the orthogonality and +FIG. 7. +The same as Fig. 5, but for the n = 4 exercise +considered in Section VII. The training results in valid closed +algebras with one (not shown), two, three, four and six gen- +erators, but not five or seven. +closure losses turned on. Representative loss curves are +shown in Figure 7. +We observe that a closed algebra +is found in four of those cases, namely Ng = 2, 3, 4, 6, +which we now discuss in turn (the trivial case of Ng = 1 +is of course always possible and will not be specifically +discussed from now on). +A. +Two generator subalgebras +One of our found examples of a closed subalgebra with +Ng = 2 generators is shown in Figure 8. The top panels +indicate that the two found generators can be roughly +approximated as +J1 ≈ +� +� +� +0 1 0 0 +−1 0 0 0 +0 0 0 0 +0 0 0 0 +� +� +� = O12, +(31a) +J2 ≈ +� +� +� +0 0 +0 0 +0 0 +0 0 +0 0 +0 1 +0 0 −1 0 +� +� +� = O34, +(31b) +in which we recognize the rotation generators O12 and +O34 from (26). As seen from the matrix forms in (31), +these two generators commute, since the two rotations +are done in completely different planes. Therefore, the +algebra formed by J1 and J2 is Abelian, which is inde- +pendently verified by the bottom panel in Figure 8. +One should keep in mind that we do not control the +overall orientation of the generators found by our proce- +dure. The example in Figure 8 was judiciously chosen +to be easily recognizable in terms of the canonical gener- + +Generator 1 +Generator 2 +Generator 3 +1.00 +1 +1 +1 +0.75 +0.50 +0.25 +2 +2 +2 +0.00 +0.25 +0.50 +3 - +3 +3 +0.75 +1.00 +1 +2 +m +1 +2 +m +1 +2 +3Structure Constants +1.00 +0.75 +12 +0.50 +0.25 +31 +0.00 +Bra +0.25 +-0.50 +23 - +0.75 +1.00 +1 +2 +3 +Generatorn = 4 [3(4)] +10l +Ng = 2 +102, +Ng = 3 +OSS +Ng = 4 +10~5, +Ng = 5 +Ng = 6 +10~8 +10~11 +0 +100 +200 +300 +400 +500 +600 +Epochs10 +FIG. 8. Top row: the successfully learned generators for the +n = 4, Ng = 2 exercise considered in Section VII A. Bottom +panel: the corresponding structure constants. The vanishing +of the structure constants indicates that this is an Abelian +subgroup. +FIG. 9. The same as Figure 8, but for a different training +run, still with Ng = 2. +ators (26). A generic training run typically returns the +generator set in some random orientation, which, how- +ever, still preserves the commutation properties. +One +such generic example is shown in Figure 9. This time, the +two found generators J1 and J2 are more general linear +combinations of the six canonical generators O12, O13, +O14, O23, O24, and O34 of the O(4) group. Neverthe- +less, the found generators J1 and J2 still commute and +form an Abelian two-dimensional subalgebra of the full +symmetry group. +FIG. 10. Top row: the successfully learned generators for the +n = 4, Ng = 3 exercise considered in Section VII B. Bottom +panel: the corresponding structure constants, which can be +identified as those of the SO(3) algebra (see text). +B. +Three generator subalgebras +Next we discuss the discovered subalgebras with three +generators (Ng = 3). Since SO(4) contains SO(3) as a +subgroup,3 we know that such subalgebras should exist, +and indeed, we find such solutions, as seen in Figure 7. +As mentioned above, a generic training run produces the +three generators in a random orientation. +For ease of +interpretation, in Figure 10 we show a judiciously cho- +sen example, in which the generators depicted in the top +panels can be recognized to be approximately +J1 ≈ O24, +J2 ≈ O23, +J3 ≈ O34. +(32) +This algebra involves rotations primarily in the last three +feature dimensions, while the first feature is largely unaf- +fected by the symmetry. The bottom panel of Figure 10 +confirms that this is the SO(3) algebra from eq. (30). +Of course, a more generic training run results in a set of +three generators that involve all four feature dimensions, +but still have the same commutation relations. +C. +Four generator subalgebras +Figure 7 shows that our method finds an algebra with +four generators as well. To see its origin theoretically, +3 Since SO(4) = SO(3)⊗SO(3), the SO(4) algebra is a direct sum +of two separate SO(3) subalgebras. + +Generator 1 +Generator 2 +1.00 +1 +1 +0.75 +0.50 +2 +2 +0.25 +0.00 +3 +3 +-0.25 +-0.50 +4 +4 +0.75 +1.00 +1 +2 +3 +4 +1 +2 +3 +4Structure Constants +T 1.0 +ket + 0.5 + 0.0 +0.5 +1.0 +1 +2 +GeneratorGenerator 1 +Generator 2 +1.00 +1 +1 +0.75 +0.50 +2 +2 +0.25 +0.00 +3 +3 +-0.25 +0.50 +4 +4 +0.75 +1.00 +2 +¥3 +4 +2 +3 +4Structure Constants +T 1.0 +ket + 0.5 + 0.0 +0.5 +1.0 +1 +2 +GeneratorGenerator 1 +Generator 2 +Generator 3 +1.0 +1 +1 - +1 +0.5 +2 +2 +2 +0.0 +3 - +3 - +3 +-0.5 +4 +4 - +4 +-1.0 +1 +2 +3 +4 +1 +2 +3 +4 +1 +2 +w +4Structure Constants +1.00 +0.75 +12 +0.50 +0.25 +31 +0.00 +Bra +-0.25 +0.50 +23 +0.75 +1.00 +1 +2 +m +Generator11 +FIG. 11. Top row: the successfully learned generators for the +n = 4, Ng = 4 exercise considered in Section VII C. Bottom +panel: the corresponding structure constants. +define a new generator basis in terms of sums and differ- +ences of pairs of commuting generators from the original +basis (26) [53] +S1 ≡ 1 +2 (O34 + O12) , +D1≡ 1 +2 (O34 − O12) , (33a) +S2 ≡ 1 +2 (O42 + O13) , +D2≡ 1 +2 (O42 − O13) , (33b) +S3 ≡ 1 +2 (O23 + O14) , +D3≡ 1 +2 (O23 − O14) . (33c) +This change of basis decouples the algebra (27) of the +original generators as follows +[Si, Sj] = − ϵijk Sk, +(34a) +[Di, Dj] = − ϵijk Dk, +(34b) +[Si, Dj] = 0. +(34c) +Therefore, a closed algebra of four generators can be +formed either from the set of three S’s plus any one of +the D’s, or from the set of three D’s plus any one of the +S’s. In either case, three of the generators will satisfy +SO(3)-type commutation relations, while the fourth one +will commute with everyone else. +This expectation is +confirmed in Figure 11, which shows our usual represen- +tation of the found generators and their algebra for one +representative result from a training run with Ng = 4. +We observe that the found generators are approximately +J1 ≡ 1 +2 (O21 + O43) = − S1, +(35a) +FIG. 12. Top row: the successfully learned generators for the +n = 4, Ng = 6 exercise considered in Section VII D. Bottom +panel: the corresponding structure constants. +J2 ≡ 1 +2 (O32 + O14) = − D3, +(35b) +J3 ≡ 1 +2 (O13 + O42) = S2, +(35c) +J4 ≡ 1 +2 (O14 + O23) = S3. +(35d) +Therefore, J1, J3 and J4 form an SO(3) algebra as implied +by eq. (34a), and furthermore, all three of them commute +with J2, as implied by eq. (34c). This pattern is precisely +what we observe in the lower panel of Figure 11. + +Generator 1 +Generator 2 +Generator 3 +Generator 4 +1 +1 +1 +0.75 +0.50 +2 +2 +0.25 +0.00 +3 +3 +0.25 +0.5 +4 +4 +0.75 +4 +4 +i +4 +1.00Structure Constants +12 +1D +13 +0.5 +14 +0.0 +0.5 +tz +-1.0 +1 +2 +4 +GeneratorGenerator 1 +Generator 2 +Generator 3 +1DO +1 +0.75 +2 +2 +3 +3 +0.50 +3 +4 +0.25 +1 +3 +0.0 +Generator 4 +Generator 5 +Generator 6 +0.25 +2 +2 +2 +0.50 +3 +3 +3 +DO't- +1 +2 +3 +2 +3 +4 +1 +3Structure Constants +0.B +12 +13 +0.6 +14 +15 +t0 +16 +2 +0.2 +t +Ja +25 +26 +0.2 +35 +36 +0.4 +45 +0.6 +46 +56 +0.8 +Generator12 +FIG. 13. Results from our search for Lie algebras of transfor- +mations preserving the Euclidean oracle (25) in n = 2, 3, 4, 5 +dimensions and for different number of generators Ng. The +cells are color coded by the base-10 logarithm of the lowest +value of the loss attained during training. +D. +Six generator algebras +Finally, we get to the case of Ng which will reveal the +full algebra of SO(4). Figure 12 shows the result from a +representative training run seeking a closed algebra with +Ng = 6 generators. Among the set of learned generators +we can approximately recognize J1 ≈ O43, J2 ≈ −D3, +J3 ≈ S3, J4 ≈ O12, J5 ≈ O42 and J6 ≈ O13. +The analysis of the last three sections can be readily +extended to even higher dimensions (n ≥ 5). We have +checked a few more values of n and the method works +each time — we obtain valid finite symmetry transforma- +tions which preserve the oracle (25), we find the closed +algebra of the full set of n(n − 1)/2 generators of the or- +thogonal O(n) group, as well as valid subalgebras. For +fun, in Figure 26 we depict our derived 45 generators of +the SO(10) group. +In conclusion of our discussion of orthogonal groups, in +Figure 13 we summarize our results for the found closed +algebras and subalgebras for n ≤ 5 and different number +of generators Ng. Each cell in the table represents a sepa- +rate exercise at a fixed number of dimensions n and for a +fixed number of generators Ng. The cells are color coded +by the base-10 logarithm of the lowest value of the loss +attained during training. The training is terminated once +the loss reaches 10−12, therefore blue cells correspond to +successful training runs resulting in closed algebras. The +right-most blue cell in each row corresponds to the full +algebra, in this case SO(n), as indicated. The preced- +ing blue cells correspond to various subalgebras. In fact +we did not anticipate the existence of the Ng = 4 sub- +algebras in the case of n = 4 and n = 5, but our model +surprised us. +VIII. +LORENTZ TRANSFORMATIONS IN +FOUR DIMENSIONAL MINKOWSKI SPACE +In this section we consider the four-dimensional +Minkowski spacetime (t, x, y, z) (in natural units with +c = 1). The usual Lorentz transformations preserve the +quadratic form +ϕ(t, x, y, z) = t2 − x2 − y2 − z2, +(t, x, y, z) ∈ R4, (36) +hence this will be our oracle in this section. The four +input features are +x(0) = t, +x(1) = x, +x(2) = y, +x(3) = z, +(37) +where in keeping with the standard physics notation we +start labelling the features from 0. +Our analysis proceeds as usual. First, we find symme- +try transformations which are in general combinations of +boosts and rotations. Upon inspection, we verify that +they have the desired symmetry properties +F0i = Fi0, +Fij = −Fji, +F00 = Fii = 0, +∀i = 1, 2, 3. +Next we analyze the algebras of generators. +Before +presenting the numerical results, for the reader’s conve- +nience we summarize some relevant information about +the mathematical structure of the Lorentz group O(1, 3). +It has six generators: the three generators of boosts Ki, +K1 = +� +� +� +0 1 0 0 +1 0 0 0 +0 0 0 0 +0 0 0 0 +� +� +� , +(38a) +K2 = +� +� +� +0 0 1 0 +0 0 0 0 +1 0 0 0 +0 0 0 0 +� +� +� , +(38b) +K3 = +� +� +� +0 0 0 1 +0 0 0 0 +0 0 0 0 +1 0 0 0 +� +� +� , +(38c) +and the three generators of rotations, Li, given by +L1 = +� +� +� +0 0 0 +0 +0 0 0 +0 +0 0 0 −1 +0 0 1 +0 +� +� +� , +(39a) +L2 = +� +� +� +0 +0 0 0 +0 +0 0 1 +0 +0 0 0 +0 −1 0 0 +� +� +� , +(39b) +L3 = +� +� +� +0 0 +0 0 +0 0 −1 0 +0 1 +0 0 +0 0 +0 0 +� +� +� , +(39c) + +2 -SO(2) +0 +3 - +SO(3) +n +-5 +4 - +SO(4) +5 +SO(5) +-10 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Ng13 +K1 − L2 K2 + L1 +K3 +L3 +L2 +L1 +X1 +X2 +X3 +X4 +X5 +X6 +X1 +0 +0 +−X1 +−X2 ++X3 ++X4 +X2 +0 +0 +−X2 ++X1 ++X4 +−X3 +X3 ++X1 ++X2 +0 +0 +−X1 − X5 X2 − X6 +X4 ++X2 +−X1 +0 +0 +−X6 ++X5 +X5 +−X3 +−X4 ++X1 + X5 +X6 +0 +−X4 +X6 +−X4 ++X3 +−X2 + X6 −X5 ++X4 +0 +TABLE I. The Lorentz algebra in terms of the generators +(43). The cells show the result from commuting one of the +generators in the leftmost column with one of the generators +in the top row. +The full Lorentz algebra can be summarized as +[Li, Lj] = ϵijk Lk, +(40) +[Li, Kj] = ϵijk Kk, +(41) +[Ki, Kj] = −ϵijk Lk. +(42) +The subgroup structure is most easily seen in a different +basis, +X1 = K1 − L2, +(43a) +X2 = K2 + L1, +(43b) +X3 = K3, +(43c) +X4 = L3, +(43d) +X5 = L2, +(43e) +X6 = L1. +(43f) +The resulting algebra in the Xi basis is summarized in +Table I. By inspection, we see that the Lorentz algebra +has several non-trivial subalgebras. +A. +Two generator subalgebras +There are several two-dimensional subalgebras: +• Abelian subalgebras. There are two abelian subal- +gebras: the first one is generated by the genera- +tor set {X1, X2}, while the second is generated by +{X3, X4}. A representative example of the former +case is shown in Figure 14. We recognize the two +FIG. 14. +Top row: successfully learned generators for the +n = 4, Ng = 2 exercise considered in Section VIII A, using +the pseudo-Euclidean oracle (36). Bottom panel: the corre- +sponding structure constants. This example shows an abelian +subalgebra. +FIG. 15. +The same as Fig. 14, but showing a non-abelian +subalgebra found by our method. +found generators to be approximately +J1 ≈ − (K3 − L1) , +(44a) +J2 ≈ − (K1 + L3) . +(44b) +This result is in agreement with eqs. (43a) and +(43b) — the transformations are combinations of +boosts along and rotations about two of the axes, +in this case x and z. +• Nonabelian subalgebras. As seen in Table I, the Lie +algebra of the Lorentz group has two nonabelian +subalgebras, generated by {X1, X3} and {X2, X3}, + +Generator 1 +Generator 2 +1.00 +1 +1 +0.75 +0.50 +2 +2 +0.25 +0.00 +3 - +3 +-0.25 +0.50 +4 - +4 +0.75 +2 +3 +4 +1 +2 +3 +4 +1.00Structure Constants +1.00 +0.75 +0.50 +Bracket +0.25 +12 + 0.00 +0.25 +0.50 +0.75 +1.00 +1 +2 +GeneratorGenerator 1 +Generator 2 +1.00 +1 +0.75 +1. +0.50 +2 +0.25 +0.00 +E +0.25 +0.50 +4 - +4 +0.75 +2 +E +4 +1 +2 +m +4 +1.00Structure Constants +1.00 +0.75 +0.50 +Bracket +0.25 +12 +0.00 +0.25 +0.50 +0.75 +1.00 +1 +2 +Generator14 +FIG. 16. +Top row: the successfully learned generators for +the n = 4, Ng = 3 exercise considered in Section VIII B, +using the pseudo-Euclidean oracle (36). Bottom panel: the +corresponding structure constants, which can be identified as +those of an SO(3) type subalgebra. +respectively. A representative training example for +this nonabelian case is shown in Figure 15. The +top panels show that the two found generators are +approximately +J1 ≈ − K3 = − X3, +(45a) +J2 ≈ K1 − L2 = X1, +(45b) +while the bottom panel confirms that their Lie bracket +is approximately given by [J1, J2] ≈ [−X3, X1] = −X1 = +− J2, as expected from Table I. +B. +Three generator subalgebras +Table I reveals several three-dimensional subalgebras: +• SO(3). The set {X4, X5, X6} generates a subalgebra +isomorphic to the Lie algebra of the SO(3) group +of rotations in three dimensions, see Section VI. +A representative training example is shown in Fig- +ure 16. The generators can be recognized as +J1 ≈ L3, +(46a) +J2 ≈ L1, +(46b) +J3 ≈ L2. +(46c) +In addition, we also found examples with subalge- +bras isomorphic to so(3) consisting of one rotation +and two boosts, or one boost and two rotations. +FIG. 17. The same as Fig. 16, but showing an example of a +Hom(2) subalgebra (47). +• Hom(2). The set {X1, X2, X3} generates a subalge- +bra isomorphic to the Lie algebra of Hom(2), the +group of Euclidean homotheities: +[X1, X2] = 0, [X3, X1] = X1, [X2, X3] = −X2. +(47) +A representative training example is shown in Fig- +ure 17. From the panels in the top row we recognize +the found three generators as +J1 ≈ + K2 − L1, +(48a) +J2 ≈ − K1 − L2, +(48b) +J3 ≈ + K3. +(48c) +The bottom panel in Figure 17 confirms that their +algebra is approximately that of eq. (47). +• E(2). +The set {X1, X2, X4} generates a subalge- +bra isomorphic to the Lie algebra of E(2), the Eu- +clidean group: +[X1, X2] = 0, [X4, X1] = X2, [X2, X4] = X1. +(49) +A representative example is shown in Figure 18, +from which we recognize the found three generators +as +J1 ≈ − K3 − L2, +(50a) +J2 ≈ − L1, +(50b) +J3 ≈ − K2 + L3. +(50c) +As expected, two of the generators involve combi- +nations of boosts along and rotations about two of + +Generator 1 +Generator 2 +Generator 3 +1.00 +1 +1 +1 +0.75 +0.50 +2 +2 +2 +0.25 +0.00 +3 +3 +3 +0.25 +0.50 +4 +4 +0.75 +1.00 +1 +2 +3 +4 +1 +4 +1 +2 +3 +4Structure Constants +1.00 +0.75 +12 +0.50 +Bracket +0.25 +31 +0.00 +0.25 +0.50 +23 - +-0.75 +-1.00 +1 +2 +GeneratorGenerator 1 +Generator 2 +Generator 3 +1.00 +1 +1 +0.75 +0.50 +2 +2 +2 +0.25 +0.00 +3 +3 +0.25 +0.50 +4 +4 +4 +0.75 +2 +1 +1.00 +1 +2 +3 +4 +1 +3 +2 +3 +4Structure Constants +1.00 +0.75 +12 +0.50 +Bracket +0.25 +31 +0.00 +0.25 +0.50 +23 +0.75 +-1.00 +1 +2 +Generator15 +FIG. 18. The same as Fig. 16, but showing an example of a +E(2) subalgebra (49). +the axes, in this case y and z, while the third gener- +ator is a rotation about the remaining axis, namely +x. The bottom panel in Figure 18 shows that the +resulting algebra is approximately that of Eq. (49). +• Bianchi VIIa. +The set {X1, X2, X3 + aX4} with +a ̸= 0 generates a Bianchi VIIa subalgebra. A rep- +resentative training example is shown in Figure 19, +from which we recognize the found three generators +as +J1 ≈ − K3 − L3 = −(X3 + X4), +(51a) +J2 ≈ + K1 − L2 = X1, +(51b) +J3 ≈ − K2 − L1 = − X2. +(51c) +This set corresponds to a Bianchi VIIa subalgebra +with a = 1, whose structure constants are indeed +consistent with the lower panel in Figure 19. +• SL(2,R). The set {X1, X3, X5} generates a subalge- +bra isomorphic to the Lie algebra of SL(2, R), the +group of isometries of the hyperbolic plane: +[X1, X3] = − X1, +(52a) +[X5, X1] = − X3, +(52b) +[X3, X5] = − X1 − X5. +(52c) +A representative training example is shown in Fig- +ure 20, from which we recognize the found three +generators as +J1 ≈ K1 − L3, +(53a) +FIG. 19. +The same as Fig. 16, but showing an example of a +Bianchi VIIa subalgebra. +FIG. 20. +The same as Fig. 16, but showing an example of +an SL(2,R) subalgebra (52). +J2 ≈ − L3, +(53b) +J3 ≈ K2. +(53c) +It is easy to verify that their structure constants +given in the lower panel of Figure 20 are consistent +with Eq. (52). + +Generator 1 +Generator 2 +Generator 3 +1.00 +1 +1 +1 +0.75 +0.50 +2 +2 +2 +0.25 +0.00 +3 +E +3 +-0.25 +0.50 +4 +4- +4 +0.75 +1 +2 +3 +1 +1.00 +2 +3 +4 +1 +4 +2 +3 +4Structure Constants +1.00 +0.75 +12 +0.50 +Bracket +0.25 +31 +0.00 +0.25 +0.50 +23 - +0.75 +-1.00 +1 +2 +3 +GeneratorGenerator 1 +Generator 2 +Generator 3 +1.00 +1 +1 +1 +0.75 +0.50 +2 +2 +2 +0.25 +0.00 +3 +3 +3 +-0.25 +-0.50 +4 - +0.75 +1 +1 +1.00 +2 +3 +4 +3 +4 +1 +2 +4Structure Constants +1.00 +0.75 +12 +0.50 +0.25 +Bracket +31 +0.00 +0.25 +0.50 +23 +0.75 +1.00 +1 +2 +3 +GeneratorGenerator 1 +Generator 2 +Generator 3 +1.00 +1 +1 +1 +0.75 +0.50 +2 +2 +2 +0.25 +0.00 +3 +3 +3 +-0.25 +0.50 +4 +4 +4 +0.75 +2 +1 +2 +1.00 +1 +2 +3 +4 +7 +4 +E +4Structure Constants +1.00 +0.75 +12 +0.50 +Bracket +0.25 +31 +0.00 +-0.25 +0.50 +23 +-0.75 +-1.00 +1 +2 +Generator16 +FIG. 21. +Top row: the successfully learned generators for +the n = 4, Ng = 4 exercise considered in Section VIII C, +using the pseudo-Euclidean oracle (36). Bottom panel: the +corresponding structure constants. +C. +Four generator subalgebras +The only four-dimensional subalgebra is generated by +{X1, X2, X3, X4}. A representative example is shown in +Figure 21. The four found generators are given by +J1 ≈ L3 = X4, +(54a) +J2 ≈ K2 + L1 = X2, +(54b) +J3 ≈ − K1 + L2 = − X1, +(54c) +J4 ≈ − K3 = − X3. +(54d) +The lower panel of Figure 21 illustrates the corresponding +structure constants. The results are consistent with Ta- +ble I. For example, J1 and J4 commute because [X3, X4] = +[K3, L3] = 0. +Similarly, J2 and J3 commute due to +[X1, X2] = 0. The results shown in the remaining rows in +the lower panel of Figure 21 can be analogously verified +with the help of Table I. +D. +Six generator algebras +Our +method +is +capable +of +finding +the +full +six- +dimensional algebra as well. The result is shown in Fig- +ure 22, and can be roughly identified as +J1 ≈ − L1, +J2 ≈ − K2, +J3 ≈ − L3, +(55a) +J4 ≈ + K1, +J5 ≈ − K3, +J6 ≈ − L2. +(55b) +FIG. 22. +Top row: the successfully learned generators for +the n = 4, Ng = 6 exercise considered in Section VIII D, +using the pseudo-Euclidean oracle (36). Bottom panel: the +corresponding structure constants. +The corresponding structure constants are shown in the +lower panel of Figure 22. +IX. +SQUEEZE MAPPING IN TWO +DIMENSIONS +Consider again two dimensions, but now let the oracle +return the product of the two input features: +ϕ(x) = x(1) x(2). +(56) +This oracle function is illustrated in Figure 23 as a color +heatmap representing the oracle values in the (x(1), x(2)) +Cartesian plane. We see that the set of points with the +same oracle values form hyperbolas. + +Generator 1 +Generator 2 +Generator 3 +Generator 4 +1.00 +0.75 +1 +1 +1 +0.50 +2 +2 +2 +2 +0.25 +0.00 +3 - +3 +3 +-0.25 +0.50 +4 - +4 +4 +4- +0.75 +2 +3 +4 +1 +2 +i +3 +F4 +1 +2 +4. +4 +1.00Structure Constants +1.00 +12 +0.75 +13 +0.50 +0.25 +14 +Bracket +0.00 +23 +-0.25 +24 +0.50 +0.75 +34 +1 +2 +m +4 +1.00 +GeneratorGenerator 1 +Generator 2 +Generator 3 +1DO +1 +1 +0.75 +2 +2 +2. +3 +3 +3 +0.50 +4 +4 +0.25 +i +2 +4 +2 +4 +i +3 +0.00 +Generator 4 +Generator 5 +Generator 6 +1 +1 +1 +0.25 +2. +2 +2 +0.50 +3 +m +3 +0.75 +4 +4 +4 - +i +2 +3 +2 +i +2 +3 +1.00 +4 +4Structure Constants +12 +13 +0.6 +14 +15 +t0 +16 +21 +0.2 +Ja +25 +0.D +26 +0.2 +35 +36 +45 +0.4 +46 +56 +0.6 +i2 +34 +56 +Generator17 +FIG. 23. Color heatmap showing the values of the oracle (56) +in the (x(1), x(2)) Cartesian plane. The superimposed vector +field represents the symmetry found by our method. +Proceeding as before, our method finds the symmetry +transformation illustrated with the vector field in Fig- +ure 23. The corresponding generator is +J = +� +−1 0 +0 1 +� +. +(57) +We verified that in this example the method is unable to +find more than one generator. +These results are precisely what one would expect. The +symmetry which preserves the oracle (56) is the squeeze +mapping +F = +� +1 +ℓ 0 +0 ℓ +� +, +(58) +where ℓ is a scale factor. Considering infinitesimal trans- +formations (18) with ℓ = 1 + ε immediately leads to the +generator (57). +X. +DISCONTINUOUS ORACLES +A. +Piecewise Linear Oracle +Our setup is not limited to only continuous oracle func- +tions like the ones discussed so far in the preceeding sec- +tions. Our method can also work with piecewise-defined +functions like +ϕ(x) = +� +−x(2), +for x(1) < 0, ++x(1), +for x(1) ≥ 0. +(59) +The resulting oracle function is illustrated with the +color map in Figure 24. Since the oracle is now a function +FIG. 24. The symmetry generated by the oracle (59) with a +shallow method with no hidden layers and no bias (top panel) +or a deep method with three hidden layers and bias (bottom +panel). +which is not continuously differentiable, our parametriza- +tion of the symmetry transformation needs to be properly +generalized. +The advantage of using a neural network as a universal +function approximator is highlighted in Figure 24. In the +top panel we use no hidden layers, while in the bottom +panel we use a deep learning architecture with three hid- +den layers. The found symmetry transformation in each +case is then shown with the vector field and superimposed +on the oracle color map. The results are noticeably dif- +ferent, particularly near the locations of discontinuity in +the oracle function. The deep-learning approach in the +bottom panel correctly identifies a transformation which +preserves the oracle values everywhere within the consid- +ered domain. In contrast, the shallow approach in the +top panel is unable to adjust the transformation near the +discontinuity, which leads to locations near the boundary +x(1) = 0 where the arrows run across the equipotential +contours, violating the conservation law. + +2 +t +t +t +X +3 +2 +1 +(x)b) +0 +0 +X +-1 +-2 +-3 +-2 +-4 +-2 +-1 +0 +1 +2 ++(1)2 +1.80 +1.35 +0.90 +0.45 +(X)d) +0 +0.00 +X +-0.45 +-0.90 +-1.35 +-1.80 +-2 +-2 +-1 +0 +1 +2 +×(1)2 +1.80 +1.35 +0.90 +0.45 +(x)d) +0 +0.00 +X +-0.45 +-0.90 +-1.35 +-1.80 +-2 +-2 +-1 +0 +1 +2 +×(1)18 +FIG. 25. The symmetry generated by the L1 oracle (60) with +a shallow method with no hidden layers and no bias (top +panel) or a deep method with three hidden layers and bias +(bottom panel). +B. +Manhattan Distance Oracle +In this subsection we consider one more example of an +oracle in two dimensions (n = 2), which, while continu- +ous, is not continuously differentiable: +ϕ(x) = |x(1)| + |x(2)|. +(60) +The results from our procedure are shown in Figure 25 +in complete analogy to Figure 24. In the top panel we +use no hidden layers, while in the bottom panel we use a +deep learning architecture with three hidden layers and +bias. We observe that the deep-learning approach can +again correctly handle the discontinuities, always gener- +ating transformations along, but never across, the con- +tours of equal oracle function values. +XI. +CONCLUSIONS +In this paper, we studied a fundamental problem in +data science which is commonly encountered in many +fields: what is the symmetry of a labeled dataset, and +how to identify its group structure? For this purpose, +we designed a deep-learning method which models the +generic symmetry transformation and its generators with +a fully connected neural network. We then constructed +suitable loss functions which ensure that the applied +transformations are symmetries and that the correspond- +ing set of generators forms a closed (sub)algebra. +An +important advantage of our approach is that we do not +require any advance knowledge of what symmetries can +be expected in the data, i.e., instead of testing for sym- +metries from a predefined list of possibilities, we learn +the symmetry directly. +Our procedure is very general and is universally appli- +cable in a wide variety of situations. The centerpiece of +our analysis is an oracle ϕ(x) which defines the conserved +quantity. The oracle can be completely general, as illus- +trated with several examples in the paper. For example, +it can be a continuous bilinear function, as in the case +of the Euclidean or Minkowski distances and the squeeze +mapping; the corresponding symmetries were discussed +in Sections V-IX. It can also be a discontinuous func- +tion (Section X A) or a function which is not smooth and +continuously differentiable (Section X B). +In the process of deriving the full set of symmetries, +the method also allows us to analyze the complete sub- +group structure of the symmetry group. +In the paper +we worked out explicitly the subgroup structure of the +rotation groups SO(2) (Section V), SO(3) (Section VI) +and SO(4) (Section VII), as well as the Lorentz group +SO(1, 3) ((Section VIII)). As one last tour de force ex- +ample, we successfully applied our method to the case of +Euclidean length-preserving rotations in n = 10 dimen- +sions. The resulting 45 generators of the corresponding +group SO(10) commonly used in grand unification [54] +are shown in Figure 26. +The symmetries discussed in this paper have impor- +tant implications for simulation-based inference, and in +particular parameter retrieval from observations. +In a +typical inverse problem scenario, the (possibly multi- +dimensional) labels y play the role of observed variables, +and the features x play the role of input parameters. +The parameter retrieval task is to infer x given y. In the +presence of a symmetry this inversion is not unique, but +results in a whole family of valid input parameters, all re- +lated by a symmetry. Therefore, knowing that there is a +symmetry in the data to begin with can be very useful in +parameter retrieval algorithms, especially when the data +is very high dimensional and the symmetry is difficult to +see by the human eye. +Note that the same approach can be extended or mod- +ified to solve other common problems and tasks in group + +2 +4.05 +3.60 +3.15 +2.70 +2.25 +(x)d) +0 +X +1.80 +1.35 +0.90 +0.45 +-2 +0.00 +-2 +-1 +0 +1 +2 ++(1)2 +4.05 +3.60 +3.15 +2.70 +2.25 +(x)d) +0 +X +1.80 +1.35 +0.90 +0.45 +-2 +0.00 +-2 +-1 +0 +1 +2 +x(1)19 +FIG. 26. The set of Ng = 45 generators found by our method in the case of the SO(10) group (length-preserving rotations in +n = 10 dimensions). Due to the complexity of the problem, the learning rate for this example was reduced to 0.01. +theory and/or BSM model building not discussed in this +paper. +• Derivation of the Cartan subalgebra. +In our ap- +proach, this can be trivially accomplished by set- +ting a[αβ]γ = 0 in (15), which will force all the +learned generators to commute. +• Direct product decompositions. +We can also look +for symmetries whose algebras can be expressed as +direct sums of two separate subalgebras — we just +have to include closure terms in the loss function +for each of the two individual subalgebras. +• Internal symmetries. Note that our approach can +be readily generalized to look for internal symme- +tries [41], which can prove useful in model building +in high energy theory. +• Canonical basis for the generators. +Since our +method is basis-independent, the generators are ob- +tained in a generic basis, where the structure con- +stants may be difficult to recognize. We could, how- +ever, aid the program in finding a canonical basis +of generators by including a term in the loss func- +tion which encourages sparsity among the structure +constants. +In this paper we focused on idealized examples with +no noise or statistical fluctuations in the determination +of the labels. It will be instructive to test our method +on noisy datasets, including real experimental data. We +postpone this study to a future publication. +In conclusion, our study opens the door for using a +machine learning approach in the study of Lie groups and +their properties and further bridges the fields of formal +mathematics and theoretical computer science. +ACKNOWLEDGMENTS +We thank A. Davis, S. Gleyzer, K. Kong, S. Mrenna, +H. Prosper and P. Shyamsundar for useful discussions. +We thank P. Ramond for group theory insights and in- +spiration. This work is supported in part by the U.S. De- +partment of Energy award number DE-SC0022148. +[1] David J. 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C 790927, 315–321 (1979), arXiv:1306.4669 [hep- +th]. + diff --git a/fNE5T4oBgHgl3EQfhA-Z/content/tmp_files/load_file.txt b/fNE5T4oBgHgl3EQfhA-Z/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3fee72d874d83f83be84212b0b08f289202101c4 --- /dev/null +++ b/fNE5T4oBgHgl3EQfhA-Z/content/tmp_files/load_file.txt @@ -0,0 +1,1162 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf,len=1161 +page_content='Deep Learning Symmetries and Their Lie Groups, Algebras, and Subalgebras from First Principles Roy Forestano,1, ∗ Konstantin T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Matchev,1, † Katia Matcheva,1, ‡ Alexander Roman,1, § Eyup Unlu,1, ¶ and Sarunas Verner1, ∗∗ 1Institute for Fundamental Theory, Physics Department, University of Florida, Gainesville, FL 32611, USA (Dated: January 12, 2023) We design a deep-learning algorithm for the discovery and identification of the continuous group of symmetries present in a labeled dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' We use fully connected neural networks to model the symmetry transformations and the corresponding generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' We construct loss functions that ensure that the applied transformations are symmetries and that the corresponding set of generators forms a closed (sub)algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Our procedure is validated with several examples illustrating different types of conserved quantities preserved by symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' In the process of deriving the full set of symmetries, we analyze the complete subgroup structure of the rotation groups SO(2), SO(3), and SO(4), and of the Lorentz group SO(1, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Other examples include squeeze mapping, piecewise discontinuous labels, and SO(10), demonstrating that our method is completely general, with many possible applications in physics and data science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Our study also opens the door for using a machine learning approach in the mathematical study of Lie groups and their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' CONTENTS I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Introduction 1 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Setup and notations 3 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Deep Learning Approach 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Invariance 4 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Infinitesimality 4 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Orthogonality 5 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Closure 5 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Linear Algebra Approach 5 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Length-preserving Transformations in Two Dimensions 6 VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Length-preserving Transformations in Three Dimensions 8 VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Length-preserving Transformations in Four Dimensions 9 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Two generator subalgebras 9 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Three generator subalgebras 10 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Four generator subalgebras 10 ∗ roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='forestano@ufl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='edu † matchev@ufl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='edu ‡ matcheva@ufl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='edu § alexroman@ufl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='edu ¶ eyup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='unlu@ufl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='edu ∗∗ verner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='s@ufl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='edu D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Six generator algebras 12 VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Lorentz Transformations in Four Dimensional Minkowski Space 12 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Two generator subalgebras 13 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Three generator subalgebras 14 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Four generator subalgebras 16 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Six generator algebras 16 IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Squeeze Mapping in Two Dimensions 16 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Discontinuous Oracles 17 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Piecewise Linear Oracle 17 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Manhattan Distance Oracle 18 XI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Conclusions 18 Acknowledgments 19 References 19 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' INTRODUCTION Symmetries play a fundamental role in modern physics [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Physical systems with continuous symmetries exhibit conservation laws that are universally applicable and in- dispensable in understanding the system’s behavior and evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' In particle physics, symmetries provide an or- ganizing principle behind the observed particle zoo and its interactions, and guide model-builders in the search for viable extensions of the Standard Model (SM)[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' At the same time, the mathematical study of symmetries is arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='05638v1 [hep-ph] 13 Jan 2023 2 interesting in its own right and has a rich history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Over the last decade, there has been increased inter- est in applications of machine learning (ML) to high- dimensional physics data as a sensitive tool for event simulation, data analysis, and statistical inference [3–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' More recently, ML is also being used to facilitate tasks that traditionally have fallen within the domain of the- orists, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=', performing symbolic computations [6, 7] or deriving analytical formulas by training a symbolic re- gression on synthetic data [8–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Applications of ML to the study of symmetries have been pursued by a number of groups in different contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' One line of work investigates how a given symmetry is reflected in a learned representation of the data [21, 22] or in the ML architecture itself, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=', in the embedding layer of a neural network (NN) [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Several proposals attempt to design special ML architectures (equivariant NNs) which have a desired symmetry property built in from the outset [24–30] and test their performance [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Incorporating the symmetry directly into the ML model makes it more economical (in terms of learned represen- tations), interpretable and trainable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The approach can be extended to cover discrete (permutation) symmetries as well [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Such efforts pave the way for data-driven blind searches for new physics which stress-test the data for violations of a well-established symmetry of the SM [34–36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' More recently, machine learning is also being applied to address more formal theoretical questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' For example, a good understanding of the symmetries present in the problem can reveal conserved quantities [37, 38] or hint at a more fundamental unified picture [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' ML has been used to discover the symmetry of a potential [23, 40, 41], to decide whether a given pair of inputs is related by sym- metry or not [42], to distinguish between scale-invariant and conformal symmetries [43], and to explore the string landscape [44–46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Recent work made use of Generative Adversarial Networks to learn transformations that pre- serve probability distributions [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' ML applications have also found their way into group theory, which provides the abstract mathematical language of symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' For example, recent work used ML to compute tensor prod- ucts and branching rules of irreducible representations of Lie groups [48] and to obtain Lie group generators of a symmetry present in the data [49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The main goal of this paper is to design a deep- learning method that mimics the traditional theorist’s thinking and is capable of discovering and categorizing the full set of (continuous) symmetries in a given dataset from first principles, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=', without any prior assumptions or prejudice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The only input to our procedure is a labeled dataset {x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' y} like the one in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (1) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' An oracle ϕ(x) = y can then be learned from the dataset, or alternatively, can be provided externally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' With those ingredients, we go through the following objectives: Discovery of symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' In the first step, de- scribed in Section III A, we learn to generate a sym- metry transformation, x f→ x′, which preserves the oracle values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The transformation f is encoded in a neural network trained on the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' In general, there will be many possible symmetry transforma- tions f, and their study and categorization from a group theory point of view is the main goal of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Discovery of symmetry generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Having learned how to generate arbitrary (finite) symme- try transformations f, we then focus on infinitesi- mal transformations δf, which give us in turn the symmetry generators J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Discovery of Lie subalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' By adding suit- able terms to the loss function, our procedure re- quires that the learned set of generators {Jα} forms a closed algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' This allows us to discover subal- gebras of the symmetry group, and identify them by their structure constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Since the number of generators Ng is a free input, in cases when the training is unsuccessful (as quantified by the loss), we can rule out the existence of Ng-dimensional subalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Discovery of the full Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The maxi- mum value of Ng which gives a vanishing loss, in- dicates the dimension of the full Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The corresponding learned set of generators describes the full symmetry group of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Identification of the symmetry group and its subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The learned sets of generators found in the previous two steps are then used to obtain the structure constants of the respective full alge- bra and its subalgebras, and thus to identify the corresponding symmetry group and its subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Our study complements and extends previous related work in [23, 40, 41, 49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' We note that our procedure is completely general, and does not anticipate what sym- metries might be present in the dataset — instead, the symmetries are learned from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' In addition, our method is basis-independent since we do not choose a specific convenient basis for the learned transformations and generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Consequently, our results will not always match the nice canonical forms of the generators found in the group theory textbooks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Nevertheless, as the exam- ples below explicitly demonstrate, all our learned trans- formations and generators will satisfy the defining prop- erties of the respective symmetry groups and subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' In Section II we in- troduce the setup for our analysis and the corresponding notation and conventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The main steps of our deep- learning procedure are outlined in Section III, which also explains and motivates the necessary ingredients for the 3 loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' For readers who are not yet fully comfort- able with a deep-learning approach, Section IV outlines an analogous linear algebra approach that often (but not always — see Section X for counterexamples) can accom- plish similar objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Each of the remaining sections contains a separate completely worked-out example illus- trating our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The examples are distinguished by the choice of oracle and the dimensionality of the fea- ture space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' In Sections V-VII we choose an oracle that returns the Euclidean distance in feature space, whose dimensionality, in turn, is chosen to be n = 2 in Sec- tion V, n = 3 in Section VI, and n = 4 in Section VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' In Section VIII we focus on the Lorentz group, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=', the oracle computes the pseudo-Euclidean distance in four- dimensional Minkowski space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' In Section IX we consider the squeeze transformations in n = 2 dimen- sions whereby the oracle returns the product of the two features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' To demonstrate the universal applicability of our technique, in Section X we show two examples of discontinuous oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' We summarize and conclude in Section XI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' SETUP AND NOTATIONS Our starting point is a labeled dataset containing m samples of n features and a target label y: m samples � � � � � � � � � � � x(1) 1 , x(2) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' , x(n) 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' y1 x(1) 2 , x(2) 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' , x(n) 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' y2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' x(1) m , x(2) m , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' , x(n) m ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' ym .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (1) In ML parlance, the dataset (1) is an m × n dataframe with n features and m samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' In what follows, we use the sample index a lot more often than the feature index, thus we use an explicit subscript for the sample index and hide the feature index in the boldface vector notation x: x ≡ {x(1), x(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' , x(n)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (2) This allows us to write the input features in a compact form as {xi} ≡ {x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' , xm} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (3) In order to study the symmetries of the data (1), we need to know the function y(x), which can be given ana- lytically or numerically in terms of an oracle ϕ : Rn → R capable of computing the corresponding output target labels y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' , ym: yi = ϕ(xi) , i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' , m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (4) This leads to two basic scenarios: The function y(x) is known analytically, and that same function has been used as in (4) to compute the labels in (1) exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' This case is of interest to theorists, and this is the approach adopted in this paper as well — for each of our examples below, we specify the relevant analytic oracle ϕ(x) and pro- ceed to study the resulting symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Note, how- ever, that we never take advantage of the knowl- edge of the analytical form of the oracle, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=', we do not differentiate or symbolically manipulate in any other way the function ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' For our purposes, we only use the oracle numerically — our method treats it simply as a black box, which, given the values of x, can produce the numerical value of the label y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The functional dependence y(x) is a priori unknown and the dataset (1) is all that is available to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' This is the typical case encountered by data scien- tists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Now, one needs to go through a preliminary step of first creating the oracle ϕ (usually in the form of a neural network trained on the dataset (1)), which is capable of computing and reporting the values of y = ϕ(x) to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' This is a standard regression task which is of no interest here since it can be accomplished using one of the many estab- lished ML regression techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' We can therefore safely assume that this preliminary step has already been completed and we have such a numerical ora- cle y = ϕ(x) already available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Since we are only using the oracle numerically, from our point of view there is no real difference between the above two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' In what follows the oracle ϕ(x) will be used in the exact same way, regardless of its origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Given this general setup, our main task is to derive the symmetry transformation f : Rn → Rn x′ = f(x) , (5) which preserves the ϕ-induced labels of our dataset (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' In other words, we want to find the function f(x) for which ϕ(x′ i) ≡ ϕ(f(xi)) = ϕ(xi), ∀i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' , m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (6) The particular instantiation of the symmetry f(x) will depend on the initialization of our parameters, so by re- peating the procedure with different initializations, we will in principle obtain a whole family of symmetry trans- formations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Next, we focus on infinitesimal symmetry transforma- tions and proceed to study the corresponding set of gener- ators {Jα}, with α = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' , Ng, where we use lowercase Greek letters to label the generators of symmetry trans- formations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' A given set of generators {Jα} forms a Lie algebra if the closure condition is satisfied, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=', if all Lie brackets � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' � can be represented as linear combinations of the generators in the set: � Jα, Jβ � = Ng � γ=1 a[αβ]γJγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (7) 4 The coefficients a[αβ]γ are the structure constants (anti- symmetric in their first two indices, as implied by the square brackets) whose values will reveal the symmetry group present in our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' In principle, the number of generators Ng is a hyperpa- rameter that must be specified ahead of time (similarly to the choice of the number of clusters in certain cluster- ing algorithms like K-means).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Therefore, when we find a closed algebra at a given Ng value, we are only guar- anteed that it is a subalgebra, and we must proceed to try out higher values for Ng as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The full algebra will then correspond to the maximum value of Ng for which a closed algebra of distinct generators is found to exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' DEEP LEARNING APPROACH In our approach, we model the output function f with a neural network (NN) FW with n neurons in the out- put layer, corresponding to the n transformed features of the data point x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The trainable network parameters (weights and biases) will be generically denoted with W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' During training, they will evolve and converge to the cor- responding trained values � W of the parameters of the trained network F� W, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=', the hat symbol will denote the result of the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Once the parameters � W are found, we can find the structure constants using standard meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' We choose a suitable loss function that ensures the desired properties of the trained network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The following subsections discuss the individual contributions to the loss function, which in our implementation are combined and minimized simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The neural network FW is implemented as a sequential feed-forward neural network in PyTorch [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The ex- amples in Sections V-VIII are simple enough to be done with no hidden layers, no bias, and no activation func- tion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=', with linear transformations (see Section IV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' For the examples in the later sections, we do add hidden layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Optimizations are performed with the Adam op- timizer with a learning rate between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='03 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The loss functions were designed to achieve a fast and efficient training process without the need for extensive hyperpa- rameter tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The training data (3) was typically on the order of a few hundred points and was sampled from a standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' An alternative approach to predicting the generators from the model parameters, which utilizes an identi- cal loss function, can be carried out where a NN, F : {Gα, a} → { �Gα, ˆa}, takes a set of randomly initialized n×n generators {Gα} and an Nb ×Ng structure constant array {a}, flattens each individual generator and the structure constant array, and proceeds to converge the elements to the desired set of generators {� Gα} and struc- ture constant array {ˆa}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Here Nb = �Ng 2 � = Ng(Ng−1) 2 denotes the number of unique one-directional Lie brack- ets for a given number of generators Ng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' This alternative approach can be implemented as a module list of sequential neural network layers consist- ing of two hidden layers for the generators and a single sequential layer consisting of two hidden layers for the structure constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Each layer consists of a bias and the hidden layers use the Rectified Linear Unit (ReLU) activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The optimizer, average learning rate, model hyperparameters, and training data gener- ation were identical to the alternative implementation described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Whereas the original method feeds the data directly into the network, the data in this approach is fed into the loss function to be transformed by the model’s predicted generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Invariance The invariance under the transformation (5) is en- forced by requiring that the labels before and after the transformation remain the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' For this purpose, we include the following mean squared error (MSE) term in the loss function L: Linv(W, {xi}) = 1 m m � i=1 [ϕ (FW(xi)) − ϕ(xi)]2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (8) A NN trained with this loss function will find an ar- bitrarily general (finite) symmetry transformation F� W parametrized by the values of the trained network pa- rameters � W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' In order to find multiple symmetries, the process can be repeated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Alternatively, several networks can be trained concurrently, by modifying the loss function to ensure that the resulting transformations are sufficiently distinct (see Section III C below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Infinitesimality In order to focus on the generators of the possible set of symmetry transformations, we restrict ourselves to in- finitesimal transformations δF in the vicinity of the iden- tity transformation I: δF ≡ I + ε GW , (9) where ε is an infinitesimal parameter and the parameters W of the new neural network G will be forced to be finite, which ensures that (9) is an infinitesimal transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The loss function (8) can then be rewritten as Linf(W, {xi}) = 1 mε2 m � i=1 [ϕ(xi + εGW(xi)) − ϕ(xi)]2 , (10) 5 where we have introduced an explicit factor of ε2 in the denominator to account for the fact that generic transfor- mations scale as ε [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Once we minimize the loss func- tion, the trained NN G� W will represent a corresponding generator J = G� W , (11) where � W ≡ arg min W � L(W, {xi}) � (12) are the learned values of the NN parameters which min- imize the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The result for � W, and therefore, for J, will in principle depend on the starting values W0 of the network parameters at initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' If we now repeat the training Ng times under different initial con- ditions W0 (or with different values of the hyperparam- eters), we will obtain a set of Ng (in general distinct) generators {Jα}, α = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' , Ng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Orthogonality In order to make the set of generators {Jα}, α = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' , Ng, found in the previous step maximally differ- ent, we introduce the following additional orthogonality term to the loss function Lortho(W, {xi}) = Ng � α<β �� p W(p) α W(p) β �2 , (13) where the index p runs over the individual NN parame- ters W(p) α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Closure In order to test whether a certain set of distinct gener- ators {Jα}, α = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' , Ng, found in the previous steps, generates a group, we need to check the closure of the algebra (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' This can be done in several ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The most principled would be to minimize Lclosure(a[αβ]γ) = � α<β Tr � CT [αβ]C[αβ] � , (14) with respect to the candidate structure constant param- eters a[αβ]γ, where the closure mismatch is defined by C[αβ](a[αβ]γ) ≡ � Jα, Jβ � − Ng � γ=1 a[αβ]γJγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (15) Since Lclosure is positive semi-definite, Lclosure = 0 would indicate that the algebra is closed and we are thus dealing with a genuine (sub)group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' In practice,1 we perform the minimization of (14) simultaneously with the previously discussed contributions to the loss function, by replacing Jα → GWα in (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The advantage of this simultaneous construction is that every set of generators that we obtain at any given value of Ng is already forming an algebra that is “as closed as possible”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Then, the size of the achieved total training loss is an indicator whether for that value of Ng a closed algebra exists or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Once a closed set of valid generators has been found, we can retrain the NN in a conveniently chosen canonical basis and obtain the canonical form of the set of genera- tors, whose structure constants in turn reveal the nature of the group behind the found symmetry transformations G� Wα, α = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' , Ng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Sections V-X illustrate the steps above with several examples of increasing complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' LINEAR ALGEBRA APPROACH The universal approximation theorems [52] guarantee that the deep-learning approach of the previous section can handle almost any symmetry transformation, includ- ing a highly non-linear one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' At the same time, a large class of interesting symmetries arising in physics are lin- ear transformations for which the usual formalism of lin- ear algebra would suffice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Furthermore, the analysis of the symmetry generators involves infinitesimal transfor- mations, which are represented with linear operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' For those reasons and to captivate the readers who are not yet at ease with the technical intricacies of machine learn- ing, in this section, we reformulate our analysis from the previous section in the language of linear algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' We fol- low the same notation and conventions, but replace the calligraphic font symbols representing neural networks with corresponding blackboard-bold symbols represent- ing n × n matrices acting on the n-dimensional feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' In this section we are interested in the linear subclass of the transformations (5), which are encoded in a generic matrix F x′ = F x , (16) whose n2 components Fij are determined by minimizing the loss function Linv(F, {xi}) = 1 m m � i=1 [ϕ (F xi) − ϕ(xi)]2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (17) 1 Another possible approach is to minimize the out-of-space com- ponents of the commutators with respect to the space of gener- ators, after flattening and Gram–Schmidt orthonormalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' 6 If the minimization is successful, then such a linear sym- metry exists and is represented with the learned matrix �F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' In analogy to (9), we can write the corresponding in- finitesimal linear transformation as δF(ε) = I + ε G , (18) where I is the unit n×n matrix and G is an n×n matrix whose components Gij are yet to be determined through the optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' In order to obtain a single generator matrix J, we can use a loss function analogous to (10) LJ(G, {xi}) = hinv mε2 m � i � ϕ � xi + ε G xi � − ϕ(xi) �2 + hnorm � Tr � GT G � − 2 �2 , (19) where the constants hinv and hnorm in (19) are hyperpa- rameter weights determining the relative contribution of the two terms in the loss function (19) enforcing the sym- metry invariance and finite normalization2 conditions, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The actual generator J is then obtained by minimizing (19): J = arg min G � LJ(G, {xi}) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (20) By repeating this procedure several times with differ- ent initial conditions, we obtain a different generator J each time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Alternatively, we can produce all Ng genera- tors in one go by adding together and minimizing simul- taneously Ng copies of the loss function (19): Ng � α=1 LJ(Gα, {xi}) , (21) which will lead to a set of Ng generators Jα, α = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' , Ng, and their respective infinitesimal transfor- mations δFα ≡ I + ε Jα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' At this point the generators Jα are completely decou- pled and independent of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' We can force them to be different by adding a loss term analogous to (13): Lortho(G, {xi}) = Ng � α<β � Tr(GT αGβ) �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (22) Finally, we can enforce the closure property by includ- ing a loss term analogous to (14): Lclosure(a[αβ]γ) = � α<β Tr � CT [αβ]C[αβ] � , (23) 2 In order for (18) to be an infinitesimal transformation, the matrix G needs to be finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' We choose to normalize our generators as Tr(GT αGβ) = 2δαβ, hence the factor of 2 in the second line of (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' A representative symmetry transformation preserving the oracle function (25), found by our procedure in the two- dimensional exercise in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' In this and all subsequent such figures, the arrows represent the displacements x′ − x resulting from the symmetry transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The color map illustrates the oracle function ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' where C[αβ](a[αβ]γ) ≡ � Gα, Gβ � − Ng � γ=1 a[αβ]γGγ , (24) and minimizing the total loss function with respect to the parameters a[αβ]γ as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' LENGTH-PRESERVING TRANSFORMATIONS IN TWO DIMENSIONS In this and the next two sections, we focus on trans- formations which preserve the Euclidean length of the feature vector, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=', our oracle ϕ will return ϕ(x) ≡ |x| = � n � i=1 [x(i)]2 �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (25) In this case we expect our method to discover the sym- metry of the orthogonal group O(n), whose generators can be written in terms of Kronecker deltas as (Omn)ij = δi mδj n − δj mδi n , (26) in which case the algebra is given by [Omn, Opq] = δnpOmq + δmqOnp − δmpOnq − δnqOmp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (27) Note that the generators Omn are labeled by two indices, which indicate the plane of rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='4 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='5 (x)d 0 X 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='9 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='3 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='0 2 1 0 1 2 ×(1)7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The evolution of the training loss with the num- ber of epochs for the two-dimensional exercise in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Our algorithm finds one symmetry generator (the training loss shown in blue steadily decreases) but not two different symmetry generators (the training loss shown in orange stays large).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' In this section we start with the simplest case of two di- mensions, n = 2, which should correspond to the single- generator group O(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' First we try to generate a sin- gle generic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=', not necessarily infinitesimal) symmetry transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' For this purpose, we train our network FW with the invariance loss (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' This exercise was suc- cessful and, depending on the initialization, we found var- ious symmetry transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' They all involved a ro- tation around the origin in the (x(1), x(2)) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' One representative symmetry transformation is depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Next, we turn to the algebra of symmetry generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' We begin with a single generator, Ng = 1, in which case we do not need to include the orthogonality and closure terms in the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The training is successful and the loss function is driven to zero, as seen by the blue solid line in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The resulting generator in matrix form is pictorially visualized in the top row of Figure 3, and we immediately recognize the familiar matrix O12 from (26) generating rotations in the 12-plane O12 = � 0 1 −1 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (28) Note that the generator has the expected antisymmetric property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Having found one symmetry generator, we next check if there is a second distinct generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' For this purpose, we add the orthogonality and closure terms in the loss function and repeat the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' This time the training is unsuccessful and the loss flattens after about 50 epochs, as seen by the orange solid line in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' In the second row of Figure 3 we show the result for the two candidate generators found in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' We note that while they do FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Top row: a successfully learned generator for n = 2 and Ng = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Bottom row: unsuccessfully learned “genera- tors” for n = 2 and Ng = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' In this and all subsequent such figures, each panel represents a learned generator Jα in ma- trix form, where the values of the individual elements of the matrix are indicated by the color bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' have the expected form for a generator of rotations in two dimensions, they are essentially the same transformation, and differ only by an overall sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' This implies that they fail the orthogonality condition — indeed, we find that the dominant contribution to the large total loss in that case is from the orthogonality loss (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Since the total loss is large and does not improve with further training (see the orange line for Ng = 2 in Figure 2), these two are not valid generators and should be discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' We thus conclude that there is no Lie algebra with Ng = 2 distinct generators in this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' We note in passing that upon repeated training runs in the Ng = 2 case, we sometimes find that the algo- rithm chooses to create generators which are orthogonal to each other, but are not genuine symmetry transfor- mations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=', the orthogonality loss is driven to zero, but only at the expense of a large invariance loss (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' In principle, it is difficult to predict what the machine will choose to do when presented with two mutually exclu- sive requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' For example, in some runs the two candidate generators ended up being identical instead of differing by a minus sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Also note that the normaliza- tion of the two candidate generators in the second row of Figure 3 is off — this is because the program chose to vi- olate the normalization condition as well, in order to help the orthogonality loss, which is the dominant penalty in that case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' In any case, since the failed training examples are of little theoretical value, in the following we shall only show results for the successful training runs which led to valid (sub)algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Next, we check the cases with Ng > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=" Similarly, we n= 2 10' 10 Loss 10- 5 Ng = 1 Ng = 2 10 14 0 100 200 300 400 500 600 EpochsGenerator 1 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='75 1 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='25 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='25 2 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 1 2Generator 1 Generator 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='75 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='25 2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 1 2 1 28 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The axes of the eigenvectors of the three generators found at intermediate stages of the training: after 1 epoch (top left), after 10 epochs (top right), after 100 epochs (bot- tom left), and after 300 epochs (bottom right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' For conve- nience, at the top of each panel we list the angle (in degrees) between each pair of axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' find that the training ends up in a large loss and that there is no consistent solution for a closed algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' We therefore conclude that in this n = 2 example there is only a one-parameter symmetry group with a generator given by (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Although the example of this section was rather trivial, it did outline and validate the main steps of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' More complicated examples follow in the next sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' LENGTH-PRESERVING TRANSFORMATIONS IN THREE DIMENSIONS In this section, we proceed to study symmetry trans- formations which preserve the oracle (25) in n = 3 di- mensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Once again, we find that training with the invariance loss (8) alone always leads to a valid sym- metry transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' We also observe that the matrix form F is antisymmetric, in agreement with the expecta- tion for the orthogonal group O(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Now that the data is three-dimensional, however, it is difficult to visualize the symmetry transformation directly as in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' In- stead, we choose to plot the symmetry axis (in this case the axis of rotation) defined by the real eigenvector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Fig- ure 4 illustrates the transformations found at different stages of the training for the case of Ng = 3 generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' At the top of each panel we list the relative angle in de- grees for each pair of axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Note how the symmetry axes FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' 2, but for the n = 3 exercise consid- ered in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The training results in valid closed algebras with one or three generators, but not two or four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' start out oriented at random, but the orthogonality loss term (13) gradually drives them to a mutually orthogonal configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' In order to analyze the group structure of the found symmetry transformations, we proceed to study the gen- erators of infinitesimal transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' First we try to determine the dimensionality of the full algebra, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=', the maximal number of linearly independent generators re- sulting in a closed algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Figure 5 shows loss curves for several different values of Ng: 1, 2, 3 and 4 (we do not show results for Ng ≥ 5 since they had large losses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' We observe that the training was successful only for the cases of Ng = 1 and Ng = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' This implies that the full algebra has 3 generators, in agreement with the expectation of n(n − 1)/2 generators for an orthogonal O(n) group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' At the same time, the successful training at Ng = 1 reveals a single generator subgroup of O(3) which is nothing but the O(2) discussed in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The results from a typical training run at Ng = 3 are shown in Figure 6, where the top row depicts the three successfully learned generators Jα, α = 1, 2, 3, while the bottom panel is a pictorial representation of the structure constants in matrix form as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Each row (labeled αβ = 12, 31, 23) represents one of the three possible com- mutators, whereas the columns (labeled γ = 1, 2, 3) rep- resent the found generators Jγ shown in the top panels of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Then, the entry in each cell represents the structure constant a[αβ]γ from the defining equation (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' A careful inspection of the top row in Figure 6 reveals that the three generators found in our example are ap- proximately J1 ≈ � � 0 0 1 0 0 0 −1 0 0 � � = − O31, (29a) Epoch: 0 Angles = 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='16°, 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='56°, 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='16°Epoch: 10 |Angles = 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='74°, 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='25°, 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='22°Ep0ch: 100 |Angles = 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='02°, 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='32°, 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='08°Ep0ch: 300 |Angles = 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='0°, 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='0°, 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='0°n= 3 [30(3)] 102 102, Ng= 1 SS Lo 10 0 Ng = 2 Ng = 3 10 10 10° 14 0 100 200 300 400 500 600 Epochs9 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Top row: the successfully learned generators for the n = 3, Ng = 3 exercise considered in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Bottom panel: the corresponding structure constants in matrix form (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' J2 ≈ 1 √ 2 � � 0 −1 0 1 0 −1 0 1 0 � � = − 1 √ 2 (O12 + O23), (29b) J3 ≈ 1 √ 2 � � 0 −1 0 1 0 1 0 −1 0 � � = − 1 √ 2 (O12 − O23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (29c) The bottom panel shows that the algebra of the found three generators Jα is given by [Jα, Jβ] = − ϵαβγJγ, (30) in which we recognize the usual SO(3) algebra involving the Levi-Civita permutation symbol ϵαβγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' From now on we will be using the Einstein summation convention for repeated generator-type indices α, β, γ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' LENGTH-PRESERVING TRANSFORMATIONS IN FOUR DIMENSIONS In this section, we generalize the discussion from the previous two sections to the case of n = 4 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The new twist here will be the existence of multiple non- trivial subalgebras of different dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The discovery of a single finite symmetry transfor- mation with the invariance loss (8) is straightforward and always succeeds in finding some finite rotation in four dimensions, represented with an orthogonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Therefore in this section, we focus only on the identi- fication of the (sub)algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' As before, we repeat the training for various number of multiple distinct genera- tors (Ng = 2, 3, 4, 5, 6, 7) and with the orthogonality and FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' 5, but for the n = 4 exercise considered in Section VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The training results in valid closed algebras with one (not shown), two, three, four and six gen- erators, but not five or seven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' closure losses turned on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Representative loss curves are shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' We observe that a closed algebra is found in four of those cases, namely Ng = 2, 3, 4, 6, which we now discuss in turn (the trivial case of Ng = 1 is of course always possible and will not be specifically discussed from now on).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Two generator subalgebras One of our found examples of a closed subalgebra with Ng = 2 generators is shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The top panels indicate that the two found generators can be roughly approximated as J1 ≈ � � � 0 1 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 � � � = O12, (31a) J2 ≈ � � � 0 0 0 0 0 0 0 0 0 0 0 1 0 0 −1 0 � � � = O34, (31b) in which we recognize the rotation generators O12 and O34 from (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' As seen from the matrix forms in (31), these two generators commute, since the two rotations are done in completely different planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Therefore, the algebra formed by J1 and J2 is Abelian, which is inde- pendently verified by the bottom panel in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' One should keep in mind that we do not control the overall orientation of the generators found by our proce- dure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The example in Figure 8 was judiciously chosen to be easily recognizable in terms of the canonical gener- Generator 1 Generator 2 Generator 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 1 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='25 2 2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='50 3 - 3 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 1 2 m 1 2 m 1 2 3Structure Constants 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='75 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='25 31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 Bra 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='50 23 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 1 2 3 Generatorn = 4 [3(4)] 10l Ng = 2 102, Ng = 3 OSS Ng = 4 10~5, Ng = 5 Ng = 6 10~8 10~11 0 100 200 300 400 500 600 Epochs10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Top row: the successfully learned generators for the n = 4, Ng = 2 exercise considered in Section VII A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Bottom panel: the corresponding structure constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The vanishing of the structure constants indicates that this is an Abelian subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The same as Figure 8, but for a different training run, still with Ng = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' ators (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' A generic training run typically returns the generator set in some random orientation, which, how- ever, still preserves the commutation properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' One such generic example is shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' This time, the two found generators J1 and J2 are more general linear combinations of the six canonical generators O12, O13, O14, O23, O24, and O34 of the O(4) group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Neverthe- less, the found generators J1 and J2 still commute and form an Abelian two-dimensional subalgebra of the full symmetry group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Top row: the successfully learned generators for the n = 4, Ng = 3 exercise considered in Section VII B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Bottom panel: the corresponding structure constants, which can be identified as those of the SO(3) algebra (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Three generator subalgebras Next we discuss the discovered subalgebras with three generators (Ng = 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Since SO(4) contains SO(3) as a subgroup,3 we know that such subalgebras should exist, and indeed, we find such solutions, as seen in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' As mentioned above, a generic training run produces the three generators in a random orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' For ease of interpretation, in Figure 10 we show a judiciously cho- sen example, in which the generators depicted in the top panels can be recognized to be approximately J1 ≈ O24, J2 ≈ O23, J3 ≈ O34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (32) This algebra involves rotations primarily in the last three feature dimensions, while the first feature is largely unaf- fected by the symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The bottom panel of Figure 10 confirms that this is the SO(3) algebra from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Of course, a more generic training run results in a set of three generators that involve all four feature dimensions, but still have the same commutation relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Four generator subalgebras Figure 7 shows that our method finds an algebra with four generators as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' To see its origin theoretically, 3 Since SO(4) = SO(3)⊗SO(3), the SO(4) algebra is a direct sum of two separate SO(3) subalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Generator 1 Generator 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='50 2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 3 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='50 4 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 1 2 3 4 1 2 3 4Structure Constants T 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='0 ket 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='0 1 2 GeneratorGenerator 1 Generator 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='50 2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 3 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='50 4 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 2 ¥3 4 2 3 4Structure Constants T 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='0 ket 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='0 1 2 GeneratorGenerator 1 Generator 2 Generator 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='0 1 1 - 1 0.' metadata={'source': 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+page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='25 31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 Bra 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='50 23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 1 2 m Generator11 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Top row: the successfully learned generators for the n = 4, Ng = 4 exercise considered in Section VII C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Bottom panel: the corresponding structure constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' define a new generator basis in terms of sums and differ- ences of pairs of commuting generators from the original basis (26) [53] S1 ≡ 1 2 (O34 + O12) , D1≡ 1 2 (O34 − O12) , (33a) S2 ≡ 1 2 (O42 + O13) , D2≡ 1 2 (O42 − O13) , (33b) S3 ≡ 1 2 (O23 + O14) , D3≡ 1 2 (O23 − O14) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (33c) This change of basis decouples the algebra (27) of the original generators as follows [Si, Sj] = − ϵijk Sk, (34a) [Di, Dj] = − ϵijk Dk, (34b) [Si, Dj] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (34c) Therefore, a closed algebra of four generators can be formed either from the set of three S’s plus any one of the D’s, or from the set of three D’s plus any one of the S’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' In either case, three of the generators will satisfy SO(3)-type commutation relations, while the fourth one will commute with everyone else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' This expectation is confirmed in Figure 11, which shows our usual represen- tation of the found generators and their algebra for one representative result from a training run with Ng = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' We observe that the found generators are approximately J1 ≡ 1 2 (O21 + O43) = − S1, (35a) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Top row: the successfully learned generators for the n = 4, Ng = 6 exercise considered in Section VII D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Bottom panel: the corresponding structure constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' J2 ≡ 1 2 (O32 + O14) = − D3, (35b) J3 ≡ 1 2 (O13 + O42) = S2, (35c) J4 ≡ 1 2 (O14 + O23) = S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (35d) Therefore, J1, J3 and J4 form an SO(3) algebra as implied by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (34a), and furthermore, all three of them commute with J2, as implied by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (34c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' This pattern is precisely what we observe in the lower panel of Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Generator 1 Generator 2 Generator 3 Generator 4 1 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='50 2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 3 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='5 4 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='75 4 4 i 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00Structure Constants 12 1D 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='5 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='5 tz 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='0 1 2 4 GeneratorGenerator 1 Generator 2 Generator 3 1DO 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='75 2 2 3 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='50 3 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='25 1 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='0 Generator 4 Generator 5 Generator 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='25 2 2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content="50 3 3 3 DO't- 1 2 3 2 3 4 1 3Structure Constants 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='B 12 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='6 14 15 t0 16 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='2 t Ja 25 26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='2 35 36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='4 45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='6 46 56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='8 Generator12 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Results from our search for Lie algebras of transfor- mations preserving the Euclidean oracle (25) in n = 2, 3, 4, 5 dimensions and for different number of generators Ng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The cells are color coded by the base-10 logarithm of the lowest value of the loss attained during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Six generator algebras Finally, we get to the case of Ng which will reveal the full algebra of SO(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Figure 12 shows the result from a representative training run seeking a closed algebra with Ng = 6 generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Among the set of learned generators we can approximately recognize J1 ≈ O43, J2 ≈ −D3, J3 ≈ S3, J4 ≈ O12, J5 ≈ O42 and J6 ≈ O13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The analysis of the last three sections can be readily extended to even higher dimensions (n ≥ 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' We have checked a few more values of n and the method works each time — we obtain valid finite symmetry transforma- tions which preserve the oracle (25), we find the closed algebra of the full set of n(n − 1)/2 generators of the or- thogonal O(n) group, as well as valid subalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' For fun, in Figure 26 we depict our derived 45 generators of the SO(10) group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' In conclusion of our discussion of orthogonal groups, in Figure 13 we summarize our results for the found closed algebras and subalgebras for n ≤ 5 and different number of generators Ng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Each cell in the table represents a sepa- rate exercise at a fixed number of dimensions n and for a fixed number of generators Ng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The cells are color coded by the base-10 logarithm of the lowest value of the loss attained during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The training is terminated once the loss reaches 10−12, therefore blue cells correspond to successful training runs resulting in closed algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The right-most blue cell in each row corresponds to the full algebra, in this case SO(n), as indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The preced- ing blue cells correspond to various subalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' In fact we did not anticipate the existence of the Ng = 4 sub- algebras in the case of n = 4 and n = 5, but our model surprised us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' LORENTZ TRANSFORMATIONS IN FOUR DIMENSIONAL MINKOWSKI SPACE In this section we consider the four-dimensional Minkowski spacetime (t, x, y, z) (in natural units with c = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The usual Lorentz transformations preserve the quadratic form ϕ(t, x, y, z) = t2 − x2 − y2 − z2, (t, x, y, z) ∈ R4, (36) hence this will be our oracle in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The four input features are x(0) = t, x(1) = x, x(2) = y, x(3) = z, (37) where in keeping with the standard physics notation we start labelling the features from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Our analysis proceeds as usual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' First, we find symme- try transformations which are in general combinations of boosts and rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Upon inspection, we verify that they have the desired symmetry properties F0i = Fi0, Fij = −Fji, F00 = Fii = 0, ∀i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Next we analyze the algebras of generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Before presenting the numerical results, for the reader’s conve- nience we summarize some relevant information about the mathematical structure of the Lorentz group O(1, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' It has six generators: the three generators of boosts Ki,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' K1 = � � � 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (38a) K2 = � � � 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (38b) K3 = � � � 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (38c) and the three generators of rotations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' given by L1 = � � � 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 1 0 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (39a) L2 = � � � 0 0 0 0 0 0 0 1 0 0 0 0 0 −1 0 0 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (39b) L3 = � � � 0 0 0 0 0 0 −1 0 0 1 0 0 0 0 0 0 � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (39c) 2 -SO(2) 0 3 - SO(3) n 5 4 - SO(4) 5 SO(5) 10 1 2 3 4 5 6 7 8 9 10 Ng13 K1 − L2 K2 + L1 K3 L3 L2 L1 X1 X2 X3 X4 X5 X6 X1 0 0 −X1 −X2 +X3 +X4 X2 0 0 −X2 +X1 +X4 −X3 X3 +X1 +X2 0 0 −X1 − X5 X2 − X6 X4 +X2 −X1 0 0 −X6 +X5 X5 −X3 −X4 +X1 + X5 +X6 0 −X4 X6 −X4 +X3 −X2 + X6 −X5 +X4 0 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The Lorentz algebra in terms of the generators (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The cells show the result from commuting one of the generators in the leftmost column with one of the generators in the top row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The full Lorentz algebra can be summarized as [Li, Lj] = ϵijk Lk, (40) [Li, Kj] = ϵijk Kk, (41) [Ki, Kj] = −ϵijk Lk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (42) The subgroup structure is most easily seen in a different basis, X1 = K1 − L2, (43a) X2 = K2 + L1, (43b) X3 = K3, (43c) X4 = L3, (43d) X5 = L2, (43e) X6 = L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (43f) The resulting algebra in the Xi basis is summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' By inspection, we see that the Lorentz algebra has several non-trivial subalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Two generator subalgebras There are several two-dimensional subalgebras: Abelian subalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' There are two abelian subal- gebras: the first one is generated by the genera- tor set {X1, X2}, while the second is generated by {X3, X4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' A representative example of the former case is shown in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' We recognize the two FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Top row: successfully learned generators for the n = 4, Ng = 2 exercise considered in Section VIII A, using the pseudo-Euclidean oracle (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Bottom panel: the corre- sponding structure constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' This example shows an abelian subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' 14, but showing a non-abelian subalgebra found by our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' found generators to be approximately J1 ≈ − (K3 − L1) , (44a) J2 ≈ − (K1 + L3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (44b) This result is in agreement with eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (43a) and (43b) — the transformations are combinations of boosts along and rotations about two of the axes, in this case x and z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Nonabelian subalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' As seen in Table I, the Lie algebra of the Lorentz group has two nonabelian subalgebras, generated by {X1, X3} and {X2, X3}, Generator 1 Generator 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='50 2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 3 - 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='50 4 - 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='75 2 3 4 1 2 3 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00Structure Constants 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='50 Bracket 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='25 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 1 2 GeneratorGenerator 1 Generator 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='50 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='50 4 - 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='75 2 E 4 1 2 m 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00Structure Constants 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='50 Bracket 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='25 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 1 2 Generator14 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Top row: the successfully learned generators for the n = 4, Ng = 3 exercise considered in Section VIII B, using the pseudo-Euclidean oracle (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Bottom panel: the corresponding structure constants, which can be identified as those of an SO(3) type subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' A representative training example for this nonabelian case is shown in Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The top panels show that the two found generators are approximately J1 ≈ − K3 = − X3, (45a) J2 ≈ K1 − L2 = X1, (45b) while the bottom panel confirms that their Lie bracket is approximately given by [J1, J2] ≈ [−X3, X1] = −X1 = − J2, as expected from Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Three generator subalgebras Table I reveals several three-dimensional subalgebras: SO(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The set {X4, X5, X6} generates a subalgebra isomorphic to the Lie algebra of the SO(3) group of rotations in three dimensions, see Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' A representative training example is shown in Fig- ure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The generators can be recognized as J1 ≈ L3, (46a) J2 ≈ L1, (46b) J3 ≈ L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (46c) In addition, we also found examples with subalge- bras isomorphic to so(3) consisting of one rotation and two boosts, or one boost and two rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' 16, but showing an example of a Hom(2) subalgebra (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Hom(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The set {X1, X2, X3} generates a subalge- bra isomorphic to the Lie algebra of Hom(2), the group of Euclidean homotheities: [X1, X2] = 0, [X3, X1] = X1, [X2, X3] = −X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (47) A representative training example is shown in Fig- ure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' From the panels in the top row we recognize the found three generators as J1 ≈ + K2 − L1, (48a) J2 ≈ − K1 − L2, (48b) J3 ≈ + K3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (48c) The bottom panel in Figure 17 confirms that their algebra is approximately that of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' E(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The set {X1, X2, X4} generates a subalge- bra isomorphic to the Lie algebra of E(2), the Eu- clidean group: [X1, X2] = 0, [X4, X1] = X2, [X2, X4] = X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (49) A representative example is shown in Figure 18, from which we recognize the found three generators as J1 ≈ − K3 − L2, (50a) J2 ≈ − L1, (50b) J3 ≈ − K2 + L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (50c) As expected, two of the generators involve combi- nations of boosts along and rotations about two of Generator 1 Generator 2 Generator 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 1 1 1 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 1 2 Generator15 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' 16, but showing an example of a E(2) subalgebra (49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' the axes, in this case y and z, while the third gener- ator is a rotation about the remaining axis, namely x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The bottom panel in Figure 18 shows that the resulting algebra is approximately that of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Bianchi VIIa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The set {X1, X2, X3 + aX4} with a ̸= 0 generates a Bianchi VIIa subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' A rep- resentative training example is shown in Figure 19, from which we recognize the found three generators as J1 ≈ − K3 − L3 = −(X3 + X4), (51a) J2 ≈ + K1 − L2 = X1, (51b) J3 ≈ − K2 − L1 = − X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (51c) This set corresponds to a Bianchi VIIa subalgebra with a = 1, whose structure constants are indeed consistent with the lower panel in Figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' SL(2,R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The set {X1, X3, X5} generates a subalge- bra isomorphic to the Lie algebra of SL(2, R), the group of isometries of the hyperbolic plane: [X1, X3] = − X1, (52a) [X5, X1] = − X3, (52b) [X3, X5] = − X1 − X5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (52c) A representative training example is shown in Fig- ure 20, from which we recognize the found three generators as J1 ≈ K1 − L3, (53a) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' 16, but showing an example of a Bianchi VIIa subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' 16, but showing an example of an SL(2,R) subalgebra (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' J2 ≈ − L3, (53b) J3 ≈ K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (53c) It is easy to verify that their structure constants given in the lower panel of Figure 20 are consistent with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Generator 1 Generator 2 Generator 3 1.' metadata={'source': 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+page_content='75 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='50 Bracket 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='25 31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='50 23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 1 2 Generator16 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Top row: the successfully learned generators for the n = 4, Ng = 4 exercise considered in Section VIII C, using the pseudo-Euclidean oracle (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Bottom panel: the corresponding structure constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Four generator subalgebras The only four-dimensional subalgebra is generated by {X1, X2, X3, X4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' A representative example is shown in Figure 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The four found generators are given by J1 ≈ L3 = X4, (54a) J2 ≈ K2 + L1 = X2, (54b) J3 ≈ − K1 + L2 = − X1, (54c) J4 ≈ − K3 = − X3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (54d) The lower panel of Figure 21 illustrates the corresponding structure constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The results are consistent with Ta- ble I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' For example, J1 and J4 commute because [X3, X4] = [K3, L3] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Similarly, J2 and J3 commute due to [X1, X2] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The results shown in the remaining rows in the lower panel of Figure 21 can be analogously verified with the help of Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Six generator algebras Our method is capable of finding the full six- dimensional algebra as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The result is shown in Fig- ure 22, and can be roughly identified as J1 ≈ − L1, J2 ≈ − K2, J3 ≈ − L3, (55a) J4 ≈ + K1, J5 ≈ − K3, J6 ≈ − L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (55b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Top row: the successfully learned generators for the n = 4, Ng = 6 exercise considered in Section VIII D, using the pseudo-Euclidean oracle (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Bottom panel: the corresponding structure constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The corresponding structure constants are shown in the lower panel of Figure 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' SQUEEZE MAPPING IN TWO DIMENSIONS Consider again two dimensions, but now let the oracle return the product of the two input features: ϕ(x) = x(1) x(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (56) This oracle function is illustrated in Figure 23 as a color heatmap representing the oracle values in the (x(1), x(2)) Cartesian plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' We see that the set of points with the same oracle values form hyperbolas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Generator 1 Generator 2 Generator 3 Generator 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='75 1 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='50 2 2 2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 3 - 3 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='50 4 - 4 4 4- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='75 2 3 4 1 2 i 3 F4 1 2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00Structure Constants 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='75 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='25 14 Bracket 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='25 24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='75 34 1 2 m 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 GeneratorGenerator 1 Generator 2 Generator 3 1DO 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='75 2 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' 3 3 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='50 4 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='25 i 2 4 2 4 i 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 Generator 4 Generator 5 Generator 6 1 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' 2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='50 3 m 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='75 4 4 4 - i 2 3 2 i 2 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 4 4Structure Constants 12 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='6 14 15 t0 16 21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='2 Ja 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='D 26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='2 35 36 45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='4 46 56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='6 i2 34 56 Generator17 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Color heatmap showing the values of the oracle (56) in the (x(1), x(2)) Cartesian plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The superimposed vector field represents the symmetry found by our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Proceeding as before, our method finds the symmetry transformation illustrated with the vector field in Fig- ure 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The corresponding generator is J = � −1 0 0 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (57) We verified that in this example the method is unable to find more than one generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' These results are precisely what one would expect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The symmetry which preserves the oracle (56) is the squeeze mapping F = � 1 ℓ 0 0 ℓ � , (58) where ℓ is a scale factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Considering infinitesimal trans- formations (18) with ℓ = 1 + ε immediately leads to the generator (57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' DISCONTINUOUS ORACLES A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Piecewise Linear Oracle Our setup is not limited to only continuous oracle func- tions like the ones discussed so far in the preceeding sec- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Our method can also work with piecewise-defined functions like ϕ(x) = � −x(2), for x(1) < 0, +x(1), for x(1) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (59) The resulting oracle function is illustrated with the color map in Figure 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Since the oracle is now a function FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The symmetry generated by the oracle (59) with a shallow method with no hidden layers and no bias (top panel) or a deep method with three hidden layers and bias (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' which is not continuously differentiable, our parametriza- tion of the symmetry transformation needs to be properly generalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The advantage of using a neural network as a universal function approximator is highlighted in Figure 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' In the top panel we use no hidden layers, while in the bottom panel we use a deep learning architecture with three hid- den layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The found symmetry transformation in each case is then shown with the vector field and superimposed on the oracle color map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The results are noticeably dif- ferent, particularly near the locations of discontinuity in the oracle function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The deep-learning approach in the bottom panel correctly identifies a transformation which preserves the oracle values everywhere within the consid- ered domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' In contrast, the shallow approach in the top panel is unable to adjust the transformation near the discontinuity, which leads to locations near the boundary x(1) = 0 where the arrows run across the equipotential contours, violating the conservation law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' 2 t t t X 3 2 1 (x)b) 0 0 X 1 2 3 2 4 2 1 0 1 2 +(1)2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='45 (X)d) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='80 2 2 1 0 1 2 ×(1)2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='45 (x)d) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='80 2 2 1 0 1 2 ×(1)18 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The symmetry generated by the L1 oracle (60) with a shallow method with no hidden layers and no bias (top panel) or a deep method with three hidden layers and bias (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Manhattan Distance Oracle In this subsection we consider one more example of an oracle in two dimensions (n = 2), which, while continu- ous, is not continuously differentiable: ϕ(x) = |x(1)| + |x(2)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' (60) The results from our procedure are shown in Figure 25 in complete analogy to Figure 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' In the top panel we use no hidden layers, while in the bottom panel we use a deep learning architecture with three hidden layers and bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' We observe that the deep-learning approach can again correctly handle the discontinuities, always gener- ating transformations along, but never across, the con- tours of equal oracle function values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' XI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' CONCLUSIONS In this paper, we studied a fundamental problem in data science which is commonly encountered in many fields: what is the symmetry of a labeled dataset, and how to identify its group structure?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' For this purpose, we designed a deep-learning method which models the generic symmetry transformation and its generators with a fully connected neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' We then constructed suitable loss functions which ensure that the applied transformations are symmetries and that the correspond- ing set of generators forms a closed (sub)algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' An important advantage of our approach is that we do not require any advance knowledge of what symmetries can be expected in the data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=', instead of testing for sym- metries from a predefined list of possibilities, we learn the symmetry directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Our procedure is very general and is universally appli- cable in a wide variety of situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The centerpiece of our analysis is an oracle ϕ(x) which defines the conserved quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The oracle can be completely general, as illus- trated with several examples in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' For example, it can be a continuous bilinear function, as in the case of the Euclidean or Minkowski distances and the squeeze mapping;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' the corresponding symmetries were discussed in Sections V-IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' It can also be a discontinuous func- tion (Section X A) or a function which is not smooth and continuously differentiable (Section X B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' In the process of deriving the full set of symmetries, the method also allows us to analyze the complete sub- group structure of the symmetry group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' In the paper we worked out explicitly the subgroup structure of the rotation groups SO(2) (Section V), SO(3) (Section VI) and SO(4) (Section VII), as well as the Lorentz group SO(1, 3) ((Section VIII)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' As one last tour de force ex- ample, we successfully applied our method to the case of Euclidean length-preserving rotations in n = 10 dimen- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The resulting 45 generators of the corresponding group SO(10) commonly used in grand unification [54] are shown in Figure 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The symmetries discussed in this paper have impor- tant implications for simulation-based inference, and in particular parameter retrieval from observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' In a typical inverse problem scenario, the (possibly multi- dimensional) labels y play the role of observed variables, and the features x play the role of input parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The parameter retrieval task is to infer x given y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' In the presence of a symmetry this inversion is not unique, but results in a whole family of valid input parameters, all re- lated by a symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Therefore, knowing that there is a symmetry in the data to begin with can be very useful in parameter retrieval algorithms, especially when the data is very high dimensional and the symmetry is difficult to see by the human eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Note that the same approach can be extended or mod- ified to solve other common problems and tasks in group 2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='60 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='70 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='25 (x)d) 0 X 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='45 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 2 1 0 1 2 +(1)2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='60 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='70 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='25 (x)d) 0 X 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='45 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='00 2 1 0 1 2 x(1)19 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' The set of Ng = 45 generators found by our method in the case of the SO(10) group (length-preserving rotations in n = 10 dimensions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Due to the complexity of the problem, the learning rate for this example was reduced to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' theory and/or BSM model building not discussed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Derivation of the Cartan subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' In our ap- proach, this can be trivially accomplished by set- ting a[αβ]γ = 0 in (15), which will force all the learned generators to commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Direct product decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' We can also look for symmetries whose algebras can be expressed as direct sums of two separate subalgebras — we just have to include closure terms in the loss function for each of the two individual subalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Internal symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Note that our approach can be readily generalized to look for internal symme- tries [41], which can prove useful in model building in high energy theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Canonical basis for the generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Since our method is basis-independent, the generators are ob- tained in a generic basis, where the structure con- stants may be difficult to recognize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' We could, how- ever, aid the program in finding a canonical basis of generators by including a term in the loss func- tion which encourages sparsity among the structure constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' In this paper we focused on idealized examples with no noise or statistical fluctuations in the determination of the labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' It will be instructive to test our method on noisy datasets, including real experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' We postpone this study to a future publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' In conclusion, our study opens the door for using a machine learning approach in the study of Lie groups and their properties and further bridges the fields of formal mathematics and theoretical computer science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' ACKNOWLEDGMENTS We thank A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Davis, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Gleyzer, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Kong, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Mrenna, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Prosper and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Shyamsundar for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' We thank P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' Ramond for group theory insights and in- spiration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content=' This work is supported in part by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE5T4oBgHgl3EQfhA-Z/content/2301.05638v1.pdf'} +page_content='S.' metadata={'source': 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b/gdAzT4oBgHgl3EQfavyA/content/tmp_files/2301.01374v1.pdf.txt @@ -0,0 +1,1623 @@ +arXiv:2301.01374v1 [math.AG] 3 Jan 2023 +ON HYPERGEOMETRIC DUALITY CONJECTURE +LEV BORISOV AND ZENGRUI HAN +Abstract. We give an explicit formula for the duality, previously con- +jectured by Horja and Borisov, of two systems of GKZ hypergeometric +PDEs. We prove that in the appropriate limit this duality can be identi- +fied with the inverse of the Euler characteristics pairing on cohomology +of certain toric Deligne-Mumford stacks, by way of Γ-series cohomology +valued solutions to the equations. +Contents +1. +Introduction +1 +2. +Pairing of solutions +3 +3. +Pairing of the Gamma series +8 +4. +Euler characteristic pairing +16 +5. +Extensions and open questions +20 +References +22 +1. Introduction +Let C be a finite rational polyhedral cone in a lattice N = ZrkN. We +assume that all ray generators of C lie on a primitive hyperplane deg(·) = 1 +where deg : N → Z is a linear function. This data encodes an affine toric +variety X = Spec C[N ∨ ∩ C∨], with the hyperplane condition equivalent to +X being Gorenstein, i.e. having trivial dualizing sheaf. +Let {vi}n +i=1 be a set of n lattice points in C which includes all of its ray +generators, with deg(vi) = 1 for all i. One can construct crepant resolutions +PΣ → X by looking at subdivisions Σ of C based on triangulations that +involve some of the points vi. Typically, PΣ is a smooth Deligne-Mumford +stack rather than a smooth variety, with the rare exception of when all cones +in Σ are unimodular. +A particular case of Kawamata-Orlov K → D conjecture asserts that +the derived categories of coherent sheaves on PΣ are independent of the +choice of Σ. In fact, it is expected that there is an isotrivial family of tri- +angulated categories which interpolates between the categories in question. +This rather mysterious family is well understood at the level of complexified +Grothendieck K-groups. Namely, these should correspond to solutions of +1 + +2 +LEV BORISOV AND ZENGRUI HAN +a certain version of the Gel’fand-Kapranov-Zelevinsky system of hypergeo- +metric PDEs. In fact, due to non-compactness of X and PΣ, there are two +such systems, denoted by bbGKZ(C, 0) and bbGKZ(C◦, 0), conjecturally +dual to each other [1]. +In the appropriate limit that corresponds to the +triangulation Σ, solutions to these systems can be identified with usual and +compactly supported orbifold cohomology of PΣ by means of two special Γ- +series. In this paper we settle positively the duality conjecture of [1]. In fact, +our duality formula is simple enough to hope that it may provide hints as to +how one could try to construct the aforementioned triangulated categories. +We will now set up the notations and review the better-behaved GKZ +hypergeometric systems. +Definition 1.1. Consider the system of partial differential equations on +the collection of functions {Φc(x1, . . . , xn)} in complex variables x1, . . . , xn, +indexed by the lattice points in C: +∂iΦc = Φc+vi, +n +� +i=1 +⟨µ, vi⟩xi∂iΦc + ⟨µ, c⟩Φc = 0 +for all µ ∈ N ∨, c ∈ C and i = 1, . . . , n. +We denote this system by +bbGKZ(C, 0). Similarly by considering lattice points in the interior C◦ only, +we can define bbGKZ(C◦, 0). +This system gives a holonomic system of PDEs. It follows from the general +theory of holonomic D-modules that its rank (i.e., the dimension of the +solution space) is finite. For more background on this, we refer to [9]. In +contrast to the usual GKZ system where rank jumps may occur at non- +generic parameters (see [10]), it is proved in [3] that the better-behaved +GKZ systems always have the expected rank which is equal to the normalized +volume of the convex hull of ray generators of the cone C. +It has been previously conjectured in [1] that the systems bbGKZ(C, 0) +and bbGKZ(C◦, 0) are dual to each other, in the sense that there is a pairing +⟨·, ·⟩ between solutions Φ = (Φc) and Ψ = (Ψd) thereof in the form +⟨Φ, Ψ⟩ = +� +c,d +pc,d(x)ΦcΨd, +where pc,d are polynomials in x, with only finitely many of them nonzero. +This pairing should be constant in x and could be viewed as the duality +of the local systems of solutions. A nontrivial example of this duality has +been verified in [1] and the rk(N) = 2 case has been settled affirmatively +in [2]. +Moreover, in certain regions of x that roughly correspond to the +complexified K¨ahler cones of PΣ, one can construct solutions of bbGKZ(C, 0) +and bbGKZ(C◦, 0) with values in certain cohomology or K-theory groups +of PΣ. Then it was conjectured in [1] that the above pairing should give (up +to a constant) the inverse of a certain Euler characteristic pairing on these + +ON HYPERGEOMETRIC DUALITY CONJECTURE +3 +spaces. In this paper we are able to verify both statements and thus prove +Conjecture 7.3 of [1] in full generality. +Specifically, the following formula provides the pairing in question. Let +v ∈ C◦ be an element in general position. For a subset I ⊆ {1, . . . , n} of +size rkN we consider the cone σI = � +i∈I R≥0vi. We define the coefficients +ξc,d,I for c + d = vI as +ξc,d,I = +� +(−1)deg(c), if dim σI = rk N and both c + εv and d − εv ∈ σ◦ +I +0, otherwise. +Here the condition has to hold for all sufficiently small ε > 0. As usual, +we denote by VolI the absolute value of the determinant of the matrix of +coefficients of vi, i ∈ I in a basis of N (i.e., the normalized volume of I). +We can now formulate the first result of this paper. +Theorem 2.4. For any pair of solutions (Φc) and (Ψd) of bbGKZ(C, 0) and +bbGKZ(C◦, 0) respectively, the pairing +⟨Φ, Ψ⟩ = +� +c,d,I +ξc,d,I VolI +�� +i∈I +xi +� +ΦcΨd +is a constant. +As was mentioned before, for a regular triangulation Σ there is a descrip- +tion of solutions to bbGKZ(C, 0) and bbGKZ(C◦, 0) in terms of the Gamma +series Γ = (Γc) and Γ◦ = (Γ◦ +d) with values in certain orbifold cohomology +spaces H and Hc associated to PΣ, considered in [1]. Then the second main +result of the paper is the following. +Theorem 4.2. The constant pairing ⟨Γ, Γ◦⟩ is equal up to a constant factor +to the inverse of the Euler characteristic pairing χ(−, −) : H ⊗ Hc → C. +The paper is organized as follows. In Section 2 we prove the above Theo- +rem 2.4. In Section 3 we introduce the spaces H and Hc, the solutions Γ and +Γc with values in them and compute the pairing of Theorem 2.4 on them. +We also calculate the asymptotic behavior of the series and their pairing +in the large K¨ahler limit, which is used in the next section. In Section 4 +we prove that this pairing is the inverse of the Euler characteristic pairing +between H and Hc. This, in particular, implies that the pairing of Theorem +2.4 is nondegenerate. Finally, in Section 5 we explain some easy extensions +of our results and state some open questions. +2. Pairing of solutions +The goal of this section is to define a pairing between the solution spaces +of the better-behaved GKZ systems associated to C and C◦. We first study +a particular class of pairings and find a sufficient condition to make it give a +constant for any pair of solutions of better-behaved GKZ systems. Then we +provide a special example of this pairing, inspired by the fan displacement + +4 +LEV BORISOV AND ZENGRUI HAN +formula for the resolution of the diagonal in toric varieties, due to Fulton +and Sturmfels [6]. +To state the first main result of this section, we first introduce some +notations. Suppose J is a subset of {1, 2, . . . , n} with |J| = rk N + 1. We +will call such subset spanning if {vi, i ∈ J} spans NR over R. For a spanning +set J there is a unique (up to multiplication by a constant factor) linear +relation among the vectors {vi}i∈J +� +i∈J +aivi = 0. +We introduce sgn : J → {0, ±1} by sgn(j) being −1, 0 or 1 if ai is negative, +zero or positive, respectively. This gives a decomposition J = J+ ⊔ J− ⊔ J0 +of the spanning set J. Note that while sgn depends on the choice of scaling +of the above linear relation, the expressions sgn(j1) sgn(j2) are well-defined. +The following lemma will be used later in this section. +For a subset +I ⊆ {1, . . . , n} of size rk N we denote by VolI the normalized volume of the +convex hull of the origin and vi, i ∈ I. +Lemma 2.1. Let I ⊂ {1, . . . , n} be such that {vi, i ∈ I} form a basis of +NR. Suppose that I contains 1 and consider j ̸∈ I. Consider the spanning +set J = I ∪ {j}. Let µ denote the unique linear function that takes value +VolI on v1 and 0 on vi, i ∈ I \ 1. Then µ(vj) = − sgn(1) sgn(j)VolJ\1 for the +sgn defined for J. +Proof. Up to sign, we can think of the linear function µ as taking a wedge +product with Λi∈I\1vi. Thus, µ(vj) = ±VolJ\1 and we just need to determine +the sign. Since {vi, i ∈ I} form a basis, the coefficient aj in the relation +� +i∈J aivi = 0 is nonzero and we may consider it to be 1, which ensures +sgn(j) = 1. We apply µ to � +i∈J aivi = 0 to get a1VolI + µ(vj) = 0. This +implies that a1 and µ(vj) have opposite signs, and the definition of sgn(1) +finishes the argument. +□ +Motivated by our previous work [2], we will look at pairings ⟨·, ·⟩ that only +have monomial terms xI = � +i∈I xi for subsets I of {1, · · · , n} of size rk N. +The following proposition provides a sufficient condition on the pairing being +a constant. +Proposition 2.2. Let {ξc,d,I} be a collection of complex numbers for all +c ∈ C, d ∈ C◦, I ⊆ {1, . . . , n} such that c + d = � +i∈I vi and |I| = rk N. +Suppose that +0 = +� +j∈J +sgn(j) +� +ξc−vj,d,J\jχ(c − vj ∈ C) + ξc,d−vj,J\jχ(d − vj ∈ C◦) +� +holds for all c ∈ C, d ∈ C◦ and all spanning subsets J ⊆ {1, 2, . . . , n} with +|J| = rk N + 1 and � +i∈J vi = c + d. +Here χ denotes the characteristic + +ON HYPERGEOMETRIC DUALITY CONJECTURE +5 +function (1 if the statement is true and 0 if it is false). Then +⟨Φ, Ψ⟩ = +� +|I|=rkN +� +c+d=vI +ξc,d,IVolIxIΦcΨd +is a constant for any pair of solutions (Φ, Ψ). +Proof. Without loss of generality, it suffices to show that ∂1⟨Φ, Ψ⟩ = 0. We +compute it as follows +∂1⟨Φ, Ψ⟩ = +� +|I|=rk N, +c+d=vI +ξc,d,IVolIxI(Φc+v1Ψd + ΦcΨd+v1) ++ +� +1∈I,|I|=rk N, +c+d=vI +ξc,d,IVolIxI\1ΦcΨd +(2.1) +and now use relations on Φ and Ψ to manipulate the second sum. For each +term, let µ be the linear function given by +µ(v) = v ∧ (Λj∈I\1vj) +under the standard identification of ΛrkNNR ∼= R where we choose the order +of {vj : j ∈ I\1} in the wedge product such that µ(v1) = VolI is positive. +Note that µ(vj) = 0 for all j ∈ I \ 1. +We use µ(c) + µ(d) = µ(� +i∈I vi) = VolI and add appropriate multiples +of equations for Φc and Ψd with this µ to get +ΦcΨdVolI = − +� +j /∈I\1 +xjµ(vj)(Φc+vjΨd + ΦcΨd+vj). +Thus, (2.1) can be rewritten as +∂1⟨Φ, Ψ⟩ = +� +|I|=rk N, +c+d=vI +ξc,d,IVolIxI(Φc+v1Ψd + ΦcΨd+v1) +− +� +1∈I,|I|=rk N, +c+d=vI +� +j /∈I\1 +ξc,d,Iµ(vj)xI1→j(Φc+vjΨd + ΦcΨd+vj) += +� +1̸∈I,|I|=rk N, +c+d=vI +ξc,d,IVolIxI(Φc+v1Ψd + ΦcΨd+v1) +− +� +1∈I,|I|=rk N, +c+d=vI +� +j /∈I +ξc,d,Iµ(vj)xI1→j(Φc+vjΨd + ΦcΨd+vj) +where we canceled the terms with 1 ∈ I in the first sum with j = 1 in the +second sum. Here I1→j = I\{1} ∪ {j}. Note that µ depends on the set I. +Let us now compute the coefficient at xˆIΦˆcΨ ˆd in the above expression. +This coefficient gets contributions from the first sum with I = ˆI and from +the second sum with I = ˆI ∪ 1 \ j. We observe that ˆI has size rk N and + +6 +LEV BORISOV AND ZENGRUI HAN +does not contain 1. Also note that if VolˆI = 0, then the coefficient is zero. +Indeed, in the second sum, µ(vj) = ±VolI1→j. Finally, we must have +ˆc + ˆd = vJ, J = {1} ⊔ ˆI. +We look at the set J which we know to be spanning, since it contains ˆI. By +Lemma 2.1, we see that µ(vj) = sgn(1) sgn(j)Vol ˆI. Therefore, the first line +contributes +ξˆc−v1, ˆd,ˆIVolˆIχ(ˆc − v1 ∈ C) + ξˆc, ˆd−v1,ˆIVolˆIχ( ˆd − v1 ∈ C◦) +and the second line contributes +� +j∈J\1 +sgn(1) sgn(j)ξˆc−vj, ˆd,J\j VolˆI χ(ˆc − vj ∈ C) ++ +� +j∈J\1 +sgn(1) sgn(j)ξˆc, ˆd−vj,J\jVolˆIχ( ˆd − vj ∈ C◦). +We observe that if sgn(1) = 0, then {vi, i ∈ J \ 1} do not span NR, so +VolˆI = 0 and the statement trivially holds. Thus we can introduce sgn(1)2 +into the first term to have the coefficient at xˆIΦˆcΨ ˆd equal +sgn(1) VolˆI +� +j∈J +sgn(j) +� +ξˆc−vj, ˆd,J\jχ(ˆc − vj ∈ C) + ξˆc, ˆd−vj,J\jχ( ˆd − vj ∈ C◦) +� +, +and the claim follows. +□ +Remark 2.3. After some sign changes, one can rephrase the condition of +Proposition 2.2 as dξ = 0 for an appropriate element ξ ∈ C[C] ⊗ C[C◦] ⊗ +ΛrkN(⊕n +i=1Cei) with the differential +d = +n +� +i=1 +[vi] ⊗ 1 ⊗ (ei∧) + +n +� +j=1 +1 ⊗ [vj] ⊗ (ej∧) +on ξ ∈ C[C]⊗C[C◦]⊗Λ•(⊕n +i=1Cei). We do not pursue this direction further +in the paper. +Now we give an explicit formula of the pairing ⟨−, −⟩ between solutions +of the better-behaved GKZ systems bbGKZ(C, 0) and bbGKZ(C◦, 0). We +prove that ⟨Φ, Ψ⟩ is a constant for any pair of solutions Φ and Ψ by using +Proposition 2.2. +Fix a choice of a generic vector v ∈ C◦. For a set I of size rkN we consider +the cone σI = � +i∈I R≥0vi. We define the coefficients ξc,d,I for c + d = vI as +ξc,d,I = +� +(−1)deg(c), if dim σI = rk N and both c + εv and d − εv ∈ σ◦ +I +0, otherwise. +(2.2) +Here the condition has to hold for all sufficiently small ε > 0. It is clear that +ξ is well-defined as long as the vector v is chosen sufficiently generic. Note + +ON HYPERGEOMETRIC DUALITY CONJECTURE +7 +that ξc,d,I ̸= 0 implies that both c and d lie in the maximum-dimensional +cone σI (but not necessarily in its interior). +We are now ready to tackle the main result of this section. +Theorem 2.4. For any pair of solutions (Φc) and (Ψd) of bbGKZ(C, 0) and +bbGKZ(C◦, 0) respectively, the pairing +⟨Φ, Ψ⟩ = +� +c,d,I +ξc,d,I VolI +�� +i∈I +xi +� +ΦcΨd +is a constant. +Proof. We prove this theorem by showing that these coefficients ξc,d,I satisfy +the conditions in Proposition 2.2, namely +0 = +� +j∈J +ξc−vj,d,J\j sgn(j)χ (c − vj ∈ C) + +� +j∈J +ξc,d−vj,J\j sgn(j)χ (d − vj ∈ C◦) +for all spanning subsets J ⊆ {1, 2, · · · , n} with |J| = rk N + 1, and all +c ∈ C, d ∈ C◦ with c + d = � +i∈J vi. +We first observe that the conditions c − vj ∈ C and d − vj ∈ C◦ in the +equations above are redundant. Indeed, to ensure that ξc−vj,d,J\j ̸= 0 we +must have c − vj + εv ∈ σ◦ +J\j, which implies that c − vj ∈ σJ\j ⊆ C. For +the second term, to ensure that ξc,d−vj,J\j ̸= 0 we must have d − vj − εv ∈ +σ◦ +J\j ⊆ C◦ (since J\j is a maximal cone), which implies d ∈ C◦. Thus, it +suffices to consider the equations +0 = +� +j∈J+⊔J− +ξc−vj,d,J\j sgn(j) + +� +j∈J+⊔J− +ξc,d−vj,J\j sgn(j) +(2.3) +for ξ defined in (2.2). The nonzero terms occur for the indices j such that +both c − vj + εv and d − εv lie in σ◦ +J\j, or both c + εv and d − vj − εv lie in +σ◦ +J\j. +We consider the equation in the variables ai +� +i∈J +aivi = c + εv. +The solution set to this equation is an affine line lc+εv in the space RrkN+1. +A contribution to the first term of (2.3) happens when there is a point on +lc+εv with aj = 1 and all other ai lie in (0, 1) due to the definition of the +coefficient ξ. Similarly, a contribution to the second term happens for aj = 0 +and all other ai lie in (0, 1). +Recall from Lemma 2.1 that we have a decomposition J = J+ ⊔ J− ⊔ J0. +For i ∈ J0, the value of ai on the line lc+εv is constant. Since v is generic, we +may assume it to be non-integer. Thus, it either prohibits any contributions +to (2.3) (if ai ̸∈ (0, 1)) or provides no restrictions. Therefore, we may now +assume that the latter happens for all i ∈ J0. + +8 +LEV BORISOV AND ZENGRUI HAN +The key idea of the proof is to consider the line segments +Si = lc+εv ∩ {0 ≤ ai ≤ 1} +on lc+εv for all i ∈ J+ ⊔ J−. The nonzero contributions to (2.3) happen +exactly for the endpoints of a line segment Sj that lie strictly inside all other +segments. The assumption that εv is generic implies that the endpoints of +different Si do not coincide. Indeed, if it were the case, then c + εv would +lie in a shift of the span of rkN − 1 of v-s by a lattice element, and we may +assure that this does not happen. Consider now S = � +i∈J+⊔J− Si. If S is +empty then there are no contribution, since this point would not lie in the +interior of other Si. So it suffices to consider the case when S is a segment +[p, q]. It is clear that the only points that could contribute to (2.3) are p +and q. In particular, there are at most two nonzero terms in (2.3). We will +show that they always cancel each other. +We also note that the orientation of the segment Si (i.e., the direction in +which the parameter ai increases) on the line lc+εv is determined by sgn(i), +since the vector along the line is given by the nontrivial linear relation on +vk,k∈J. If both p and q are the ai = 1 and aj = 1 ends of the segments Si +and Sj, then the segments must have opposite orientations on lc+εv (since +they both should point towards the other point). This means that sgn(i) = +− sgn(j) and the two terms of (2.3) cancel. Similarly, they cancel if p and q +are the ai = 0 and aj = 0 ends of Si and Sj. +Now suppose that p and q correspond to ai = 0 and aj = 1 ends of +Si and Sj (in this case it is possible to have i = j). In this case the two +segments must have the same orientation, and then the factor (−1)deg c in +the definition of ξ ensures that the two terms cancel each other. +□ +Remark 2.5. As v varies, we get a finite number of different formulas for +the pairing. It is also possible to take a more uniform choice of the pairing +by integrating over v of degree 1 (ignoring the contributions of measure zero +set of nongeneric v). However, there does not appear to be any advantage +in doing so. We will later see that the pairing is in fact independent of the +choice of v. +3. Pairing of the Gamma series +In this section we compute the pairing from the previous one on the +cohomology-valued solutions to the better-behaved GKZ systems provided +by the Γ series. We will show in the next section that the result is the dual of +the intersection pairing which provides the proof of Conjecture 7.3 from [1]. +We consider a regular triangulation Σ of the cone C whose vertices are +among these vectors {vi}n +i=1 and its corresponding toric Deligne-Mumford +stack PΣ. +Remark 3.1. It will be convenient for us to abuse notation and denote +by I both a subset of {1, . . . , n} and the corresponding cone � +i∈I R≥0vi. + +ON HYPERGEOMETRIC DUALITY CONJECTURE +9 +Similarly, Σ denotes both a simplicial complex on {1, . . . , n} and the corre- +sponding simplicial fan in NR which refines C and its faces. +Definition 3.2. For each cone σ ∈ Σ we define Box(σ) to be the set of +lattice points γ which can be written as γ = � +i∈σ γivi with 0 ≤ γi < 1. We +denote the union of all Box(σ) by Box(Σ). To each element γ ∈ Box(Σ) we +associate a twisted sector of PΣ corresponding to the minimal cone σ(γ) in +Σ containing γ. We define the dual of a twisted sector γ = � γivi by +γ∨ = +� +γi̸=0 +(1 − γi)vi. +or equivalently, the unique element in Box(σ(γ)) that satisfies +γ∨ = −γ mod +� +i∈σ +Zvi +Remark 3.3. The dual of γ = 0 is itself. Clearly, we have σ(γ) = σ(γ∨) +and (γ∨)∨ = γ. +Twisted sectors are themselves smooth toric DM stacks and the following +propositions describe a Stanley-Reisner type presentation of the spaces of +cohomology and cohomology with compact support of their coarse moduli +spaces, see [1]. +Proposition 3.4. As usual, Star(σ(γ)) denotes the set of cones in Σ that +contain σ(γ). Cohomology space Hγ of the twisted sector γ is naturally iso- +morphic to the quotient of the polynomial ring C[Di : i ∈ Star(σ(γ))\σ(γ)] +by the ideal generated by the relations +� +j∈J +Dj, J ̸∈ Star(σ(γ)), +and +� +i∈Star(σ(γ))\σ(γ) +µ(vi)Di, µ ∈ Ann(vi, i ∈ σ(γ)). +We can also view Hγ as a module over the polynomial ring C[D1, . . . , Dn] by +declaring Di = 0 for i ̸∈ Star(σ(γ)) and solving (uniquely) for Di, i ∈ σ(γ) +to satisfy the linear relations �n +i=1 µ(vi)Di = 0 for all µ ∈ N ∨. +Proposition 3.5. Cohomology space with compact support Hc +γ (viewed as +a module over Hγ) is generated by FI for I ∈ Star(σ(γ)) such that σ◦ +I ⊆ C◦ +with relations +DiFI − FI∪{i} for i ̸∈ I, I ∪ {i} ∈ Star(σ(γ)) +and DiFI for i ̸∈ I, I ∪ {i} ̸∈ Star(σ(γ)) +Similarly, it is given a structure of a module over C[D1, . . . , Dn]. +Definition 3.6. The orbifold cohomology H of the smooth toric DM stack +PΣ is defined as the direct sum � +γ Hγ over all twisted sectors. Similarly, +the orbifold cohomology with compact support Hc is defined as � +γ Hc +γ. We +denote by 1γ the generator of Hγ. + +10 +LEV BORISOV AND ZENGRUI HAN +There is a natural perfect pairing between H and Hc called Euler charac- +teristic pairing. Its origin is the eponymous pairing on certain Grothendick +K-groups, which is then translated to the cohomology via the Chern char- +acter, see [1]. We will not be using the original definition, but rather the +following formula for the Euler characteristic pairing, which is proved in [2]. +Proposition 3.7. The Euler characteristic pairing χ : H ⊗ Hc → C on the +toric DM stack PΣ is given by +χ(a, b) = χ(⊕γaγ, ⊕γbγ) = +� +γ +1 +| Box(σ(γ))| +� +γ∨ Td(γ∨)a∗ +γbγ∨ +Here ∗ : H → H is the duality map given by (1γ)∗ = 1γ∨ and (Di)∗ = −Di, +and Td(γ) is the Todd class of the twisted sector γ which is defined as +Td(γ) = +� +i∈Star σ(γ)\σ(γ) Di +� +i∈Star σ(γ)(1 − e−Di). +The linear function +� +: Hc +γ → C takes values +1 +VolI on each generator FI, +where VolI denotes the volume of the cone σI in the quotient fan Σ/σ(γ). +It takes value zero on all elements of Hc +γ of lower degree. +Let Σ be a regular (=projective) subdivision of C based on some of the vi. +Let ψi be the real numbers such that Σ reads off the lower boundary of the +convex hull of the origin and {(vi, ψi), 1 ≤ i ≤ n} in NR⊕R. We assume that +ψi are generic so this convex hull is simplicial. We denote by ψ the strictly +convex piecewise linear function on C whose graph is the aforementioned +lower boundary. It takes values ψi on all vi which generate rays in Σ and +has lower values than ψi on other vi. Its key property is that for any finite +collection wi ∈ C and αi ∈ R>0 there holds +ψ( +� +i +αiwi) ≤ +� +i +αiψ(wi) +with equality if and only if there exists a cone in Σ which contains all of the +wi. +Recall from [1] the following solution to the equations bbGKZ(C, 0) with +values in H = � +γ Hγ. We define +Γc(x1, . . . , xn) = +� +γ +� +l∈Lc,γ +n +� +i=1 +x +li+ Di +2πi +i +Γ(1 + li + Di +2πi) +(3.1) +where the direct sum is taken over twisted sectors γ = � +j∈σ(γ) γjvj and the +set Lc,γ is the set of solutions to �n +i=1 livi = −c with li − γi ∈ Z for all i. +The numerator is defined by picking a branch of log(xi). +We will first prove that for each c ∈ C ∩ N the series for Γ converges +absolutely and uniformly on compacts for x such that the (− log |xi|) are +in an appropriate shift of the cone of values on vi of convex Σ-piecewise + +ON HYPERGEOMETRIC DUALITY CONJECTURE +11 +linear functions. The proof was skipped in [1] because it is essentially the +same as that in [4], but we will present it here, both for completeness and +to facilitate arguments about the asymptotic behavior of Γc. +Proposition 3.8. We denote by CΣ the cone of the secondary fan that +corresponds to Σ, i.e. the cone of (ψi) ∈ Rn that give rise to Σ. For each +c ∈ C ∩N there exists ˆψ ∈ Rn such that the series (3.1) converges absolutely +and uniformly on compacts in the region of Cn +{(− log |x1|, . . . , − log |xn|) ∈ ˆψ + CΣ, arg(x) ∈ (−π, π)n}. +(3.2) +Proof. An immediate observation is that we can ignore the factor +n +� +i=1 +x +Di +2πi +i += +n +� +i=1 +e +Di log xi +2πi +because it does not depend on l and is bounded on compacts in the region +(3.2). +It suffices to understand what happens for a fixed γ. Note that while the +summation takes place over an affine lattice Lc,γ, the nonzero contributions +only occur for (l1, . . . , ln) such that the set +I(l) = {i, li ∈ Z<0} ⊔ σ(γ) +is a cone σ in Σ, because each li ∈ Z<0 contributes a factor Di due to a +pole of Γ at a nonpositive integer. Consequently, it suffices to bound the +summation over the subset Lc,γ,σ of Lc,γ with the additional property that +the above defined I(l) is a subset of some fixed maximum-dimensional cone +σ of Σ that contains σ(γ). For any such l ∈ Lc,γ,σ we have +� +i,li<0 +(−li)vi = +� +i,li≥0 +livi + c. +Let us denote by ψ the Σ-piecewise linear convex function that corresponds +to (− ˆψi − log |xi|) by the assumption on x. Since the vi on the left hand +side of the above equation lie in σ ∈ Σ, we have +� +i,li<0 +(−li)(− ˆψi − log |xi|) = ψ( +� +i,li<0 +(−li)vi) ≤ +� +i,li≥0 +liψ(vi) + ψ(c) += +� +i,li≥0 +li(− ˆψi − log |xi|) + ψ(c) +and therefore +n +� +i=1 +li log |xi| ≤ − +n +� +i=1 +li ˆψi + ψ(c). +(3.3) +This leads to an upper bound +��� +� +i=1 +xli +i +��� ≤ eψ(c)e− �n +i=1 li ˆψi. +(3.4) + +12 +LEV BORISOV AND ZENGRUI HAN +Crucially, since all vi have degree 1, we see that � +i li = − deg c. Thus, +we can apply the key estimate of [4, Lemma A.4] which states that for any +δ > 0 and any collection of real numbers ai, bi for i = 1, . . . , n with +| +� +i +ai| ≤ δ, +� +i +|bi| ≤ δ +there exists a constant A such that +��� +n +� +i=1 +1 +Γ(ai + ibi) +��� ≤ A(4n) +�n +i=1 |ai|. +By the Cauchy’s formula for partial derivatives, this implies an upper bound +of the form A1(A2) +�n +i=1 |li| on the coefficients on all monomials in Di of +bounded degree of the function +n +� +i=1 +1 +Γ(1 + li + Di +2πi) +. +Together with (3.4), we conclude that in any Euclidean norm on Hγ the +absolute value of each term of the series is bounded by +��� +n +� +i=1 +xli +i +Γ(1 + li + Di +2πi) +��� ≤ A1(A2) +�n +i=1 |li|��� +� +i=1 +xli +i +��� ≤ A3(A2) +�n +i=1 |li|e− �n +i=1 li ˆψi. +(3.5) +We observe that the set Lc,γ,σ is the set of lattice points in a shift +of a (lower-dimensional) polyhedral cone Cσ in Rn given by the equality +�n +i=1 livi = 0 and inequalities li ≥ 0 for all i ̸∈ σ. We may assume ˆψ to give +a strictly Σ-convex function. It then follows that for any ray generator l of +Cσ there holds +� +i +li ˆψi > 0. +Indeed, by convexity for ˆψ for any l ∈ Cσ we have the inequality � +i li ˆψi ≥ 0 +(the proof is the same as that of (3.3)) which holds even if ˆψ is deformed +slightly, so it can only be equality for l = 0. As a consequence, there is a +constant r such that +n +� +i=1 +|li| ≤ r( +� +i +li ˆψi) +on Cσ. +Therefore, we can replace ˆψ by a large enough multiple of itself and use +(3.5) to get on any compact subset of the region (3.2) +��� +n +� +i=1 +xli +i +Γ(1 + li + Di +2πi) +��� ≤ A4e−A5 +�n +i=1 li ˆψi + +ON HYPERGEOMETRIC DUALITY CONJECTURE +13 +for some A5 > 0. Since the number of terms in Lc,γ,σ with �n +i=1 li ˆψi ∈ +[m, m+1) is bounded by a polynomial in m, we get the desired convergence. +□ +There is a similarly defined Γ-series solution Γ◦ of bbGKZ(C◦, 0), with +values in Hc = � +γ Hc +γ. We define +Γ◦ +c(x1, . . . , xn) = +� +γ +� +l∈Lc,γ +n +� +i=1 +x +li+ Di +2πi +i +Γ(1 + li + Di +2πi) +�� +i∈σ +D−1 +i +� +Fσ +where σ is the set of i with li ∈ Z<0. +Proposition 3.9. The series Γ◦ converges uniformly on compacts in the +region (3.2) for an appropriate choice of ˆψ. +Proof. The idea of the proof are the same as that of Proposition 3.8 and we +leave the details to the reader. +□ +Our next goal is to understand the asymptotic behavior of +Γc(t−ψ(v1)x1, . . . , t−ψ(vn)xn) +for real t → +∞. We can assume xi to be generic nonzero complex numbers, +so that for large enough t we fall within the range of convergence of Γ. +For each c we consider the minimum cone σ(c) of Σ that contains c. We +have c = � +j∈σ(c) cjvj. It defines a twisted sector γ(c) = � +j∈σ(c){cj}vj. +We also consider the dual twisted sector γ∨(c) = � +j∈σ(c),cj̸∈Z(1 − {cj})vj. +There is a special element +−c = +� +i∈I +(−ci)vi +(3.6) +in Lc,γ∨(c). +Lemma 3.10. As t → +∞, we have for c ∈ C ∩ N and γ ̸= γ∨(c) the γ +summand of Γc(t−ψ(v1)x1, . . . , t−ψ(vn)xn) is o(tψ(c)). For γ = γ∨(c) we have +Γc(t−ψ(v1)x1, . . .) = tψ(c) +n +� +i=1 +e +Di +2πi (log xi−ψ(vi) log t) +n +� +i=1 +x−ci +i +Γ(1 − ci + Di +2πi) +(1 + o(1)). +Proof. Let γ = � +j∈σ(γ) γjvj. Let (li) be an element of Lc,γ. The contribu- +tion to Γc(t−ψ(v1)x1, . . .) is only nonzero if the set of i for which li ∈ Z<0 +together with σ(γ) is a cone in Σ. Consequently, i for which li are negative +lie in a cone of Σ. Therefore, +� +li<0 +(−li)ψ(vi) = ψ( +� +li<0 +(−li)vi) = ψ(c + +� +li>0 +livi) ≤ +� +li>0 +liψ(vi) + ψ(c), +(3.7) + +14 +LEV BORISOV AND ZENGRUI HAN +which implies +− +n +� +i=1 +liψ(vi) ≤ ψ(c). +(3.8) +Now notice that the equality in (3.8) holds if and only if the minimal cone +of � +li<0(−li)vi is a cone in Σ which contains c and all vi with li > 0. This +cone would then contain c and all vi for which li ̸= 0. This means that +li = −ci, which implies that γ = γ∨(c). This gives the claimed asymptotic +contribution. +It is not enough to bound the asymptotic behavior of each individual term +as t → ∞, one also needs to ensure that the rest of the terms together do +not contribute to anything larger than o(tψ(c)). This follows either from the +estimates of Proposition 3.8 or simply from the fact that we have absolute +convergence at x and then all other terms decay faster. Indeed, if we have +an absolutely convergent series � +i≥0 ai and then consider � +i≥0 aitαi with +α0 − αi larger than some positive ε, then as t → ∞ we have +� +i≥0 +aitαi = a0tα0(1 + o(1)) +because +��� +� +i>0 +aitαi−α0 +��� ≤ t−ε � +i>0 +|ai|. +We can apply it to our situation since (li) are in a countable set and there +exists ε > 0 so that for all other terms the inequality (3.8) is strict by at +least ε. The logarithmic terms � +i(t−ψ(v1)xi) +Di +2πi can be absorbed by a slight +change of ε. +□ +We can state a similar result for Γ◦. For d ∈ C◦ we consider the element +of Ld,γ∨(d) +−d = +� +i∈σ(d) +(−di)vi. +Lemma 3.11. As t → +∞, we have for c ∈ C ∩ N and γ ̸= γ∨(d) the γ +summand of Γ◦ +d(t−ψ(v1)x1, . . . , t−ψ(vn)xn) is o(tψ(d)). For γ = γ∨(d) we have +Γd(t−ψ(v1)x1, . . .) = tψ(d) +n +� +i=1 +e +Di +2πi (log xi−ψ(vi) log t) +n +� +i=1 +x−di +i +Γ(1 − di + Di +2πi) + + � +i∈σ(d) +D−1 +i + + Fσ(d)(1 + o(1)). +Proof. The proof is analogous to that of Lemma 3.10 and is left to the +reader. +□ + +ON HYPERGEOMETRIC DUALITY CONJECTURE +15 +Now we use this information about the asymptotic behavior of Γ and Γ◦ +to compute the constant ⟨Γ, Γ◦⟩ = � +c,d,I ξc,d,I VolI +�� +i∈I xi +� +Γc ⊗ Γ◦ +d where +ξ are defined in Theorem 2.4. +As in Section 2, let I be a subset of {1, . . . , n} of size rkN, which may +or may not be a cone in Σ. Let c and d be such that c + d = � +i∈I vi and +c + εv, d − εv ∈ � +i∈I R≥0vi for small ε > 0. The following observation is +key. +Proposition 3.12. Under the above assumptions on c, d, I we have +lim +t→+∞ +n +� +i=1 +(t−ψ(vi)xi)Γc(t−ψ(v1)x1, . . .)Γ◦ +d(t−ψ(v1)x1, . . .) = 0 +unless γ(d) = γ∨(c) and I contains σ(γ(c)). +Proof. Since c and d are contained in � +i∈I R≥0vi and c + d = vI, we have +c = +� +i∈I +αivi, d = +� +i∈I +(1 − αi)vi +with αi ∈ [0, 1]. Convexity of ψ implies that +ψ(c) ≤ +� +i +αiψ(vi), ψ(d) ≤ +� +i +(1 − αi)ψ(vi) +(3.9) +which leads to ψ(c) + ψ(d) − � +i ψ(vi) ≤ 0, so we can use Propositions 3.10 +and 3.11 to see that the leading power of t is nonpositive. +In fact, it is +negative, unless the inequalities in (3.9) are equalities, which means that +the subset of I for which αi > 0 is a cone in Σ, and similarly for the subset +of αi < 1. This implies the claim. +□ +Proposition 3.13. If γ(c) = γ∨, γ(d) = γ = � +i∈I γivi, then we define Ic +to be the subset of I such that the coefficients ci of c are equal to 1 and +similarly for Id. The asymptotic behavior as t → ∞ is +n +� +i=1 +(t−ψ(vi)xi)Γc(t−ψ(v1)x1, . . .)Γ◦ +d(t−ψ(v1)x1, . . .) = o(1) + +1 +(2πi)rk N−|σ(γ)| +· +DIc +� +i∈σ(γ) Γ(γi + Di +2πi) � +i∈Star(σ(γ))\σ(γ) Γ(1 + Di +2πi) +n +� +i=1 +e +Di +2πi (log xi−ψ(vi) log t) +� +FId +� +i∈σ(γ) Γ(1 − γi + Di +2πi) � +i∈Star(σ(γ))\σ(γ) Γ(1 + Di +2πi) +n +� +i=1 +e +Di +2πi (log xi−ψ(vi) log t) +in Hγ ⊗ Hc +γ∨. +Proof. The proof of Proposition 3.12 shows that the only contribution other +than o(1) can come from the terms that give better than o(tψ(c)) and o(tψ(d)) +contributions to the asymptotic behavior of Φc and Φd. So by Propositions + +16 +LEV BORISOV AND ZENGRUI HAN +3.10 and 3.11 the only contributions come from elements of Lc,γ∨ and Ld,γ +given by +−c = +� +i∈I +(−ci)vi, −d = +� +i∈I +(ci − 1)vi. +For i ∈ σ(γ) we note that γi = 1 − ci if ci ∈ (0, 1). For i ∈ Ic we use +1 +Γ(1 − ci + Di +2πi) += +1 +Γ( Di +2πi) += +Di +2πi +Γ(1 + Di +2πi) +and similarly for i ∈ Id, and the result follows. +□ +Now we recall that ⟨Γ, Γ◦⟩ is constant. +Corollary 3.14. The constant pairing ⟨Γ, Γ◦⟩ lies in � +γ Hγ ⊗ Hc +γ∨ and is +given by +1 +(2πi)rk N +� +γ +� +c∈C,d∈C◦ +|I|=rkN +ξc,d,I VolI(2πi)|σ(γ)| DIc +�Γγ +⊗ FId +�Γγ∨ +where �Γγ = � +i∈σ(γ) Γ(γi + Di +2πi) � +i∈Star(σ(γ))\σ(γ) Γ(1 + Di +2πi) and similarly for +�Γγ∨. There also holds for each k +0 = +� +γ +� +c∈C,d∈C◦ +|I|=rk N +ξc,d,I VolI(2πi)|σ(γ)|� +Dk +DIc +�Γγ +� +⊗ FId +�Γγ∨ ++ +� +γ +� +c∈C,d∈C◦ +|I|=rkN +ξc,d,I VolI(2πi)|σ(γ)| DIc +�Γγ +⊗ +� +Dk +FId +�Γγ∨ +� +. +Proof. Proposition 3.13 gives the asymptotic behavior of ⟨Γ, Γ◦⟩ as a poly- +nomial in log xi. However, we also know it is a constant by Theorem 2.4. +The first statement of the proposition is reading off the constant term of +the polynomial and the second statement is reading off the coefficient by +log xk. +□ +4. Euler characteristic pairing +Now we are ready to prove that the pairing of Gamma series ⟨Γ, Γ◦⟩ is +inverse to the Euler characteristic pairing on PΣ. Before we state the main +theorem of this section, we have the following useful observation, which is an +orbifold analog of the relationship between the Γ-class and the Todd class +of a smooth manifold. Recall that ∗ is the duality map on H defined in +Proposition 3.7. +Lemma 4.1. (�Γγ)∗�Γγ∨ = (2πi)|σ(γ)|(−1)deg γ∨ Td(γ∨). + +ON HYPERGEOMETRIC DUALITY CONJECTURE +17 +Proof. We can expand (�Γγ)∗�Γγ∨ as +� +i∈σ(γ) +Γ(γi + Di +2πi)∗Γ(1 − γi + Di +2πi) +· +� +i∈Star(σ(γ))\σ(γ) +Γ(1 + Di +2πi)∗Γ(1 + Di +2πi) += +� +i∈σ(γ) +Γ(γi − Di +2πi)Γ(1 − γi + Di +2πi) +· +� +i∈Star(σ(γ))\σ(γ) +Γ(1 − Di +2πi)Γ(1 + Di +2πi). +We use the identity Γ(z)Γ(1 − z) = − 2πi eπiz +1−e2πiz to rewrite the first product as +(−2πi)|σ(γ)|e +� +i∈σ(γ) πiγie− 1 +2 +� +i∈σ(γ) Di � +i∈σ(γ) +1 +1 − e2πiγi−Di . +For the second product, we use Γ(1− +z +2πi)Γ(1+ +z +2πi) = ze− z +2 +1−e−z to rewrite it as +e− 1 +2 +� +i∈Star(σ(γ))\σ(γ) Di +� +i∈Star(σ(γ))\σ(γ) +Di +1 − e−Di . +Putting the two formulas together, we get +(�Γγ)∗�Γγ∨ = (2πi)|σ(γ)|(−1)deg γ∨e− 1 +2 +� +i∈Star(σ(γ)) Di +� +i∈Star(σ(γ))\σ(γ) Di +� +i∈Star(σ(γ)) 1 − e−Di += (2πi)|σ(γ)|(−1)deg γ∨ Td(γ∨) +where we used � +i∈Star(σ(γ)) Di = �n +i=1 Di = 0. +□ +Now we can state and prove the main theorem of this section. Recall that +we defined the pairing ⟨·, ·⟩ on solutions of the better-behaved GKZ systems. +When we apply it to Γ and Γ◦, we get a constant element of H ⊗ Hc. +Theorem 4.2. The constant pairing ⟨Γ, Γ◦⟩ is equal up to a constant factor +to the inverse of the Euler characteristic pairing χ(−, −) : H ⊗ Hc → C. +Proof. It’s clear that we can consider each twisted sector individually. For +a fixed γ, the statement is equivalent to the assertion that +� +c∈C,d∈C◦ +|I|=rkN +ξc,d,I VolI(2πi)|σ(γ)|χ +� +P, FId +�Γγ∨ +� +DIc +�Γγ += P +holds for all classes P ∈ Hγ. Since the class �Γγ is invertible in Hγ, dividing +by it induces an automorphism on the cohomology, hence it suffices to prove +� +c∈C,d∈C◦ +|I|=rkN +ξc,d,I VolI(2πi)|σ(γ)|χ +� +P +�Γγ +, FId +�Γγ∨ +� +DIc +�Γγ += P +�Γγ +(4.1) +for all P. We prove this by induction on the degree of P. + +18 +LEV BORISOV AND ZENGRUI HAN +The base case deg P = 0 corresponds to P = 1γ. Since +χ +� +1γ +�Γγ +, FId +�Γγ∨ +� += 0 +unless |Id| = rkN − |σ(γ)|, the equation becomes +� +|Id|=rk N−|σ(γ)| +ξγ∨,γ+vId,Id⊔σ(γ) VolId⊔σ(γ)(2πi)|σ(γ)|χ +� +1γ +�Γγ +, FId +�Γγ∨ +� +1γ +�Γγ += 1γ +�Γγ +. +(4.2) +Then by definition of χ and Lemma 4.1, we have +χ +� +1γ +�Γγ +, FId +�Γγ∨ +� += +1 +| Box(σ(γ))| +� +γ∨ Td(γ∨) +� +1 +�Γγ +�∗ FId +�Γγ∨ += +1 +| Box(σ(γ))| +� +γ∨ +FId +(�Γγ)∗�Γγ∨ +Td(γ∨) += +1 +| Box(σ(γ))| +� +γ∨ +FId +(2πi)|σ(γ)|(−1)deg γ∨ += +(−1)deg γ∨ +(2πi)|σ(γ)| VolId | Box(σ(γ))| +here VolId denotes the volume of the cone σId in the quotient fan Σ/σ(γ). +Note that we have +VolId⊔σ(γ) = VolId | Box(σ(γ))| +hence (4.2) becomes +� +|Id|=rk N−|σ(γ)| +(−1)deg γ∨ξγ∨,γ+vId,Id⊔σ(γ) = 1. +If we perturb γ∨ by εv, then it will fall in the interior of exactly one maximal +cone in Σ, and the corresponding coefficient ξ is the only nonzero term in +the sum above (recall the definition of ξc,d,I in Theorem 2.4), which is equal +to +(−1)deg γ∨(−1)deg γ∨ = 1 +So the base case is proved. +Now we assume the equality (4.1) holds for all classes of degree less than +m. Since the cohomology Hγ is generated as an algebra by classes Dk, it +suffices to prove the identity +� +c∈C,d∈C◦ +|I|=rkN +ξc,d,I VolI(2πi)|σ(γ)|χ +� +DkP +�Γγ +, FId +�Γγ∨ +� +DIc = DkP + +ON HYPERGEOMETRIC DUALITY CONJECTURE +19 +for each DkP where P ∈ Hγ is of degree m−1. Since Dk is skew-symmetric +with respect to the χ pairing, the above statement can be rewritten as +DkP = − +� +c∈C,d∈C◦ +|I|=rk N +ξc,d,I VolI(2πi)|σ(γ)|χ +� +P +�Γγ +, DkFId +�Γγ∨ +� +DIc. +On the other hand, we can multiply the induction assumption for P by Dk +to get +� +c∈C,d∈C◦ +|I|=rkN +ξc,d,I VolI(2πi)|σ(γ)|χ +� +P +�Γγ +, FId +�Γγ∨ +� +Dk DIc = DkP. +Compare these two identities. It suffices to show +0 = +� +c∈C,d∈C◦ +|I|=rkN +ξc,d,I VolI +� +Dk · DIc +�Γγ +� +⊗ FId +�Γγ∨ ++ +� +c∈C,d∈C◦ +|I|=rkN +ξc,d,I VolI +DIc +�Γγ +⊗ +� +Dk · FId +�Γγ∨ +� +(4.3) +which follows from Corollary 3.14. +□ +Remark 4.3. Theorem 4.2 implies, in particular, that the pairing of The- +orem 2.4 is nondegenerate and is independent of v. We are not aware of a +direct proof of this fact. +We conclude this section by an explanation of our motivation behind the +definition of the coefficients ξc,d,I in Theorem 2.4. This definition is inspired +by the following fan displacement resolution of diagonal formula of Fulton- +Sturmfels [6]. +Proposition 4.4. Let X be the toric variety corresponds to a complete fan +Σ in a lattice N, denote the diagonal embedding X ֒→ X × X by δ. Let +σ ∈ Σ be any cone and v a generic point in N, then the diagonal class +decomposes as +[δ(V (σ))] = +� +σ1,σ2 +mσ +σ1,σ2 · [V (τ1) × V (τ2)] +where mσ +σ1,σ2 = [N : Nσ1 +Nσ2] and the sum is over all cones σ1, σ2 ∈ Σ with +codim σ1 + codim σ2 = codim σ and σ ⊆ σ1, σ2 such that (v + σ1) ∩ σ2 ̸= ∅. +Note that the coefficient mσ +σ1,σ2 is exactly the volume Volσ1∪σ2 of the cone +spanned by σ1 and σ2. This formula cannot be applied to our case directly, +since the toric varieties they worked with are complete while ours are not. +Nevertheless we have the following relationship between the definition of +ξc,d,I and the conditions occurred in Fulton-Sturmfels formula. + +20 +LEV BORISOV AND ZENGRUI HAN +Proposition 4.5. Let c, d ∈ σI and v be a generic point in C◦. Then both +c + εv and d − εv lies in σ◦ +I for all sufficiently small ε > 0 if and only if +(v + σ(c)) ∩ σ(d) ̸= ∅ +where σ(c) denotes the minimal cone of Σ that contains c. +Proof. Assume both c+εv and d−εv lies in σ◦ +I. Then we can write c+εv = +� +i∈I sivi where all si ∈ (0, 1). Recall that I = σ(c)∪σ(d) = Ic⊔Id⊔σ(γ(c)), +this equation can be rewritten into the form v = v1−v2, where v1 ∈ σ(c) and +v2 ∈ σ(d), which is equivalent to the second statement. The other direction +can be proved similarly. +□ +Remark 4.6. We believe our methods should allow one to give a new proof +of the Fulton-Sturmfels formula, which could be done by restricting our +results to the twisted sectors that are compact. We do not go into details +further in this paper. +5. Extensions and open questions +There is a more general version of the better-behaved GKZ systems which +includes a parameter β ∈ NC, with β = 0 case being the one we considered +so far. Namely, the torus homogeneity equations of Definition 1.1 read +n +� +i=1 +⟨µ, vi⟩xi∂iΦc + ⟨µ, c − β⟩Φc = 0 +and similarly for Ψd. Much of what we did in this paper is applicable to the +pair of better behaved GKZ systems with parameters ±β. For instance, we +readily observe that our argument in Section 2 goes through for arbitrary +parameter β to give a pairing between spaces of solutions to bbGKZ(C, β) +and bbGKZ(C◦, −β). +We would like to see what happens in the limit given by a regular sub- +division Σ for a generic β. While there are certain versions of H and Hc +considered in [8] it will be easier for our purposes to simply write Vol(∆) +linearly independent solutions given by Γ-series, essentially along the lines +of the solutions of the original GKZ paper [7]. +Let Σ be a regular subdivision of C. For each maximum-dimensional cone +σ we consider Vol(σ) linearly independent solutions in the large K¨ahler limit +of PΣ, in bijection with the elements γ of N/ � +i∈σ Zvi. Namely, we define +the set Lc,γ,σ;β ⊂ Cn by +n +� +i=1 +livi = β − c +and the properties li ∈ Z for all i ̸∈ σ and c+� +i̸∈σ livi = −γ mod � +i∈σ Zvi. +Then for each γ we define a solution Φγ,σ of bbGKZ(C, β) by +Φγ,σ +c +(x1, . . . , xn) = +� +l∈Lc,γ,σ;β +n +� +i=1 +xli +i +Γ(1 + li). + +ON HYPERGEOMETRIC DUALITY CONJECTURE +21 +We define Γ-series solutions Ψγ,σ to bbGKZ(C◦, −β) in the same way by +Ψγ,σ +d (x1, . . . , xn) = +� +l∈Ld,γ,σ;−β +n +� +i=1 +xli +i +Γ(1 + li). +Note that in the case of generic β every solution of bbGKZ(C◦, −β) can be +uniquely extended to solutions of bbGKZ(C, −β). It is not hard to show +that these Φc and Ψd converge uniformly on compacts in the region (3.2) +for an appropriate choice of ˆψ. Moreover, as σ and γ vary, we get bases +of the space of solutions, with linear independence assured by them lying +in different eigenspaces of the monodromy operators for small loops around +xi = 0. +Monodromy considerations imply that for the pairing ⟨·, ·⟩ of Section 2 +we have ⟨Φγ,σ, Ψγ′,σ′⟩ = 0 unless σ = σ′ and γ = −γ′ mod � +i∈σ Zvi. In the +latter case, the constant contribution will happen for li + l′ +i = 0 for i ̸∈ I +and li + l′ +i = −1 for i ∈ I. If any of li, l′ +i is a negative integer, then the +corresponding term vanishes, due to a pole of Γ, so we may assume that +they are nonnegative for i ̸∈ σ, which then implies that +I = σ; li + l′ +i = −1, for i ∈ σ; li = l′ +i = 0 for i ̸∈ σ. +This implies that c = −γ mod � +i∈σ Zvi and d = γ mod � +i∈σ Zvi. +We claim that for any γ there exists exactly one pair (c, d) in σ satisfying +this constraint and ξc,d,σ ̸= 0. The definition of the coefficients ξ of the +pairing implies that we must also have c + d = � +i∈σ vi with c + εv and +d − εv in the corresponding cone � +i∈σ R≥0vi for all small ε > 0. We can +write β, v and γ uniquely as +β = +� +i∈σ +βivi, v = +� +i∈σ +sivi, γ = +� +i +γivi +with γi ∈ [0, 1). It is then easy to see that ξc,d,σ is nonzero if and only if +c = +� +{i:γi̸=0} +(1 − γi)vi + +� +{i:γi=0,si<0} +vi, +d = +� +{i:γi̸=0} +γivi + +� +{i:γi=0,si>0} +vi. +Thus for γi ̸= 0 we have li = βi − 1 + γi, l′ +i = −βi − γi. For γi = 0 and +si > 0 we have li = βi, l′ +i = −1 − βi and for γi = 0 and si < 0 we have +li = −1 + βi, l′ +i = −βi. In particular, +deg(c) = − deg(γ) + rkN − #{i : γi = 0, si > 0}. + +22 +LEV BORISOV AND ZENGRUI HAN +Therefore the pairing is given by +⟨Φγ,σ, Ψ−γ,σ⟩ = (−1)deg(c) Vol(σ) +� +γi̸=0 +1 +Γ(βi + γi)Γ(1 − βi − γi) +� +γi=0,si>0 +1 +Γ(1 + βi)Γ(−βi) +� +γi=0,si<0 +1 +Γ(βi)Γ(1 − βi) += (−1)deg(c) Vol(σ) +� +γi̸=0 +e2πi(βi+γi) − 1 +2πi eπi(βi+γi) +� +γi=0,si>0 +e2πi(βi+1) − 1 +2πi eπi(βi+1) +� +γi=0,si<0 +e2πiβi − 1 +2πi eπiβi += (−1)deg(c) Vol(σ) +(2πi)rkN +e−πi � +i∈σ(βi+γi) +� +γi=0,si>0 +eπi � +i∈σ +(e2πi(βi+γi) − 1) += Vol(σ) +(2πi)rkN e−πi deg(β)−2πi deg(γ) � +i∈σ +(1 − e2πi(βi+γi)) += e−πi deg(β)Vol(σ) +(2πi)rkN +� +i∈σ +(1 − e2πi(βi+γi)). +Remark 5.1. An immediate consequence of the above calculation is that +the pairing ⟨·, ·⟩ is non-degenerate for a generic β. +Further directions. +We conclude this section by stating some open +problems related to our construction, in no particular order. +• Is the pairing of this paper nondegenerate for all β? We know this +to be the case for β = 0 and β generic, and it seems likely to be +always true. +• We would like to settle the analytic continuation conjecture of [1] to +extend the main result of [4] to the better-behaved GKZ systems. +One consequence of Theorem 4.2 is that it should be enough to just +work with the usual K-theory and the compactly supported version +should follow from duality. +• What is the HMS counterpart of our pairing from the point of view +of Fukaya-Seidel categories for the mirror potential? Our formula for +the pairing is quite simple, so presumably so should be the mirror +version of it. We refer to [5], [11] for background. +• Solutions to bbGKZ systems come with a lattice structure inherited +from the K-theory of PΣ (it is independent of Σ). Can this structure +be locally defined outside of the region of convergence of any Γ-series? +References +[1] L. Borisov, R.P. Horja, Applications of homological mirror symmetry to hypergeomet- +ric systems: duality conjectures. Advances in Mathematics 271 (2015): 153–187. + +ON HYPERGEOMETRIC DUALITY CONJECTURE +23 +[2] L. Borisov, Z. Han, C. Wang, On duality of certain GKZ hypergeometric systems. +Asian Journal of Mathematics 25.1 (2021): 65–88. +[3] L. Borisov, R.P. Horja, On the better behaved version of the GKZ hypergeometric +system. Mathematische Annalen 357.2 (2013): 585–603. +[4] L. Borisov, R.P. Horja, Mellin-Barnes integrals as Fourier-Mukai transforms. Ad- +vances in Mathematics 207.2 (2006): 876–927. +[5] K. Fukaya, Y.-G. Oh, H. Ohta, K. Ono, Lagrangian intersection Floer theory: anom- +aly and obstruction, Part II. Vol. 2. American Mathematical Soc., 2010. +[6] W. Fulton, B. Sturmfels, Intersection theory on toric varieties. Topology 36 (1997), +no. 2, 335–353. +[7] I. Gelfand, +M. Kapranov, A. Zelevinsky, Generalized Euler integrals and A– +hypergeometric functions. Advances in Mathematics 84.2 (1990): 255–271. +[8] R.P. Horja, Toric Deligne-Mumford stacks and the better behaved version of the GKZ +hypergeometric system. Strings, Gauge Fields, and the Geometry Behind: The Legacy +of Maximilian Kreuzer. 2013. 329–348. +[9] R. Hotta, K. Takeuchi, T. Tanisaki, D-modules, perverse sheaves, and representation +theory. Vol. 236. Springer Science & Business Media, 2007. +[10] L. Matusevich, E. Miller, U. Walther, Homological methods for hypergeometric fami- +lies. Journal of the American Mathematical Society 18.4 (2005): 919–941. +[11] P. Seidel, Fukaya categories and Picard-Lefschetz theory. Vol. 10. European Mathe- +matical Society, 2008. +Department of Mathematics, Rutgers University, Piscataway, NJ 08854 +Email address: borisov@math.rutgers.edu +Department of Mathematics, Rutgers University, Piscataway, NJ 08854 +Email address: zh223@math.rutgers.edu + diff --git a/gdAzT4oBgHgl3EQfavyA/content/tmp_files/load_file.txt b/gdAzT4oBgHgl3EQfavyA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..31ed65857a78cfc76ec09ba942ae27cb6b7f6aa0 --- /dev/null +++ b/gdAzT4oBgHgl3EQfavyA/content/tmp_files/load_file.txt @@ -0,0 +1,675 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf,len=674 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='01374v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='AG] 3 Jan 2023 ON HYPERGEOMETRIC DUALITY CONJECTURE LEV BORISOV AND ZENGRUI HAN Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We give an explicit formula for the duality, previously con- jectured by Horja and Borisov, of two systems of GKZ hypergeometric PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We prove that in the appropriate limit this duality can be identi- fied with the inverse of the Euler characteristics pairing on cohomology of certain toric Deligne-Mumford stacks, by way of Γ-series cohomology valued solutions to the equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Pairing of solutions 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Pairing of the Gamma series 8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Euler characteristic pairing 16 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Extensions and open questions 20 References 22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Introduction Let C be a finite rational polyhedral cone in a lattice N = ZrkN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We assume that all ray generators of C lie on a primitive hyperplane deg(·) = 1 where deg : N → Z is a linear function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' This data encodes an affine toric variety X = Spec C[N ∨ ∩ C∨], with the hyperplane condition equivalent to X being Gorenstein, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' having trivial dualizing sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Let {vi}n i=1 be a set of n lattice points in C which includes all of its ray generators, with deg(vi) = 1 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' One can construct crepant resolutions PΣ → X by looking at subdivisions Σ of C based on triangulations that involve some of the points vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Typically, PΣ is a smooth Deligne-Mumford stack rather than a smooth variety, with the rare exception of when all cones in Σ are unimodular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' A particular case of Kawamata-Orlov K → D conjecture asserts that the derived categories of coherent sheaves on PΣ are independent of the choice of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' In fact, it is expected that there is an isotrivial family of tri- angulated categories which interpolates between the categories in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' This rather mysterious family is well understood at the level of complexified Grothendieck K-groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Namely, these should correspond to solutions of 1 2 LEV BORISOV AND ZENGRUI HAN a certain version of the Gel’fand-Kapranov-Zelevinsky system of hypergeo- metric PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' In fact, due to non-compactness of X and PΣ, there are two such systems, denoted by bbGKZ(C, 0) and bbGKZ(C◦, 0), conjecturally dual to each other [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' In the appropriate limit that corresponds to the triangulation Σ, solutions to these systems can be identified with usual and compactly supported orbifold cohomology of PΣ by means of two special Γ- series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' In this paper we settle positively the duality conjecture of [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' In fact, our duality formula is simple enough to hope that it may provide hints as to how one could try to construct the aforementioned triangulated categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We will now set up the notations and review the better-behaved GKZ hypergeometric systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Consider the system of partial differential equations on the collection of functions {Φc(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' , xn)} in complex variables x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' , xn, indexed by the lattice points in C: ∂iΦc = Φc+vi, n � i=1 ⟨µ, vi⟩xi∂iΦc + ⟨µ, c⟩Φc = 0 for all µ ∈ N ∨, c ∈ C and i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We denote this system by bbGKZ(C, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Similarly by considering lattice points in the interior C◦ only, we can define bbGKZ(C◦, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' This system gives a holonomic system of PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' It follows from the general theory of holonomic D-modules that its rank (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=', the dimension of the solution space) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' For more background on this, we refer to [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' In contrast to the usual GKZ system where rank jumps may occur at non- generic parameters (see [10]), it is proved in [3] that the better-behaved GKZ systems always have the expected rank which is equal to the normalized volume of the convex hull of ray generators of the cone C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' It has been previously conjectured in [1] that the systems bbGKZ(C, 0) and bbGKZ(C◦, 0) are dual to each other, in the sense that there is a pairing ⟨·, ·⟩ between solutions Φ = (Φc) and Ψ = (Ψd) thereof in the form ⟨Φ, Ψ⟩ = � c,d pc,d(x)ΦcΨd, where pc,d are polynomials in x, with only finitely many of them nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' This pairing should be constant in x and could be viewed as the duality of the local systems of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' A nontrivial example of this duality has been verified in [1] and the rk(N) = 2 case has been settled affirmatively in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Moreover, in certain regions of x that roughly correspond to the complexified K¨ahler cones of PΣ, one can construct solutions of bbGKZ(C, 0) and bbGKZ(C◦, 0) with values in certain cohomology or K-theory groups of PΣ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Then it was conjectured in [1] that the above pairing should give (up to a constant) the inverse of a certain Euler characteristic pairing on these ON HYPERGEOMETRIC DUALITY CONJECTURE 3 spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' In this paper we are able to verify both statements and thus prove Conjecture 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='3 of [1] in full generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Specifically, the following formula provides the pairing in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Let v ∈ C◦ be an element in general position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' For a subset I ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' , n} of size rkN we consider the cone σI = � i∈I R≥0vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We define the coefficients ξc,d,I for c + d = vI as ξc,d,I = � (−1)deg(c), if dim σI = rk N and both c + εv and d − εv ∈ σ◦ I 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Here the condition has to hold for all sufficiently small ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' As usual, we denote by VolI the absolute value of the determinant of the matrix of coefficients of vi, i ∈ I in a basis of N (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=', the normalized volume of I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We can now formulate the first result of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' For any pair of solutions (Φc) and (Ψd) of bbGKZ(C, 0) and bbGKZ(C◦, 0) respectively, the pairing ⟨Φ, Ψ⟩ = � c,d,I ξc,d,I VolI �� i∈I xi � ΦcΨd is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' As was mentioned before, for a regular triangulation Σ there is a descrip- tion of solutions to bbGKZ(C, 0) and bbGKZ(C◦, 0) in terms of the Gamma series Γ = (Γc) and Γ◦ = (Γ◦ d) with values in certain orbifold cohomology spaces H and Hc associated to PΣ, considered in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Then the second main result of the paper is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' The constant pairing ⟨Γ, Γ◦⟩ is equal up to a constant factor to the inverse of the Euler characteristic pairing χ(−, −) : H ⊗ Hc → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' In Section 2 we prove the above Theo- rem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' In Section 3 we introduce the spaces H and Hc, the solutions Γ and Γc with values in them and compute the pairing of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='4 on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We also calculate the asymptotic behavior of the series and their pairing in the large K¨ahler limit, which is used in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' In Section 4 we prove that this pairing is the inverse of the Euler characteristic pairing between H and Hc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' This, in particular, implies that the pairing of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='4 is nondegenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Finally, in Section 5 we explain some easy extensions of our results and state some open questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Pairing of solutions The goal of this section is to define a pairing between the solution spaces of the better-behaved GKZ systems associated to C and C◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We first study a particular class of pairings and find a sufficient condition to make it give a constant for any pair of solutions of better-behaved GKZ systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Then we provide a special example of this pairing, inspired by the fan displacement 4 LEV BORISOV AND ZENGRUI HAN formula for the resolution of the diagonal in toric varieties, due to Fulton and Sturmfels [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' To state the first main result of this section, we first introduce some notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Suppose J is a subset of {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' , n} with |J| = rk N + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We will call such subset spanning if {vi, i ∈ J} spans NR over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' For a spanning set J there is a unique (up to multiplication by a constant factor) linear relation among the vectors {vi}i∈J � i∈J aivi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We introduce sgn : J → {0, ±1} by sgn(j) being −1, 0 or 1 if ai is negative, zero or positive, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' This gives a decomposition J = J+ ⊔ J− ⊔ J0 of the spanning set J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Note that while sgn depends on the choice of scaling of the above linear relation, the expressions sgn(j1) sgn(j2) are well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' The following lemma will be used later in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' For a subset I ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' , n} of size rk N we denote by VolI the normalized volume of the convex hull of the origin and vi, i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Let I ⊂ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' , n} be such that {vi, i ∈ I} form a basis of NR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Suppose that I contains 1 and consider j ̸∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Consider the spanning set J = I ∪ {j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Let µ denote the unique linear function that takes value VolI on v1 and 0 on vi, i ∈ I \\ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Then µ(vj) = − sgn(1) sgn(j)VolJ\\1 for the sgn defined for J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Up to sign, we can think of the linear function µ as taking a wedge product with Λi∈I\\1vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Thus, µ(vj) = ±VolJ\\1 and we just need to determine the sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Since {vi, i ∈ I} form a basis, the coefficient aj in the relation � i∈J aivi = 0 is nonzero and we may consider it to be 1, which ensures sgn(j) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We apply µ to � i∈J aivi = 0 to get a1VolI + µ(vj) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' This implies that a1 and µ(vj) have opposite signs, and the definition of sgn(1) finishes the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' □ Motivated by our previous work [2], we will look at pairings ⟨·, ·⟩ that only have monomial terms xI = � i∈I xi for subsets I of {1, · · · , n} of size rk N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' The following proposition provides a sufficient condition on the pairing being a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Let {ξc,d,I} be a collection of complex numbers for all c ∈ C, d ∈ C◦, I ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' , n} such that c + d = � i∈I vi and |I| = rk N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Suppose that 0 = � j∈J sgn(j) � ξc−vj,d,J\\jχ(c − vj ∈ C) + ξc,d−vj,J\\jχ(d − vj ∈ C◦) � holds for all c ∈ C, d ∈ C◦ and all spanning subsets J ⊆ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' , n} with |J| = rk N + 1 and � i∈J vi = c + d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Here χ denotes the characteristic ON HYPERGEOMETRIC DUALITY CONJECTURE 5 function (1 if the statement is true and 0 if it is false).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Then ⟨Φ, Ψ⟩ = � |I|=rkN � c+d=vI ξc,d,IVolIxIΦcΨd is a constant for any pair of solutions (Φ, Ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Without loss of generality, it suffices to show that ∂1⟨Φ, Ψ⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We compute it as follows ∂1⟨Φ, Ψ⟩ = � |I|=rk N, c+d=vI ξc,d,IVolIxI(Φc+v1Ψd + ΦcΨd+v1) + � 1∈I,|I|=rk N, c+d=vI ξc,d,IVolIxI\\1ΦcΨd (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='1) and now use relations on Φ and Ψ to manipulate the second sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' For each term, let µ be the linear function given by µ(v) = v ∧ (Λj∈I\\1vj) under the standard identification of ΛrkNNR ∼= R where we choose the order of {vj : j ∈ I\\1} in the wedge product such that µ(v1) = VolI is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Note that µ(vj) = 0 for all j ∈ I \\ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We use µ(c) + µ(d) = µ(� i∈I vi) = VolI and add appropriate multiples of equations for Φc and Ψd with this µ to get ΦcΨdVolI = − � j /∈I\\1 xjµ(vj)(Φc+vjΨd + ΦcΨd+vj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Thus, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='1) can be rewritten as ∂1⟨Φ, Ψ⟩ = � |I|=rk N, c+d=vI ξc,d,IVolIxI(Φc+v1Ψd + ΦcΨd+v1) − � 1∈I,|I|=rk N, c+d=vI � j /∈I\\1 ξc,d,Iµ(vj)xI1→j(Φc+vjΨd + ΦcΨd+vj) = � 1̸∈I,|I|=rk N, c+d=vI ξc,d,IVolIxI(Φc+v1Ψd + ΦcΨd+v1) − � 1∈I,|I|=rk N, c+d=vI � j /∈I ξc,d,Iµ(vj)xI1→j(Φc+vjΨd + ΦcΨd+vj) where we canceled the terms with 1 ∈ I in the first sum with j = 1 in the second sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Here I1→j = I\\{1} ∪ {j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Note that µ depends on the set I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Let us now compute the coefficient at xˆIΦˆcΨ ˆd in the above expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' This coefficient gets contributions from the first sum with I = ˆI and from the second sum with I = ˆI ∪ 1 \\ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We observe that ˆI has size rk N and 6 LEV BORISOV AND ZENGRUI HAN does not contain 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Also note that if VolˆI = 0, then the coefficient is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Indeed, in the second sum, µ(vj) = ±VolI1→j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Finally, we must have ˆc + ˆd = vJ, J = {1} ⊔ ˆI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We look at the set J which we know to be spanning, since it contains ˆI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='1, we see that µ(vj) = sgn(1) sgn(j)Vol ˆI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Therefore, the first line contributes ξˆc−v1, ˆd,ˆIVolˆIχ(ˆc − v1 ∈ C) + ξˆc, ˆd−v1,ˆIVolˆIχ( ˆd − v1 ∈ C◦) and the second line contributes � j∈J\\1 sgn(1) sgn(j)ξˆc−vj, ˆd,J\\j VolˆI χ(ˆc − vj ∈ C) + � j∈J\\1 sgn(1) sgn(j)ξˆc, ˆd−vj,J\\jVolˆIχ( ˆd − vj ∈ C◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We observe that if sgn(1) = 0, then {vi, i ∈ J \\ 1} do not span NR, so VolˆI = 0 and the statement trivially holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Thus we can introduce sgn(1)2 into the first term to have the coefficient at xˆIΦˆcΨ ˆd equal sgn(1) VolˆI � j∈J sgn(j) � ξˆc−vj, ˆd,J\\jχ(ˆc − vj ∈ C) + ξˆc, ˆd−vj,J\\jχ( ˆd − vj ∈ C◦) � , and the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' After some sign changes, one can rephrase the condition of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='2 as dξ = 0 for an appropriate element ξ ∈ C[C] ⊗ C[C◦] ⊗ ΛrkN(⊕n i=1Cei) with the differential d = n � i=1 [vi] ⊗ 1 ⊗ (ei∧) + n � j=1 1 ⊗ [vj] ⊗ (ej∧) on ξ ∈ C[C]⊗C[C◦]⊗Λ•(⊕n i=1Cei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We do not pursue this direction further in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Now we give an explicit formula of the pairing ⟨−, −⟩ between solutions of the better-behaved GKZ systems bbGKZ(C, 0) and bbGKZ(C◦, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We prove that ⟨Φ, Ψ⟩ is a constant for any pair of solutions Φ and Ψ by using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Fix a choice of a generic vector v ∈ C◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' For a set I of size rkN we consider the cone σI = � i∈I R≥0vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We define the coefficients ξc,d,I for c + d = vI as ξc,d,I = � (−1)deg(c), if dim σI = rk N and both c + εv and d − εv ∈ σ◦ I 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='2) Here the condition has to hold for all sufficiently small ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' It is clear that ξ is well-defined as long as the vector v is chosen sufficiently generic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Note ON HYPERGEOMETRIC DUALITY CONJECTURE 7 that ξc,d,I ̸= 0 implies that both c and d lie in the maximum-dimensional cone σI (but not necessarily in its interior).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We are now ready to tackle the main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' For any pair of solutions (Φc) and (Ψd) of bbGKZ(C, 0) and bbGKZ(C◦, 0) respectively, the pairing ⟨Φ, Ψ⟩ = � c,d,I ξc,d,I VolI �� i∈I xi � ΦcΨd is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We prove this theorem by showing that these coefficients ξc,d,I satisfy the conditions in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='2, namely 0 = � j∈J ξc−vj,d,J\\j sgn(j)χ (c − vj ∈ C) + � j∈J ξc,d−vj,J\\j sgn(j)χ (d − vj ∈ C◦) for all spanning subsets J ⊆ {1, 2, · · · , n} with |J| = rk N + 1, and all c ∈ C, d ∈ C◦ with c + d = � i∈J vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We first observe that the conditions c − vj ∈ C and d − vj ∈ C◦ in the equations above are redundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Indeed, to ensure that ξc−vj,d,J\\j ̸= 0 we must have c − vj + εv ∈ σ◦ J\\j, which implies that c − vj ∈ σJ\\j ⊆ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' For the second term, to ensure that ξc,d−vj,J\\j ̸= 0 we must have d − vj − εv ∈ σ◦ J\\j ⊆ C◦ (since J\\j is a maximal cone), which implies d ∈ C◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Thus, it suffices to consider the equations 0 = � j∈J+⊔J− ξc−vj,d,J\\j sgn(j) + � j∈J+⊔J− ξc,d−vj,J\\j sgn(j) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='3) for ξ defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' The nonzero terms occur for the indices j such that both c − vj + εv and d − εv lie in σ◦ J\\j, or both c + εv and d − vj − εv lie in σ◦ J\\j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We consider the equation in the variables ai � i∈J aivi = c + εv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' The solution set to this equation is an affine line lc+εv in the space RrkN+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' A contribution to the first term of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='3) happens when there is a point on lc+εv with aj = 1 and all other ai lie in (0, 1) due to the definition of the coefficient ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Similarly, a contribution to the second term happens for aj = 0 and all other ai lie in (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Recall from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='1 that we have a decomposition J = J+ ⊔ J− ⊔ J0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' For i ∈ J0, the value of ai on the line lc+εv is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Since v is generic, we may assume it to be non-integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Thus, it either prohibits any contributions to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='3) (if ai ̸∈ (0, 1)) or provides no restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Therefore, we may now assume that the latter happens for all i ∈ J0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' 8 LEV BORISOV AND ZENGRUI HAN The key idea of the proof is to consider the line segments Si = lc+εv ∩ {0 ≤ ai ≤ 1} on lc+εv for all i ∈ J+ ⊔ J−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' The nonzero contributions to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='3) happen exactly for the endpoints of a line segment Sj that lie strictly inside all other segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' The assumption that εv is generic implies that the endpoints of different Si do not coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Indeed, if it were the case, then c + εv would lie in a shift of the span of rkN − 1 of v-s by a lattice element, and we may assure that this does not happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Consider now S = � i∈J+⊔J− Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' If S is empty then there are no contribution, since this point would not lie in the interior of other Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' So it suffices to consider the case when S is a segment [p, q].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' It is clear that the only points that could contribute to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='3) are p and q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' In particular, there are at most two nonzero terms in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We will show that they always cancel each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We also note that the orientation of the segment Si (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=', the direction in which the parameter ai increases) on the line lc+εv is determined by sgn(i), since the vector along the line is given by the nontrivial linear relation on vk,k∈J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' If both p and q are the ai = 1 and aj = 1 ends of the segments Si and Sj, then the segments must have opposite orientations on lc+εv (since they both should point towards the other point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' This means that sgn(i) = − sgn(j) and the two terms of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='3) cancel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Similarly, they cancel if p and q are the ai = 0 and aj = 0 ends of Si and Sj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Now suppose that p and q correspond to ai = 0 and aj = 1 ends of Si and Sj (in this case it is possible to have i = j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' In this case the two segments must have the same orientation, and then the factor (−1)deg c in the definition of ξ ensures that the two terms cancel each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' As v varies, we get a finite number of different formulas for the pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' It is also possible to take a more uniform choice of the pairing by integrating over v of degree 1 (ignoring the contributions of measure zero set of nongeneric v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' However, there does not appear to be any advantage in doing so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We will later see that the pairing is in fact independent of the choice of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Pairing of the Gamma series In this section we compute the pairing from the previous one on the cohomology-valued solutions to the better-behaved GKZ systems provided by the Γ series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We will show in the next section that the result is the dual of the intersection pairing which provides the proof of Conjecture 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='3 from [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We consider a regular triangulation Σ of the cone C whose vertices are among these vectors {vi}n i=1 and its corresponding toric Deligne-Mumford stack PΣ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' It will be convenient for us to abuse notation and denote by I both a subset of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' , n} and the corresponding cone � i∈I R≥0vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' ON HYPERGEOMETRIC DUALITY CONJECTURE 9 Similarly, Σ denotes both a simplicial complex on {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' , n} and the corre- sponding simplicial fan in NR which refines C and its faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' For each cone σ ∈ Σ we define Box(σ) to be the set of lattice points γ which can be written as γ = � i∈σ γivi with 0 ≤ γi < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We denote the union of all Box(σ) by Box(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' To each element γ ∈ Box(Σ) we associate a twisted sector of PΣ corresponding to the minimal cone σ(γ) in Σ containing γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We define the dual of a twisted sector γ = � γivi by γ∨ = � γi̸=0 (1 − γi)vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' or equivalently, the unique element in Box(σ(γ)) that satisfies γ∨ = −γ mod � i∈σ Zvi Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' The dual of γ = 0 is itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Clearly, we have σ(γ) = σ(γ∨) and (γ∨)∨ = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Twisted sectors are themselves smooth toric DM stacks and the following propositions describe a Stanley-Reisner type presentation of the spaces of cohomology and cohomology with compact support of their coarse moduli spaces, see [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' As usual, Star(σ(γ)) denotes the set of cones in Σ that contain σ(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Cohomology space Hγ of the twisted sector γ is naturally iso- morphic to the quotient of the polynomial ring C[Di : i ∈ Star(σ(γ))\\σ(γ)] by the ideal generated by the relations � j∈J Dj, J ̸∈ Star(σ(γ)), and � i∈Star(σ(γ))\\σ(γ) µ(vi)Di, µ ∈ Ann(vi, i ∈ σ(γ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We can also view Hγ as a module over the polynomial ring C[D1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' , Dn] by declaring Di = 0 for i ̸∈ Star(σ(γ)) and solving (uniquely) for Di, i ∈ σ(γ) to satisfy the linear relations �n i=1 µ(vi)Di = 0 for all µ ∈ N ∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Cohomology space with compact support Hc γ (viewed as a module over Hγ) is generated by FI for I ∈ Star(σ(γ)) such that σ◦ I ⊆ C◦ with relations DiFI − FI∪{i} for i ̸∈ I, I ∪ {i} ∈ Star(σ(γ)) and DiFI for i ̸∈ I, I ∪ {i} ̸∈ Star(σ(γ)) Similarly, it is given a structure of a module over C[D1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' , Dn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' The orbifold cohomology H of the smooth toric DM stack PΣ is defined as the direct sum � γ Hγ over all twisted sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Similarly, the orbifold cohomology with compact support Hc is defined as � γ Hc γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We denote by 1γ the generator of Hγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' 10 LEV BORISOV AND ZENGRUI HAN There is a natural perfect pairing between H and Hc called Euler charac- teristic pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Its origin is the eponymous pairing on certain Grothendick K-groups, which is then translated to the cohomology via the Chern char- acter, see [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We will not be using the original definition, but rather the following formula for the Euler characteristic pairing, which is proved in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' The Euler characteristic pairing χ : H ⊗ Hc → C on the toric DM stack PΣ is given by χ(a, b) = χ(⊕γaγ, ⊕γbγ) = � γ 1 | Box(σ(γ))| � γ∨ Td(γ∨)a∗ γbγ∨ Here ∗ : H → H is the duality map given by (1γ)∗ = 1γ∨ and (Di)∗ = −Di, and Td(γ) is the Todd class of the twisted sector γ which is defined as Td(γ) = � i∈Star σ(γ)\\σ(γ) Di � i∈Star σ(γ)(1 − e−Di).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' The linear function � : Hc γ → C takes values 1 VolI on each generator FI, where VolI denotes the volume of the cone σI in the quotient fan Σ/σ(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' It takes value zero on all elements of Hc γ of lower degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Let Σ be a regular (=projective) subdivision of C based on some of the vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Let ψi be the real numbers such that Σ reads off the lower boundary of the convex hull of the origin and {(vi, ψi), 1 ≤ i ≤ n} in NR⊕R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We assume that ψi are generic so this convex hull is simplicial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We denote by ψ the strictly convex piecewise linear function on C whose graph is the aforementioned lower boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' It takes values ψi on all vi which generate rays in Σ and has lower values than ψi on other vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Its key property is that for any finite collection wi ∈ C and αi ∈ R>0 there holds ψ( � i αiwi) ≤ � i αiψ(wi) with equality if and only if there exists a cone in Σ which contains all of the wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Recall from [1] the following solution to the equations bbGKZ(C, 0) with values in H = � γ Hγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We define Γc(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' , xn) = � γ � l∈Lc,γ n � i=1 x li+ Di 2πi i Γ(1 + li + Di 2πi) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='1) where the direct sum is taken over twisted sectors γ = � j∈σ(γ) γjvj and the set Lc,γ is the set of solutions to �n i=1 livi = −c with li − γi ∈ Z for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' The numerator is defined by picking a branch of log(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We will first prove that for each c ∈ C ∩ N the series for Γ converges absolutely and uniformly on compacts for x such that the (− log |xi|) are in an appropriate shift of the cone of values on vi of convex Σ-piecewise ON HYPERGEOMETRIC DUALITY CONJECTURE 11 linear functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' The proof was skipped in [1] because it is essentially the same as that in [4], but we will present it here, both for completeness and to facilitate arguments about the asymptotic behavior of Γc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We denote by CΣ the cone of the secondary fan that corresponds to Σ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' the cone of (ψi) ∈ Rn that give rise to Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' For each c ∈ C ∩N there exists ˆψ ∈ Rn such that the series (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='1) converges absolutely and uniformly on compacts in the region of Cn {(− log |x1|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' , − log |xn|) ∈ ˆψ + CΣ, arg(x) ∈ (−π, π)n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='2) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' An immediate observation is that we can ignore the factor n � i=1 x Di 2πi i = n � i=1 e Di log xi 2πi because it does not depend on l and is bounded on compacts in the region (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' It suffices to understand what happens for a fixed γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Note that while the summation takes place over an affine lattice Lc,γ, the nonzero contributions only occur for (l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' , ln) such that the set I(l) = {i, li ∈ Z<0} ⊔ σ(γ) is a cone σ in Σ, because each li ∈ Z<0 contributes a factor Di due to a pole of Γ at a nonpositive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Consequently, it suffices to bound the summation over the subset Lc,γ,σ of Lc,γ with the additional property that the above defined I(l) is a subset of some fixed maximum-dimensional cone σ of Σ that contains σ(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' For any such l ∈ Lc,γ,σ we have � i,li<0 (−li)vi = � i,li≥0 livi + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Let us denote by ψ the Σ-piecewise linear convex function that corresponds to (− ˆψi − log |xi|) by the assumption on x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Since the vi on the left hand side of the above equation lie in σ ∈ Σ, we have � i,li<0 (−li)(− ˆψi − log |xi|) = ψ( � i,li<0 (−li)vi) ≤ � i,li≥0 liψ(vi) + ψ(c) = � i,li≥0 li(− ˆψi − log |xi|) + ψ(c) and therefore n � i=1 li log |xi| ≤ − n � i=1 li ˆψi + ψ(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='3) This leads to an upper bound ��� � i=1 xli i ��� ≤ eψ(c)e− �n i=1 li ˆψi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='4) 12 LEV BORISOV AND ZENGRUI HAN Crucially, since all vi have degree 1, we see that � i li = − deg c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Thus, we can apply the key estimate of [4, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='4] which states that for any δ > 0 and any collection of real numbers ai, bi for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' , n with | � i ai| ≤ δ, � i |bi| ≤ δ there exists a constant A such that ��� n � i=1 1 Γ(ai + ibi) ��� ≤ A(4n) �n i=1 |ai|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' By the Cauchy’s formula for partial derivatives, this implies an upper bound of the form A1(A2) �n i=1 |li| on the coefficients on all monomials in Di of bounded degree of the function n � i=1 1 Γ(1 + li + Di 2πi) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Together with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='4), we conclude that in any Euclidean norm on Hγ the absolute value of each term of the series is bounded by ��� n � i=1 xli i Γ(1 + li + Di 2πi) ��� ≤ A1(A2) �n i=1 |li|��� � i=1 xli i ��� ≤ A3(A2) �n i=1 |li|e− �n i=1 li ˆψi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='5) We observe that the set Lc,γ,σ is the set of lattice points in a shift of a (lower-dimensional) polyhedral cone Cσ in Rn given by the equality �n i=1 livi = 0 and inequalities li ≥ 0 for all i ̸∈ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We may assume ˆψ to give a strictly Σ-convex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' It then follows that for any ray generator l of Cσ there holds � i li ˆψi > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Indeed, by convexity for ˆψ for any l ∈ Cσ we have the inequality � i li ˆψi ≥ 0 (the proof is the same as that of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='3)) which holds even if ˆψ is deformed slightly, so it can only be equality for l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' As a consequence, there is a constant r such that n � i=1 |li| ≤ r( � i li ˆψi) on Cσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Therefore, we can replace ˆψ by a large enough multiple of itself and use (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='5) to get on any compact subset of the region (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='2) ��� n � i=1 xli i Γ(1 + li + Di 2πi) ��� ≤ A4e−A5 �n i=1 li ˆψi ON HYPERGEOMETRIC DUALITY CONJECTURE 13 for some A5 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Since the number of terms in Lc,γ,σ with �n i=1 li ˆψi ∈ [m, m+1) is bounded by a polynomial in m, we get the desired convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' □ There is a similarly defined Γ-series solution Γ◦ of bbGKZ(C◦, 0), with values in Hc = � γ Hc γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We define Γ◦ c(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' , xn) = � γ � l∈Lc,γ n � i=1 x li+ Di 2πi i Γ(1 + li + Di 2πi) �� i∈σ D−1 i � Fσ where σ is the set of i with li ∈ Z<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' The series Γ◦ converges uniformly on compacts in the region (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='2) for an appropriate choice of ˆψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' The idea of the proof are the same as that of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='8 and we leave the details to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' □ Our next goal is to understand the asymptotic behavior of Γc(t−ψ(v1)x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' , t−ψ(vn)xn) for real t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We can assume xi to be generic nonzero complex numbers, so that for large enough t we fall within the range of convergence of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' For each c we consider the minimum cone σ(c) of Σ that contains c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We have c = � j∈σ(c) cjvj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' It defines a twisted sector γ(c) = � j∈σ(c){cj}vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We also consider the dual twisted sector γ∨(c) = � j∈σ(c),cj̸∈Z(1 − {cj})vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' There is a special element −c = � i∈I (−ci)vi (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='6) in Lc,γ∨(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' As t → +∞, we have for c ∈ C ∩ N and γ ̸= γ∨(c) the γ summand of Γc(t−ψ(v1)x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' , t−ψ(vn)xn) is o(tψ(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' For γ = γ∨(c) we have Γc(t−ψ(v1)x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=') = tψ(c) n � i=1 e Di 2πi (log xi−ψ(vi) log t) n � i=1 x−ci i Γ(1 − ci + Di 2πi) (1 + o(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Let γ = � j∈σ(γ) γjvj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Let (li) be an element of Lc,γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' The contribu- tion to Γc(t−ψ(v1)x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=') is only nonzero if the set of i for which li ∈ Z<0 together with σ(γ) is a cone in Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Consequently, i for which li are negative lie in a cone of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Therefore, � li<0 (−li)ψ(vi) = ψ( � li<0 (−li)vi) = ψ(c + � li>0 livi) ≤ � li>0 liψ(vi) + ψ(c), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='7) 14 LEV BORISOV AND ZENGRUI HAN which implies − n � i=1 liψ(vi) ≤ ψ(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='8) Now notice that the equality in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='8) holds if and only if the minimal cone of � li<0(−li)vi is a cone in Σ which contains c and all vi with li > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' This cone would then contain c and all vi for which li ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' This means that li = −ci, which implies that γ = γ∨(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' This gives the claimed asymptotic contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' It is not enough to bound the asymptotic behavior of each individual term as t → ∞, one also needs to ensure that the rest of the terms together do not contribute to anything larger than o(tψ(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' This follows either from the estimates of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='8 or simply from the fact that we have absolute convergence at x and then all other terms decay faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Indeed, if we have an absolutely convergent series � i≥0 ai and then consider � i≥0 aitαi with α0 − αi larger than some positive ε, then as t → ∞ we have � i≥0 aitαi = a0tα0(1 + o(1)) because ��� � i>0 aitαi−α0 ��� ≤ t−ε � i>0 |ai|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We can apply it to our situation since (li) are in a countable set and there exists ε > 0 so that for all other terms the inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='8) is strict by at least ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' The logarithmic terms � i(t−ψ(v1)xi) Di 2πi can be absorbed by a slight change of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' □ We can state a similar result for Γ◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' For d ∈ C◦ we consider the element of Ld,γ∨(d) −d = � i∈σ(d) (−di)vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' As t → +∞, we have for c ∈ C ∩ N and γ ̸= γ∨(d) the γ summand of Γ◦ d(t−ψ(v1)x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' , t−ψ(vn)xn) is o(tψ(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' For γ = γ∨(d) we have Γd(t−ψ(v1)x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=') = tψ(d) n � i=1 e Di 2πi (log xi−ψ(vi) log t) n � i=1 x−di i Γ(1 − di + Di 2πi) \uf8eb \uf8ed � i∈σ(d) D−1 i \uf8f6 \uf8f8 Fσ(d)(1 + o(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' The proof is analogous to that of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='10 and is left to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' □ ON HYPERGEOMETRIC DUALITY CONJECTURE 15 Now we use this information about the asymptotic behavior of Γ and Γ◦ to compute the constant ⟨Γ, Γ◦⟩ = � c,d,I ξc,d,I VolI �� i∈I xi � Γc ⊗ Γ◦ d where ξ are defined in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' As in Section 2, let I be a subset of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' , n} of size rkN, which may or may not be a cone in Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Let c and d be such that c + d = � i∈I vi and c + εv, d − εv ∈ � i∈I R≥0vi for small ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' The following observation is key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Under the above assumptions on c, d, I we have lim t→+∞ n � i=1 (t−ψ(vi)xi)Γc(t−ψ(v1)x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' )Γ◦ d(t−ψ(v1)x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=') = 0 unless γ(d) = γ∨(c) and I contains σ(γ(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Since c and d are contained in � i∈I R≥0vi and c + d = vI, we have c = � i∈I αivi, d = � i∈I (1 − αi)vi with αi ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Convexity of ψ implies that ψ(c) ≤ � i αiψ(vi), ψ(d) ≤ � i (1 − αi)ψ(vi) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='9) which leads to ψ(c) + ψ(d) − � i ψ(vi) ≤ 0, so we can use Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='10 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='11 to see that the leading power of t is nonpositive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' In fact, it is negative, unless the inequalities in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='9) are equalities, which means that the subset of I for which αi > 0 is a cone in Σ, and similarly for the subset of αi < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' This implies the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' If γ(c) = γ∨, γ(d) = γ = � i∈I γivi, then we define Ic to be the subset of I such that the coefficients ci of c are equal to 1 and similarly for Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' The asymptotic behavior as t → ∞ is n � i=1 (t−ψ(vi)xi)Γc(t−ψ(v1)x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' )Γ◦ d(t−ψ(v1)x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=') = o(1) + 1 (2πi)rk N−|σ(γ)| DIc � i∈σ(γ) Γ(γi + Di 2πi) � i∈Star(σ(γ))\\σ(γ) Γ(1 + Di 2πi) n � i=1 e Di 2πi (log xi−ψ(vi) log t) � FId � i∈σ(γ) Γ(1 − γi + Di 2πi) � i∈Star(σ(γ))\\σ(γ) Γ(1 + Di 2πi) n � i=1 e Di 2πi (log xi−ψ(vi) log t) in Hγ ⊗ Hc γ∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' The proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='12 shows that the only contribution other than o(1) can come from the terms that give better than o(tψ(c)) and o(tψ(d)) contributions to the asymptotic behavior of Φc and Φd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' So by Propositions 16 LEV BORISOV AND ZENGRUI HAN 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='10 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='11 the only contributions come from elements of Lc,γ∨ and Ld,γ given by −c = � i∈I (−ci)vi, −d = � i∈I (ci − 1)vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' For i ∈ σ(γ) we note that γi = 1 − ci if ci ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' For i ∈ Ic we use 1 Γ(1 − ci + Di 2πi) = 1 Γ( Di 2πi) = Di 2πi Γ(1 + Di 2πi) and similarly for i ∈ Id, and the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' □ Now we recall that ⟨Γ, Γ◦⟩ is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' The constant pairing ⟨Γ, Γ◦⟩ lies in � γ Hγ ⊗ Hc γ∨ and is given by 1 (2πi)rk N � γ � c∈C,d∈C◦ |I|=rkN ξc,d,I VolI(2πi)|σ(γ)| DIc �Γγ ⊗ FId �Γγ∨ where �Γγ = � i∈σ(γ) Γ(γi + Di 2πi) � i∈Star(σ(γ))\\σ(γ) Γ(1 + Di 2πi) and similarly for �Γγ∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' There also holds for each k 0 = � γ � c∈C,d∈C◦ |I|=rk N ξc,d,I VolI(2πi)|σ(γ)|� Dk DIc �Γγ � ⊗ FId �Γγ∨ + � γ � c∈C,d∈C◦ |I|=rkN ξc,d,I VolI(2πi)|σ(γ)| DIc �Γγ ⊗ � Dk FId �Γγ∨ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='13 gives the asymptotic behavior of ⟨Γ, Γ◦⟩ as a poly- nomial in log xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' However, we also know it is a constant by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' The first statement of the proposition is reading off the constant term of the polynomial and the second statement is reading off the coefficient by log xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Euler characteristic pairing Now we are ready to prove that the pairing of Gamma series ⟨Γ, Γ◦⟩ is inverse to the Euler characteristic pairing on PΣ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Before we state the main theorem of this section, we have the following useful observation, which is an orbifold analog of the relationship between the Γ-class and the Todd class of a smooth manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Recall that ∗ is the duality map on H defined in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' (�Γγ)∗�Γγ∨ = (2πi)|σ(γ)|(−1)deg γ∨ Td(γ∨).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' ON HYPERGEOMETRIC DUALITY CONJECTURE 17 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We can expand (�Γγ)∗�Γγ∨ as � i∈σ(γ) Γ(γi + Di 2πi)∗Γ(1 − γi + Di 2πi) � i∈Star(σ(γ))\\σ(γ) Γ(1 + Di 2πi)∗Γ(1 + Di 2πi) = � i∈σ(γ) Γ(γi − Di 2πi)Γ(1 − γi + Di 2πi) � i∈Star(σ(γ))\\σ(γ) Γ(1 − Di 2πi)Γ(1 + Di 2πi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We use the identity Γ(z)Γ(1 − z) = − 2πi eπiz 1−e2πiz to rewrite the first product as (−2πi)|σ(γ)|e � i∈σ(γ) πiγie− 1 2 � i∈σ(γ) Di � i∈σ(γ) 1 1 − e2πiγi−Di .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' For the second product, we use Γ(1− z 2πi)Γ(1+ z 2πi) = ze− z 2 1−e−z to rewrite it as e− 1 2 � i∈Star(σ(γ))\\σ(γ) Di � i∈Star(σ(γ))\\σ(γ) Di 1 − e−Di .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Putting the two formulas together, we get (�Γγ)∗�Γγ∨ = (2πi)|σ(γ)|(−1)deg γ∨e− 1 2 � i∈Star(σ(γ)) Di � i∈Star(σ(γ))\\σ(γ) Di � i∈Star(σ(γ)) 1 − e−Di = (2πi)|σ(γ)|(−1)deg γ∨ Td(γ∨) where we used � i∈Star(σ(γ)) Di = �n i=1 Di = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' □ Now we can state and prove the main theorem of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Recall that we defined the pairing ⟨·, ·⟩ on solutions of the better-behaved GKZ systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' When we apply it to Γ and Γ◦, we get a constant element of H ⊗ Hc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' The constant pairing ⟨Γ, Γ◦⟩ is equal up to a constant factor to the inverse of the Euler characteristic pairing χ(−, −) : H ⊗ Hc → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' It’s clear that we can consider each twisted sector individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' For a fixed γ, the statement is equivalent to the assertion that � c∈C,d∈C◦ |I|=rkN ξc,d,I VolI(2πi)|σ(γ)|χ � P, FId �Γγ∨ � DIc �Γγ = P holds for all classes P ∈ Hγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Since the class �Γγ is invertible in Hγ, dividing by it induces an automorphism on the cohomology, hence it suffices to prove � c∈C,d∈C◦ |I|=rkN ξc,d,I VolI(2πi)|σ(γ)|χ � P �Γγ , FId �Γγ∨ � DIc �Γγ = P �Γγ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='1) for all P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We prove this by induction on the degree of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' 18 LEV BORISOV AND ZENGRUI HAN The base case deg P = 0 corresponds to P = 1γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Since χ � 1γ �Γγ , FId �Γγ∨ � = 0 unless |Id| = rkN − |σ(γ)|, the equation becomes � |Id|=rk N−|σ(γ)| ξγ∨,γ+vId,Id⊔σ(γ) VolId⊔σ(γ)(2πi)|σ(γ)|χ � 1γ �Γγ , FId �Γγ∨ � 1γ �Γγ = 1γ �Γγ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='2) Then by definition of χ and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='1, we have χ � 1γ �Γγ , FId �Γγ∨ � = 1 | Box(σ(γ))| � γ∨ Td(γ∨) � 1 �Γγ �∗ FId �Γγ∨ = 1 | Box(σ(γ))| � γ∨ FId (�Γγ)∗�Γγ∨ Td(γ∨) = 1 | Box(σ(γ))| � γ∨ FId (2πi)|σ(γ)|(−1)deg γ∨ = (−1)deg γ∨ (2πi)|σ(γ)| VolId | Box(σ(γ))| here VolId denotes the volume of the cone σId in the quotient fan Σ/σ(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Note that we have VolId⊔σ(γ) = VolId | Box(σ(γ))| hence (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='2) becomes � |Id|=rk N−|σ(γ)| (−1)deg γ∨ξγ∨,γ+vId,Id⊔σ(γ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' If we perturb γ∨ by εv, then it will fall in the interior of exactly one maximal cone in Σ, and the corresponding coefficient ξ is the only nonzero term in the sum above (recall the definition of ξc,d,I in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='4), which is equal to (−1)deg γ∨(−1)deg γ∨ = 1 So the base case is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Now we assume the equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='1) holds for all classes of degree less than m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Since the cohomology Hγ is generated as an algebra by classes Dk, it suffices to prove the identity � c∈C,d∈C◦ |I|=rkN ξc,d,I VolI(2πi)|σ(γ)|χ � DkP �Γγ , FId �Γγ∨ � DIc = DkP ON HYPERGEOMETRIC DUALITY CONJECTURE 19 for each DkP where P ∈ Hγ is of degree m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Since Dk is skew-symmetric with respect to the χ pairing, the above statement can be rewritten as DkP = − � c∈C,d∈C◦ |I|=rk N ξc,d,I VolI(2πi)|σ(γ)|χ � P �Γγ , DkFId �Γγ∨ � DIc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' On the other hand, we can multiply the induction assumption for P by Dk to get � c∈C,d∈C◦ |I|=rkN ξc,d,I VolI(2πi)|σ(γ)|χ � P �Γγ , FId �Γγ∨ � Dk DIc = DkP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Compare these two identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' It suffices to show 0 = � c∈C,d∈C◦ |I|=rkN ξc,d,I VolI � Dk · DIc �Γγ � ⊗ FId �Γγ∨ + � c∈C,d∈C◦ |I|=rkN ξc,d,I VolI DIc �Γγ ⊗ � Dk · FId �Γγ∨ � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='3) which follows from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='2 implies, in particular, that the pairing of The- orem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='4 is nondegenerate and is independent of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We are not aware of a direct proof of this fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We conclude this section by an explanation of our motivation behind the definition of the coefficients ξc,d,I in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' This definition is inspired by the following fan displacement resolution of diagonal formula of Fulton- Sturmfels [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Let X be the toric variety corresponds to a complete fan Σ in a lattice N, denote the diagonal embedding X ֒→ X × X by δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Let σ ∈ Σ be any cone and v a generic point in N, then the diagonal class decomposes as [δ(V (σ))] = � σ1,σ2 mσ σ1,σ2 · [V (τ1) × V (τ2)] where mσ σ1,σ2 = [N : Nσ1 +Nσ2] and the sum is over all cones σ1, σ2 ∈ Σ with codim σ1 + codim σ2 = codim σ and σ ⊆ σ1, σ2 such that (v + σ1) ∩ σ2 ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Note that the coefficient mσ σ1,σ2 is exactly the volume Volσ1∪σ2 of the cone spanned by σ1 and σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' This formula cannot be applied to our case directly, since the toric varieties they worked with are complete while ours are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Nevertheless we have the following relationship between the definition of ξc,d,I and the conditions occurred in Fulton-Sturmfels formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' 20 LEV BORISOV AND ZENGRUI HAN Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Let c, d ∈ σI and v be a generic point in C◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Then both c + εv and d − εv lies in σ◦ I for all sufficiently small ε > 0 if and only if (v + σ(c)) ∩ σ(d) ̸= ∅ where σ(c) denotes the minimal cone of Σ that contains c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Assume both c+εv and d−εv lies in σ◦ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Then we can write c+εv = � i∈I sivi where all si ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Recall that I = σ(c)∪σ(d) = Ic⊔Id⊔σ(γ(c)), this equation can be rewritten into the form v = v1−v2, where v1 ∈ σ(c) and v2 ∈ σ(d), which is equivalent to the second statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' The other direction can be proved similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We believe our methods should allow one to give a new proof of the Fulton-Sturmfels formula, which could be done by restricting our results to the twisted sectors that are compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We do not go into details further in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Extensions and open questions There is a more general version of the better-behaved GKZ systems which includes a parameter β ∈ NC, with β = 0 case being the one we considered so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Namely, the torus homogeneity equations of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='1 read n � i=1 ⟨µ, vi⟩xi∂iΦc + ⟨µ, c − β⟩Φc = 0 and similarly for Ψd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Much of what we did in this paper is applicable to the pair of better behaved GKZ systems with parameters ±β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' For instance, we readily observe that our argument in Section 2 goes through for arbitrary parameter β to give a pairing between spaces of solutions to bbGKZ(C, β) and bbGKZ(C◦, −β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We would like to see what happens in the limit given by a regular sub- division Σ for a generic β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' While there are certain versions of H and Hc considered in [8] it will be easier for our purposes to simply write Vol(∆) linearly independent solutions given by Γ-series, essentially along the lines of the solutions of the original GKZ paper [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Let Σ be a regular subdivision of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' For each maximum-dimensional cone σ we consider Vol(σ) linearly independent solutions in the large K¨ahler limit of PΣ, in bijection with the elements γ of N/ � i∈σ Zvi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Namely, we define the set Lc,γ,σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='β ⊂ Cn by n � i=1 livi = β − c and the properties li ∈ Z for all i ̸∈ σ and c+� i̸∈σ livi = −γ mod � i∈σ Zvi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Then for each γ we define a solution Φγ,σ of bbGKZ(C, β) by Φγ,σ c (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' , xn) = � l∈Lc,γ,σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='β n � i=1 xli i Γ(1 + li).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' ON HYPERGEOMETRIC DUALITY CONJECTURE 21 We define Γ-series solutions Ψγ,σ to bbGKZ(C◦, −β) in the same way by Ψγ,σ d (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' , xn) = � l∈Ld,γ,σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='−β n � i=1 xli i Γ(1 + li).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Note that in the case of generic β every solution of bbGKZ(C◦, −β) can be uniquely extended to solutions of bbGKZ(C, −β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' It is not hard to show that these Φc and Ψd converge uniformly on compacts in the region (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='2) for an appropriate choice of ˆψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Moreover, as σ and γ vary, we get bases of the space of solutions, with linear independence assured by them lying in different eigenspaces of the monodromy operators for small loops around xi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Monodromy considerations imply that for the pairing ⟨·, ·⟩ of Section 2 we have ⟨Φγ,σ, Ψγ′,σ′⟩ = 0 unless σ = σ′ and γ = −γ′ mod � i∈σ Zvi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' In the latter case, the constant contribution will happen for li + l′ i = 0 for i ̸∈ I and li + l′ i = −1 for i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' If any of li, l′ i is a negative integer, then the corresponding term vanishes, due to a pole of Γ, so we may assume that they are nonnegative for i ̸∈ σ, which then implies that I = σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' li + l′ i = −1, for i ∈ σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' li = l′ i = 0 for i ̸∈ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' This implies that c = −γ mod � i∈σ Zvi and d = γ mod � i∈σ Zvi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We claim that for any γ there exists exactly one pair (c, d) in σ satisfying this constraint and ξc,d,σ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' The definition of the coefficients ξ of the pairing implies that we must also have c + d = � i∈σ vi with c + εv and d − εv in the corresponding cone � i∈σ R≥0vi for all small ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We can write β, v and γ uniquely as β = � i∈σ βivi, v = � i∈σ sivi, γ = � i γivi with γi ∈ [0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' It is then easy to see that ξc,d,σ is nonzero if and only if c = � {i:γi̸=0} (1 − γi)vi + � {i:γi=0,si<0} vi, d = � {i:γi̸=0} γivi + � {i:γi=0,si>0} vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Thus for γi ̸= 0 we have li = βi − 1 + γi, l′ i = −βi − γi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' For γi = 0 and si > 0 we have li = βi, l′ i = −1 − βi and for γi = 0 and si < 0 we have li = −1 + βi, l′ i = −βi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' In particular, deg(c) = − deg(γ) + rkN − #{i : γi = 0, si > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' 22 LEV BORISOV AND ZENGRUI HAN Therefore the pairing is given by ⟨Φγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Ψ−γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='σ⟩ = (−1)deg(c) Vol(σ) � γi̸=0 1 Γ(βi + γi)Γ(1 − βi − γi) � γi=0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='si>0 1 Γ(1 + βi)Γ(−βi) � γi=0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='si<0 1 Γ(βi)Γ(1 − βi) = (−1)deg(c) Vol(σ) � γi̸=0 e2πi(βi+γi) − 1 2πi eπi(βi+γi) � γi=0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='si>0 e2πi(βi+1) − 1 2πi eπi(βi+1) � γi=0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='si<0 e2πiβi − 1 2πi eπiβi = (−1)deg(c) Vol(σ) (2πi)rkN e−πi � i∈σ(βi+γi) � γi=0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='si>0 eπi � i∈σ (e2πi(βi+γi) − 1) = Vol(σ) (2πi)rkN e−πi deg(β)−2πi deg(γ) � i∈σ (1 − e2πi(βi+γi)) = e−πi deg(β)Vol(σ) (2πi)rkN � i∈σ (1 − e2πi(βi+γi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' An immediate consequence of the above calculation is that the pairing ⟨·, ·⟩ is non-degenerate for a generic β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Further directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We conclude this section by stating some open problems related to our construction, in no particular order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Is the pairing of this paper nondegenerate for all β?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We know this to be the case for β = 0 and β generic, and it seems likely to be always true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We would like to settle the analytic continuation conjecture of [1] to extend the main result of [4] to the better-behaved GKZ systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' One consequence of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='2 is that it should be enough to just work with the usual K-theory and the compactly supported version should follow from duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' What is the HMS counterpart of our pairing from the point of view of Fukaya-Seidel categories for the mirror potential?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Our formula for the pairing is quite simple, so presumably so should be the mirror version of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' We refer to [5], [11] for background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Solutions to bbGKZ systems come with a lattice structure inherited from the K-theory of PΣ (it is independent of Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Can this structure be locally defined outside of the region of convergence of any Γ-series?' metadata={'source': 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+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' European Mathe- matical Society, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content=' Department of Mathematics, Rutgers University, Piscataway, NJ 08854 Email address: borisov@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='rutgers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='edu Department of Mathematics, Rutgers University, Piscataway, NJ 08854 Email address: zh223@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='rutgers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} +page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdAzT4oBgHgl3EQfavyA/content/2301.01374v1.pdf'} diff --git a/hNAzT4oBgHgl3EQfov3Z/content/tmp_files/2301.01603v1.pdf.txt b/hNAzT4oBgHgl3EQfov3Z/content/tmp_files/2301.01603v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b7bfadcd4ff25d1a6f7afbdba825526d74dc7187 --- /dev/null +++ b/hNAzT4oBgHgl3EQfov3Z/content/tmp_files/2301.01603v1.pdf.txt @@ -0,0 +1,997 @@ +arXiv:2301.01603v1 [gr-qc] 25 Dec 2022 +Exponential corrected thermodynamics of Born-Infeld +BTZ black holes in massive gravity +B. Pourhassan,a,b,c M. Dehghani,d S. Upadhyay, e,f,a,∗ ˙I. Sakallı,g and D. V. Singhh +aSchool of Physics, Damghan University, Damghan, 3671641167, Iran. +bPhysics Department, Istanbul Technical University, Istanbul 34469, Turkey. +cCanadian Quantum Research Center 204-3002 32 Avenue Vernon, British Columbia V1T 2L7 Canada. +dDepartment of Physics, Razi University, Kermanshah, Iran. +eDepartment of Physics, K.L.S. College, Nawada, Bihar 805110, India. +fDepartment of Physics, Magadh University, Bodh Gaya, Bihar 824234, India. +gDepartment of Physics, Eastern Mediterranean University, Famagusta 99628, North Cyprus via Mersin +10, Turkey. +hDepartment of Physics, Institute of applied Science and Humanities, GLA University, Mathura 281406 +Uttar Pradesh, India. +E-mail: b.pourhassan@du.ac.ir, m.dehghani@razi.ac.ir, +sudhakerupadhyay@gmail.com; sudhaker@associates.iucaa.in, +izzet.sakalli@emu.edu.tr, veerdsingh@gmail.com +Abstract: It is known that entropy of black hole gets correction at quantum level. Universally, +these corrections are logarithmic and exponential in nature. We analyze the impacts of these +quantum corrections on thermodynamics of Born-Infeld BTZ black hole in massive gravity by +considering both such kinds of correction. We do comparative analysis of corrected thermody- +namics with their equilibrium values. Here, we find that the exponential correction yields to the +second point of the first order phase transition. Also, quantum correction effects significantly +on the Helmholtz free energy of larger black holes. +We study the equation of state for the +exponential corrected black hole to obtain a leading order virial expansion. +Keywords: Keywords: Black hole; Born-Infeld electrodynamics; Corrected entropy; Thermo- +dynamics. +*Visiting Associate at Inter-University Centre for Astronomy and Astrophysics (IUCAA), Pune, Maharashtra +411007, India. + +Contents +1 +Introduction +1 +2 +BI BTZ black holes in massive gravity +3 +2.1 +Gravitational field equations and singular solution +3 +2.2 +Equilibrium thermodynamics +5 +3 +Quantum corrected thermodynamics due to the LC entropy +6 +4 +Quantum corrected thermodynamics due to the EC entropy +9 +5 +Conclusion +12 +1 +Introduction +Even though three dimensional gravity has no Newtonian limit, it has always been useful for +conceptual issues. This was further supported by discovery of the three dimensional Banados– +Teitelboim–Zanelli (BTZ) black hole solution [1–3]. This discovery of three dimensional BTZ +black holes really helped in recent developments in gravity, gauge and string theory. The behavior +of geodesics [5, 6] and the propagation of strings in a BTZ background [7] had already been +discussed. In contrast to Schwarzschild and Kerr solutions, the BTZ black hole is asymptotically +anti-de Sitter and has no curvature singularity at center. Due to asymptotically AdS in nature +of such BTZ black hole solution, it strengthen the idea of AdS/CFT correspondence. BTZ black +holes are widely studies [4, 8, 9]. The noncommutativity and scattering of massless planar scalar +waves had been discussed in the context of BTZ black hole [10]. +On the other hand, there exist a lot of reasoning to modify the Einstein gravity. The idea to +include massive gravitons is one of them which explains the current acceleration of the universe +without considering a cosmological constant [11, 12]. The solution of hierarchy problem hints the +existence of massive modes [13]. Massive gravity gets relevance in astrophysics also as neutron +stars can have mass more than 3M⊙ in massive gravity scenario [14]. The massive graviton +affects the gravitational waves also [15]. Various black hole solution have been studied in four- +and higher-dimensional massive gravity [16–18]. One can not ignore the possibility to explore +BTZ black hole in massive gravity. An AdS charged BTZ black hole in a massive gravity has +been explored [19]. Here, thermodynamics and phase structure have also been discussed in both +grand canonical and canonical ensembles [19]. +– 1 – + +Bekenstein, profoundly, interpreted the area of the event horizon of a black hole (A) as its +entropy (S0) [20]. This is given by +S0 = A +4l2p +, +(1.1) +where lp is the Planck length. This relationship more or less agree in the circumstances when +black holes are much larger than the Planck scale. However, for relatively smaller black hole +this relation needs correction. For such black holes, thermal fluctuations around equilibrium +becomes significant and correct the black hole entropy. There are several ways to obtain such +corrections. Recently, it has been suggested that correction terms are universal and have the +following approximate shape [21], +S = S0 + α ln S0 + γ +S0 ++ ωe−S0 + ... +(1.2) +where dots denote higher-order corrections. Here, we notice that at first-order entropy gets loga- +rithmic correction as confirmed by microstate counting in string theory as well as loop quantum +gravity [22]. +However, exponential corrected (EC) term occurs when microstate counting is +done for quantum states on the horizon only [21]. The phase transition and thermodynamics of +BTZ black holes through the Landau-Lifshitz theory has been given in Ref. [23]. The effects of +entropy correction in the noncommutative BTZ black holes is presented in Ref. [24]. Thermal +fluctuations of charged black holes in massive gravity is presented in Ref. [25]. The effects of +corrected entropy on thermodynamics of various black holes are studied [26–31]. Now, the main +goal of this paper is to study the exponential correction of the Born-Infeld BTZ black holes in +massive gravity thermodynamics. +In GR and its extensions, nonlinear electrodynamics plays a crucial role. For example, it +produces many interesting geometries such as regular black holes. The nonlinear electrodynamics +is a direct generalization of the Maxwell electrodynamics originated by Born and Infeld (BI) +in order to remove the central singularity of a point charge [32]. +BI terms also appears in +superstrings scenario more naturally [33, 34]. Black holes with BI term are quite important in +astrophysical observations [35–38]. Bardeen was first who proposed regular black hole [39] and it +was found that regular black hole with BI term only describes the black hole formation from an +initial vacuum region. Albeit significant progress made on the subject, quantum effects on the +thermodynamics of BI BTZ black hole in massive gravity are still unstudied. This proves us an +opportunity to shed light on this. The BTZ black hole is a solution of the Einstein field equations +in three dimensions that describes a black hole with negative cosmological constant. The Born- +Infeld theory is a modified theory of electromagnetism that is characterized by a non-linear +dependence of the electromagnetic field on the electric and magnetic fields. The combination +of these two concepts leads to the concept of a quantum Born-Infeld BTZ black hole, which is +a black hole that is characterized by both quantum and non-linear electromagnetic effects. In +the context of massive gravity, the thermodynamics of quantum Born-Infeld BTZ black holes +can be studied by considering the effects of the mass term for the graviton on the properties of +the black hole. This can include the effects on the black hole temperature, entropy, and other +thermodynamic quantities. It is important to note that the thermodynamics of quantum Born- +Infeld BTZ black holes in massive gravity is still a highly theoretical and complex topic that is not +– 2 – + +yet fully understood. Further research is needed to fully understand the thermodynamics of these +objects and their potential implications for our understanding of gravity and the fundamental +nature of spacetime. These are motivations behind present work. +The paper is presented systematically in following manner. In Sec. 2, we discuss the singular +solution of BI massive gravity and their equilibrium thermodynamics. The quantum corrected +thermodynamics of such black hole attributed by logarithmic corrected (LC) entropy is presented +in section 3. The corrections in thermodynamics of this black hole attributed by EC entropy is +discussed in section 4. Finally, results of paper is summarized in the last section. +2 +BI BTZ black holes in massive gravity +In this section, we shed light on the gravitational field equations and singular solution of the +three-dimensional massive gravity in presence of nonlinear electrodynamics. We also review its +thermal properties. +2.1 +Gravitational field equations and singular solution +In this section, we shed light on particular massive BTZ black hole solution [40] in presence of +BI source. Let us begin by writing the action of three-dimensional massive gravity in presence +of nonlinear electrodynamics [41, 42] +I = +1 +16π +� √−g +� +R − 2Λ + m2 +G +4 +� +i=1 +ciUi(g, f) + L(F) +� +d3x, +(2.1) +where Λ is the cosmological constant, mG is the constant parameter related to the graviton mass, +coefficients ci are some constant, and f is fixed symmetric tensor known as reference metric. +Here, symmetric polynomials Ui are given by [43, 44] +U1 = [K], +U2 = [K]2 − [K2], +U3 = [K]3 − 3[K][K2] + 2[K3], +U4 = [K]4 − 6[K]2[K2] + 8[K3][K] + 3[K2]2 − 6[K4], +(2.2) +where [K] ≡ Ka +a is eigenvalues of the 3 × 3 matrix Ka +b = √gacfcb. However, L(F) refers to the +Lagrangian of nonlinear electrodynamics expressed in terms of Maxwell’s invariant F = F abFab, +where field-strength tensor of vector field Aa has the following form: Fab = ∂aAb − ∂bAa. +Specifically, here, we are interested in following BI type Lagrangian of nonlinear electrodynamics +[45]: +L(F) = 4b2 +� +1 − +� +1 + F +2b2 +� +, +(2.3) +where b is a constant parameter called as nonlinearity parameter. In the large b limit (b → ∞), +the Lagrangian (2.3) corresponds to the Maxwell’s classical electrodynamics. +– 3 – + +For the given action (2.1), the field equations corresponding to gravitational field gab and +vector field Ab are given, respectively, by [43, 44] +Rab − 1 +2Rgab + Λgab + m2 +Gχab = 1 +2L(F)gab − 2dL(F) +dF +FacF c +b , +(2.4) +∂a +�√−g dL(F) +dF +F ab +� += 0, +(2.5) +where tensor χab related to massive graviton has the following expression [43]: +χab = − c1 +2 (U1gab − Kab) − c2 +2 +� +U2gab − 2U1Kab + 2K2 +ab +� +− c3 +2 +� +U3gab − 3U2Kab + 6U1K2 +ab − 6K3 +ab +� +− c4 +2 +� +U4gab − 4U3Kab + 12U2K2 +ab − 24U1K3 +ab + 24K4 +ab +� +. +(2.6) +In order to have the spherically symmetric solution for the field equations (2.4) and (2.5), we +make an ansatz for the line element [46, 47]: +ds2 = −f(r)dt2 + dr2 +f(r) + r2dθ2, +(2.7) +where the specific form of metric function f(r) will be determined in the presence of nonlinear +(BI) electrodynamics. +For the following reference metric fab = diag(0, 0, c2), the polynomials introduced by equa- +tion (2.2) are simplified to [19, 43], +U1 = c +r, +and +U2 = U3 = U4 = 0, +(2.8) +where c is a positive constant. +For, these values of polynomials, the components of tensor χab (2.6) identified to +χtt = −cc1 +2r gtt, χrr = cc1 +2r grr, +χθθ = 0. +(2.9) +Noting the facts that non-vanishing components of field-strength tensor are Ftr = −Frt = +−∂rAt(r) and, thus, Maxwell’s invariant can be expressed as F = −2F 2 +tr. Corresponding to +relations (2.5) and (2.7), the electromagnetic field equation leads to +Ftr(r) = q +rβ , +and +At(r) = −q ln +� r +2ℓ (1 + β) +� +. +(2.10) +Here q refers to an integration constant which is relevant for the black hole charge and β = +� +1 + +q2 +b2r2 [48]. +Corresponding to the metric (2.7), the components of gravitational field equations (2.4) take +the following differential forms: +Gtt = Grr = f ′(r) +r ++ 2Λ + 4b2 (β − 1) − cc1m2 +G +r += 0, +(2.11) +– 4 – + +Gθθ = f ′′(r) + 2Λ + 4b2 � +β−1 − 1 +� += 0. +(2.12) +Here, we observe that the components of the gravitational field equations hold the following +relation: +� +r d +dr + 1 +� +Gtt = Gθθ. +(2.13) +Since the components of field equations are interrelated, we need to solve only one (preferably +first-order one) of equations (2.11) and (2.12). This leads to the solution for metric function as +f(r) = −m − Λr2 + m2 +Gcc1r − 2b2r2 (β − 1) + q2 − 2q2 ln +� r +2ℓ (1 + β) +� +, +(2.14) +where mass parameter m is a constant of integration. +This is worth to calculate Ricci scalar and Kretschmann scalar as they play significant role +in the study of space-time singularities. Thus, the Ricci scalar and the Kretschmann scalar for +this theory of gravity are calculated, respectively, by +R = 6Λ + 4b2 �� +β−1 − 1 +� ++ 2 (β − 1) +� +− 2m2 +Gcc1 +r +, +(2.15) +RµνρλRµνρλ = 12Λ2 + 16b4 �� +β−1 − 1 +�2 + 2 (β − 1)2� +− 8b2m2 +Gcc1 +r +(β − 1) ++ 16Λb2 �� +β−1 − 1 +� ++ 2 (β − 1) +� +− 8Λm2 +Gcc1 +r ++ 2(m2 +Gcc1)2 +r2 +. +(2.16) +It is obvious from Eqs. (2.15) and (2.15) that the Ricci scalar and Kretschmann scalar take finite +value for finite r. The solution described by metric function (2.14) is not a regular solution as +the singularity is not a coordinate one but a essential singularity. This solution behaves like +AdS black holes asymptotically [48]. Above solutions are already presented by Refs. [19, 48] +2.2 +Equilibrium thermodynamics +In order to discuss the equilibrium thermodynamics of this black hole, we first compute the +black hole mass M as [46], +M = m +8 , +(2.17) +where m can be estimated by vanishing metric function f(r)|r=r+ = 0 as +m = −Λr2 + m2 +Gcc1r − 2b2r2 (β − 1) + q2 − 2q2 ln +� r +2ℓ (1 + β) +� +. +(2.18) +The Hawking temperature using surface gravity is calculated by +T = 1 +4π +� +m2 +Gcc1 − 2Λr+ − +4q2 +r+(1 + β+) +� +, +with +β+ = β|r=r+. +(2.19) +The equilibrium Hawking-Bekenstein entropy for the BI BTZ black holes in massive gravity +rainbow is calculated by +S0 = πr+ +2 . +(2.20) +– 5 – + +The electric potential Φ(r+) and conserved electric charge Q are calculated, respectively, by +Φ(r+) = −q ln +�r+ +2ℓ (1 + β+) +� +, +Q = q +2. +(2.21) +With the above calculated thermodynamical quantities at equilibrium, one can easily check the +validity of first-law of thermodynamics. +3 +Quantum corrected thermodynamics due to the LC entropy +Now, we focus to the correction in entropy as confirmed by microstate counting in string theory +as well as loop quantum gravity. The particular case of the LC entropy (1.2) is given by [22] +S(LC) = S0 + α ln S0, +(3.1) +where the expression of equilibrium area-law entropy S0 is given by equation (2.20), and α is the +correction coefficient [49]. This consideration modifies the thermodynamics for BI black holes +in massive gravity theory. From equation (1.2), the horizon radius can be expressed in terms of +LC entropy as +r+ = 2α +π W(η), +with +η = 1 +α exp +� +S(LC) +α +� +, +(3.2) +where W(x) is the well-known Lambert function satisfying the equation W(x)eW (x) = x [50]. +Now, plugging the value of r+ (3.2) into Eq. (2.17), we obtain +M(Q, S(LC)) = −αW(η) +4π2 +� +2ΛαW(η) − m2 +Gcc1π + 4b2αW(η) (βS(LC) − 1) +− 2π2Q2 +αW(η) +� +1 − 2 ln +�αW(η) +πℓ +(1 + βS(LC)) +��� +, +(3.3) +where the definition βS(LC) = +� +1 + π2Q2 +b2α2 W −2(η) is used. +The Logarithmic quantum corrected potential and temperature are calculated by +Φ(LC) = +�∂M +∂Q +� +S(LC) = Q +� +1 − 2 ln +�αW(η) +πℓ +(1 + βS(LC)) +�� +, +T (LC) = +� ∂M +∂S(LC) +� +Q += +r+ +� +m2 +Gcc1 − 2Λr+ − +4q2 +r+(1+β+) +� +4(πr+ + 2α) += +T +1 + α +S0 +. +(3.4) +The first-law of thermodynamics under the consideration of LC entropy is given by +dM(Q, S(LC), λ(LC)) = T (LC)dS(LC) + Φ(LC)dQ = TdS(LC) + Φ(LC)dQ − αdλ(LC), +(3.5) +where, +dλ(LC) = T +S0 +dS0 = T(r+)dr+ +r+ +. +(3.6) +– 6 – + +Indeed, λ(LC) can be a new thermodynamics parameter responsible for logarithmic correction +term. Here, coefficient α can play the role of its conjugate variable. Hence, the thermodynamic +relations (3.4) must be extended as +Φ(LC) = +�∂M +∂Q +� +S(LC),λ(LC) , +T (LC) = +� ∂M +∂S(LC) +� +Q,λ(LC) , +α = − +� ∂M +∂λ(LC) +� +Q,S(LC) . +(3.7) +Certainly, such relations will help to fix the value of correction coefficient for various horizon +radius. With the help of equations (2.19) and (3.6), the value of λ(LC) is simplified to +λ(LC) = 1 +4π +� +mG2cc1 ln r+ + 2 +� +2b2 − Λ +� +r+ − 4b +� +br+β+ − q ln +� 2 +r+ ++ 2b +q β+ +��� +. +(3.8) +On the other hand, Eq.(1.2) suggests that S(LC) is a function of S0 and dS(LC) = +� +1 + α +S0 +� +dS0. +This eventually leads to +dM(Q, S(LC), λ(LC)) = dM(Q, S0) = TdS0 + Φ(LC)dQ. +(3.9) +which is another form of the first-law of thermodynamics. +In order to study the effects of LC on the phase transition and stability of the BI massive +black holes, one requires the signature of specific heat. The specific heat of the BI black hole in +massive gravity can be calculated from the following definition: +C(LC) +Q += T +� +∂S(LC) +∂T +� +Q +, +(3.10) +This yields +C(LC) +Q += πβ+[m2 +Gcc1 − 4b2r+(β+ − 1) − 2Λr+] +4[2b2(β+ − 1) − Λβ+] +� +1 + 2α +πr+ +� +. +(3.11) +The vanishing numerator of above expression determines first-order phase transition point. How- +ever, vanishing numerator of specific heat shows the (divergent) second-order phase transition +point. The expression suggests that second-order phase transition occurs at the center of black +hole due to correction term. However, the first-order phase transition occurs at two points of +horizon radius. +This specific heat (3.11) can be expressed in terms of equilibrium specific heat C(0) +Q +as +C(LC) +Q += C(0) +Q +� +1 + α +S0 +� +, +(3.12) +– 7 – + +Figure 1. Specific heat C(LC) +Q +in terms of r+ for unit values of the model parameters. +where +C(0) +Q = πβ+[m2 +Gcc1 − 4b2r+(β+ − 1) − 2Λr+] +4[2b2(β+ − 1) − Λβ+] +(3.13) +is uncorrected specific heat corresponding to α = 0. +In Fig. 1 we can see typical behavior of C(LC) +Q +in terms of r+. Clearly, in the absence of loga- +rithmic correction there is only one point of second-order phase transition located at r+ = R. It +means that the black holes with horizon radius equal to R undergo second-order phase transition. +The black holes with horizon radii greater than R are locally stable while those with horizon +radii smaller than R are unstable. The logarithmic correction with positive α (i.e. α = 0.5 and +α = 1) has no any effect on the stability of BI BTZ black holes. When the logarithmic correc- +tion is considered with negative α (i.e. α = −0.5 and α = −1), in addition to the mentioned +second-order phase transition, there is a first-order phase transition too. If α = −0.5 is chosen, +the first-order phase transition occurs at r+ = R1 < R and it happens at r+ = R2 > R for +α = −1. In the case α = −0.5, the black holes with horizon radii smaller than R1 and greater +than R are stable wile in the case α = −1 those with horizon radii smaller than R and greater +than R2 remain stable. +The Helmholtz free energy is given by, +F (LC) = − +� +S(LC)dT = − +� +(S0 + α ln S0) dT = F (0) + αδF (LC), +(3.14) +where the equilibrium Helmholtz free energy is given by +F (0) = − +� +S0dT = 1 +8 +� +Λr2 ++ − 2q2 +� +2 ln +�β+ + 1 +β+ − 1 +� +− +1 +β+ + 1 +�� +, +(3.15) +– 8 – + +Figure 2. Helmholtz free energy F (LC) in terms of r+ for unit values of the model parameters. +and the correction term is given by +δF (LC) = − +� +ln S0 dT, += r+ +2π +� +Λ (ln r+ − 1) +� +Λ − 2b2� +− 2qb +r+ +ln +� 2 +r+ +� +1 + br+ +q β+ +�� +− 2b2β+ (ln r+ + 1) +� ++ T ln +� 2 +π +� +. +(3.16) +In Fig. +2, we can see typical behavior of F (LC) in terms of r+. +We can see that the +logarithmic correction, depending on the correction coefficient, may increase or decrease value +of the Helmholtz free energy. In the case of α = −1, Helmholtz free energy is completely positive +including a positive minimum (see dashed red line of Fig. 2). Solid orange line of Fig. 2 represent +the case of α = − 1 +2 [52]. +The part of the logarithmic corrected thermodynamics presented here is already presented by +the Ref. [53]. However here we succeed to determine the correction coefficient. +4 +Quantum corrected thermodynamics due to the EC entropy +In the main part of this paper we would like to consider the exponential correction on the black +hole entropy. As we know, the entropy of black hole gets exponential correction when microstate +counting is performed for quantum states residing on the horizon only [21]. Here, we consider +the entropy perturbation due to exponential term only. This is given by [54] +S(EC) = S0 + ωe−S0, +(4.1) +– 9 – + +where ω is the correction coefficient [55, 56]. This leads to +r+ = 2 +π +� +S(EC) − W(ξ) +� +, +with +ξ = ωe−S(EC), +(4.2) +where, W(ξ) is the well-known Lambert W function. +Now, inserting (4.2) into Eq. (2.17), we obtain mass in terms of EC entropy as +M(Q, S(EC)) = W(ξ) − S(EC) +2π2 +� +Λ +� +S(EC) − W(ξ) +� ++ 2b2 � +S(EC) − W(ξ) +� +(ωS(EC) − 1) +− π +2 m2 +Gcc1 − +π2Q2 +S(EC) − W(ξ) +� +1 − 2 ln +� +[S(EC) − W(ξ)] (1 + ωS(EC)) +ℓ +��� +,(4.3) +where ωS(EC) = +� +1 + π2Q2 +b2 [S(EC) − W(ξ)]−2. From the straightforward calculations, we obtain +the following expressions: +Φ(EC) = +�∂M +∂Q +� +S(EC) = Q +� +1 − 2 ln +� +[S(EC) − W(ξ)] (1 + ωS(EC)) +ℓ +�� +, +T (EC) = +� ∂M +∂S(EC) +� +Q += +T +1 − ωe−S0 . +(4.4) +By considering EC entropy, the first-law of thermodynamics for this black hole is given by +dM(Q, S(EC), λ(EC)) = T (EC)dS(EC) + Φ(EC)dQ = TdS(EC) + Φ(EC)dQ + ωdλ(EC), +(4.5) +where dλ(EC) has following expression: +dλ(EC) = Te−S0dS0. +(4.6) +Indeed, λ(EC) is introduced as a new thermodynamical variable responsible for exponential +correction term with conjugate variable ω. Hence, the thermodynamic relations (4.4) can be +extended as +Φ(EC) = +�∂M +∂Q +� +S(EC),λ(EC) , +T (EC) = +� ∂M +∂S(EC) +� +Q,λ(EC) , +ω = − +� ∂M +∂λ(EC) +� +Q,S(EC) . +(4.7) +This relation will be helpful in order to fix the value of EC parameter for various horizon radius. +Finally, Eq. (4.6) simplified to +λ(EC) = − 1 +4π +� +m2 +Gcc1 e−S0 + 4 +π +� +2b2 − Λ +� +(1 + S0) e−S0 + 8b2 +π +� +ωS0e−S0S0dS0 +� +. +(4.8) +– 10 – + +From Eq.(4.1), we note that S(EC) is a function of S0 and dS(EC) = +� +1 − ωe−S0� +dS0. This, +eventually, modifies the first-law of thermodynamics (4.5) as +dM(Q, S(EC), λ(EC)) = dM(Q, S0) = TdS0 + Φ(EC)dQ. +(4.9) +Starting from the following definition of the black hole specific heat +C(EC) +Q += T +� +∂S(EC) +∂T +� +Q +, += C(0) +Q +� +1 − ωe−S0� +, +(4.10) +where C(0) +Q , as the un-corrected specific heat corresponding to ω = 0, is given by the equation +(3.13). In Fig. 3 we have shown typical behavior of the specific heat for different values of ω by +letting mG = 2. In the absence of correction we see that a second-order phase transition may +happen at r+ = r and a first-order one at r+ = r1 > r. The BI black holes with the horizon radii +in the ranges r+ < r and r+ > r1 are stable. Note that the difference between α = 0 and ω = 0 +cases arson from different choices of mG, which reflects the impacts of massive gravitons. In the +presence of exponential correction with negative ω the situation remains unchanged. When ω is +taken positive (i.e. ω = 2 and ω = 3) stability of the black holes is affected by the exponential +quantum correction such that an additional first-order phase transition can occur. In the case +ω = 2 it appears at r+ = r0 < r and for ω = 3 at r+ = r′, with r < r′ < r1. The black holes +with ω = 2 are stable for horizon radii in the ranges r0 < r+ < r and r+ > r1, also those with +ω = 2 and horizon radii in the intervals r < r+ < r′ and r+ > r1 prefer thermal stability. +The Helmholtz free energy corresponding to EC entropy is derived as +F (EC) = − +� +S(EC)dT = − +� � +S0 + ωe−S0� +dT = F (0) + ωδF (EC), +(4.11) +in which F (0) has same form as Eq. (3.15) and +δF (EC) = − +� +e−S0dT = e−S0 +π2 +� +Λ + 2b2� ++ b2 +π +� e−S0 +β+ +dr+. +(4.12) +With this expression, one can check the effects of exponential correction on free energy. Assuming +small b, one can obtain, +F (EC) = r+ +8 + Q ln (r+) 3Q +b2r2 ++ ++ ωe− +πr+ +2 +π2 +� +2b2 + Λ − b3(πr+ + 2) +Qπ +� +. +(4.13) +Then, using the thermodynamics relation, +P = − +� +∂F (EC) +∂V +� +T +, +(4.14) +– 11 – + +Figure 3. Specific heat C(EC) +Q +in terms of r+ for mG = 2 and unit values of the model parameters. +one can obtain the pressure as the following expression, +P = +ωb2r3 ++ +� +(b2 + Λ +2 )Q − b3r+ +2 +� +e− +πr+ +2 +− πQ +� +(b2r2 ++ − 6)Q2 + +b2r3 ++ +8 +� +2πvQb2r4 ++ +, +(4.15) +where we assumed V ∝ r2 ++ (considered v as a proportionality constant). Assuming Q = 4b +π one +can rewrite the equation of state as the following form, +PV +T += 1 − B +V + f(T), +(4.16) +where B = − 6v +b2 , and f(T) is a temperature dependent function which we draw its typical +behavior in Fig. 4. +We can see that f(T) vanishes at higher temperature and higher temperature means in- +finitesimal radii where the exponential correction is dominant. +Therefore equation of state +(4.16) at small black hole phase reflects the leading order virial expansion. +5 +Conclusion +First of all, we have considered the three-dimensional massive gravity coupled with nonlinear +electrodynamics and calculated the relate equations of motion. By solving them, we obtained +a class of BI BTZ black hole solution. In order to study the space-time singularities, we have +estimated the Ricci and Kretschmann scalars which are singular at r = 0 and, take finite value +– 12 – + +Figure 4. Typical behavior of f(T ) in terms of T for unit values of the model parameters. +for finite r. This confirms that obtained black hole solution is not a regular solution as the +singularity is not due to coordinate one but it is an essential singularity. Asymptotically, this +solution behaves like AdS black holes. +Furthermore, we have discussed the thermodynamics of such system in equilibrium and, espe- +cially entropy, mass, Hawking temperature and electric potential have been calculated. It is +well-known that there exists some quantum corrections on the entropy of black holes [21]. For +instance, the logarithmic correction in entropy has been confirmed by micro-state counting in +string theory as well as loop quantum gravity [22]. However, the entropy of black hole gets ex- +ponential correction when micro-state counting is performed for quantum states residing on the +horizon only [21]. Here, we have first computed the quantum corrected thermodynamics for the +BI BTZ black hole in massive gravity due to the LC entropy. The effects of quantum correction +due to LC entropy on the stability is also studied. The calculated specific heat corresponding +to LC entropy suggests that in addition to a second-order phase transition there is first-order +phase transition which is due to consideration of LC entropy. However, the logarithmic cor- +rection has no effect on the second-order phase transition points. By this we mean that the +number and location of second-order phase transition points remain unchanged. We have also +found that logarithmic correction affects the Helmholtz free energy significantly for larger black +holes. The effects of exponential correction is also studied on the thermodynamics and stability +of BI massive BTZ black holes. 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Faizal, JHEP 2021 (2021) 1. +– 15 – + diff --git a/hNAzT4oBgHgl3EQfov3Z/content/tmp_files/load_file.txt b/hNAzT4oBgHgl3EQfov3Z/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a5768f8823c37aeedd48d686d424e581b12823c8 --- /dev/null +++ b/hNAzT4oBgHgl3EQfov3Z/content/tmp_files/load_file.txt @@ -0,0 +1,702 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf,len=701 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='01603v1 [gr-qc] 25 Dec 2022 Exponential corrected thermodynamics of Born-Infeld BTZ black holes in massive gravity B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Pourhassan,a,b,c M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Dehghani,d S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Upadhyay, e,f,a,∗ ˙I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Sakallı,g and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Singhh aSchool of Physics, Damghan University, Damghan, 3671641167, Iran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' bPhysics Department, Istanbul Technical University, Istanbul 34469, Turkey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' cCanadian Quantum Research Center 204-3002 32 Avenue Vernon, British Columbia V1T 2L7 Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' dDepartment of Physics, Razi University, Kermanshah, Iran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' eDepartment of Physics, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' College, Nawada, Bihar 805110, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' fDepartment of Physics, Magadh University, Bodh Gaya, Bihar 824234, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' gDepartment of Physics, Eastern Mediterranean University, Famagusta 99628, North Cyprus via Mersin 10, Turkey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' hDepartment of Physics, Institute of applied Science and Humanities, GLA University, Mathura 281406 Uttar Pradesh, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' E-mail: b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='pourhassan@du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='ir, m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='dehghani@razi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='ir, sudhakerupadhyay@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' sudhaker@associates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='iucaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='in, izzet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='sakalli@emu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='tr, veerdsingh@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='com Abstract: It is known that entropy of black hole gets correction at quantum level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Universally, these corrections are logarithmic and exponential in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' We analyze the impacts of these quantum corrections on thermodynamics of Born-Infeld BTZ black hole in massive gravity by considering both such kinds of correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' We do comparative analysis of corrected thermody- namics with their equilibrium values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Here, we find that the exponential correction yields to the second point of the first order phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Also, quantum correction effects significantly on the Helmholtz free energy of larger black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' We study the equation of state for the exponential corrected black hole to obtain a leading order virial expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Keywords: Keywords: Black hole;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Born-Infeld electrodynamics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Corrected entropy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Thermo- dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Visiting Associate at Inter-University Centre for Astronomy and Astrophysics (IUCAA), Pune, Maharashtra 411007, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Contents 1 Introduction 1 2 BI BTZ black holes in massive gravity 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='1 Gravitational field equations and singular solution 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='2 Equilibrium thermodynamics 5 3 Quantum corrected thermodynamics due to the LC entropy 6 4 Quantum corrected thermodynamics due to the EC entropy 9 5 Conclusion 12 1 Introduction Even though three dimensional gravity has no Newtonian limit, it has always been useful for conceptual issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' This was further supported by discovery of the three dimensional Banados– Teitelboim–Zanelli (BTZ) black hole solution [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' This discovery of three dimensional BTZ black holes really helped in recent developments in gravity, gauge and string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' The behavior of geodesics [5, 6] and the propagation of strings in a BTZ background [7] had already been discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' In contrast to Schwarzschild and Kerr solutions, the BTZ black hole is asymptotically anti-de Sitter and has no curvature singularity at center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Due to asymptotically AdS in nature of such BTZ black hole solution, it strengthen the idea of AdS/CFT correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' BTZ black holes are widely studies [4, 8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' The noncommutativity and scattering of massless planar scalar waves had been discussed in the context of BTZ black hole [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' On the other hand, there exist a lot of reasoning to modify the Einstein gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' The idea to include massive gravitons is one of them which explains the current acceleration of the universe without considering a cosmological constant [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' The solution of hierarchy problem hints the existence of massive modes [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Massive gravity gets relevance in astrophysics also as neutron stars can have mass more than 3M⊙ in massive gravity scenario [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' The massive graviton affects the gravitational waves also [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Various black hole solution have been studied in four- and higher-dimensional massive gravity [16–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' One can not ignore the possibility to explore BTZ black hole in massive gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' An AdS charged BTZ black hole in a massive gravity has been explored [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Here, thermodynamics and phase structure have also been discussed in both grand canonical and canonical ensembles [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' – 1 – Bekenstein, profoundly, interpreted the area of the event horizon of a black hole (A) as its entropy (S0) [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' This is given by S0 = A 4l2p , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='1) where lp is the Planck length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' This relationship more or less agree in the circumstances when black holes are much larger than the Planck scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' However, for relatively smaller black hole this relation needs correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' For such black holes, thermal fluctuations around equilibrium becomes significant and correct the black hole entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' There are several ways to obtain such corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Recently, it has been suggested that correction terms are universal and have the following approximate shape [21], S = S0 + α ln S0 + γ S0 + ωe−S0 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='2) where dots denote higher-order corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Here, we notice that at first-order entropy gets loga- rithmic correction as confirmed by microstate counting in string theory as well as loop quantum gravity [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' However, exponential corrected (EC) term occurs when microstate counting is done for quantum states on the horizon only [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' The phase transition and thermodynamics of BTZ black holes through the Landau-Lifshitz theory has been given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' The effects of entropy correction in the noncommutative BTZ black holes is presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Thermal fluctuations of charged black holes in massive gravity is presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' The effects of corrected entropy on thermodynamics of various black holes are studied [26–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Now, the main goal of this paper is to study the exponential correction of the Born-Infeld BTZ black holes in massive gravity thermodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' In GR and its extensions, nonlinear electrodynamics plays a crucial role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' For example, it produces many interesting geometries such as regular black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' The nonlinear electrodynamics is a direct generalization of the Maxwell electrodynamics originated by Born and Infeld (BI) in order to remove the central singularity of a point charge [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' BI terms also appears in superstrings scenario more naturally [33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Black holes with BI term are quite important in astrophysical observations [35–38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Bardeen was first who proposed regular black hole [39] and it was found that regular black hole with BI term only describes the black hole formation from an initial vacuum region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Albeit significant progress made on the subject, quantum effects on the thermodynamics of BI BTZ black hole in massive gravity are still unstudied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' This proves us an opportunity to shed light on this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' The BTZ black hole is a solution of the Einstein field equations in three dimensions that describes a black hole with negative cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' The Born- Infeld theory is a modified theory of electromagnetism that is characterized by a non-linear dependence of the electromagnetic field on the electric and magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' The combination of these two concepts leads to the concept of a quantum Born-Infeld BTZ black hole, which is a black hole that is characterized by both quantum and non-linear electromagnetic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' In the context of massive gravity, the thermodynamics of quantum Born-Infeld BTZ black holes can be studied by considering the effects of the mass term for the graviton on the properties of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' This can include the effects on the black hole temperature, entropy, and other thermodynamic quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' It is important to note that the thermodynamics of quantum Born- Infeld BTZ black holes in massive gravity is still a highly theoretical and complex topic that is not – 2 – yet fully understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Further research is needed to fully understand the thermodynamics of these objects and their potential implications for our understanding of gravity and the fundamental nature of spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' These are motivations behind present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' The paper is presented systematically in following manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' 2, we discuss the singular solution of BI massive gravity and their equilibrium thermodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' The quantum corrected thermodynamics of such black hole attributed by logarithmic corrected (LC) entropy is presented in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' The corrections in thermodynamics of this black hole attributed by EC entropy is discussed in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Finally, results of paper is summarized in the last section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' 2 BI BTZ black holes in massive gravity In this section, we shed light on the gravitational field equations and singular solution of the three-dimensional massive gravity in presence of nonlinear electrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' We also review its thermal properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='1 Gravitational field equations and singular solution In this section, we shed light on particular massive BTZ black hole solution [40] in presence of BI source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Let us begin by writing the action of three-dimensional massive gravity in presence of nonlinear electrodynamics [41, 42] I = 1 16π � √−g � R − 2Λ + m2 G 4 � i=1 ciUi(g, f) + L(F) � d3x, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='1) where Λ is the cosmological constant, mG is the constant parameter related to the graviton mass, coefficients ci are some constant, and f is fixed symmetric tensor known as reference metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Here, symmetric polynomials Ui are given by [43, 44] U1 = [K], U2 = [K]2 − [K2], U3 = [K]3 − 3[K][K2] + 2[K3], U4 = [K]4 − 6[K]2[K2] + 8[K3][K] + 3[K2]2 − 6[K4], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='2) where [K] ≡ Ka a is eigenvalues of the 3 × 3 matrix Ka b = √gacfcb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' However, L(F) refers to the Lagrangian of nonlinear electrodynamics expressed in terms of Maxwell’s invariant F = F abFab, where field-strength tensor of vector field Aa has the following form: Fab = ∂aAb − ∂bAa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Specifically, here, we are interested in following BI type Lagrangian of nonlinear electrodynamics [45]: L(F) = 4b2 � 1 − � 1 + F 2b2 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='3) where b is a constant parameter called as nonlinearity parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' In the large b limit (b → ∞), the Lagrangian (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='3) corresponds to the Maxwell’s classical electrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' – 3 – For the given action (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='1), the field equations corresponding to gravitational field gab and vector field Ab are given, respectively, by [43, 44] Rab − 1 2Rgab + Λgab + m2 Gχab = 1 2L(F)gab − 2dL(F) dF FacF c b , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='4) ∂a �√−g dL(F) dF F ab � = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='5) where tensor χab related to massive graviton has the following expression [43]: χab = − c1 2 (U1gab − Kab) − c2 2 � U2gab − 2U1Kab + 2K2 ab � − c3 2 � U3gab − 3U2Kab + 6U1K2 ab − 6K3 ab � − c4 2 � U4gab − 4U3Kab + 12U2K2 ab − 24U1K3 ab + 24K4 ab � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='6) In order to have the spherically symmetric solution for the field equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='4) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='5), we make an ansatz for the line element [46, 47]: ds2 = −f(r)dt2 + dr2 f(r) + r2dθ2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='7) where the specific form of metric function f(r) will be determined in the presence of nonlinear (BI) electrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' For the following reference metric fab = diag(0, 0, c2), the polynomials introduced by equa- tion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='2) are simplified to [19, 43], U1 = c r, and U2 = U3 = U4 = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='8) where c is a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' For, these values of polynomials, the components of tensor χab (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='6) identified to χtt = −cc1 2r gtt, χrr = cc1 2r grr, χθθ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='9) Noting the facts that non-vanishing components of field-strength tensor are Ftr = −Frt = −∂rAt(r) and, thus, Maxwell’s invariant can be expressed as F = −2F 2 tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Corresponding to relations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='7), the electromagnetic field equation leads to Ftr(r) = q rβ , and At(r) = −q ln � r 2ℓ (1 + β) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='10) Here q refers to an integration constant which is relevant for the black hole charge and β = � 1 + q2 b2r2 [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Corresponding to the metric (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='7), the components of gravitational field equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='4) take the following differential forms: Gtt = Grr = f ′(r) r + 2Λ + 4b2 (β − 1) − cc1m2 G r = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='11) – 4 – Gθθ = f ′′(r) + 2Λ + 4b2 � β−1 − 1 � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='12) Here, we observe that the components of the gravitational field equations hold the following relation: � r d dr + 1 � Gtt = Gθθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='13) Since the components of field equations are interrelated, we need to solve only one (preferably first-order one) of equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' This leads to the solution for metric function as f(r) = −m − Λr2 + m2 Gcc1r − 2b2r2 (β − 1) + q2 − 2q2 ln � r 2ℓ (1 + β) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='14) where mass parameter m is a constant of integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' This is worth to calculate Ricci scalar and Kretschmann scalar as they play significant role in the study of space-time singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Thus, the Ricci scalar and the Kretschmann scalar for this theory of gravity are calculated, respectively, by R = 6Λ + 4b2 �� β−1 − 1 � + 2 (β − 1) � − 2m2 Gcc1 r , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='15) RµνρλRµνρλ = 12Λ2 + 16b4 �� β−1 − 1 �2 + 2 (β − 1)2� − 8b2m2 Gcc1 r (β − 1) + 16Λb2 �� β−1 − 1 � + 2 (β − 1) � − 8Λm2 Gcc1 r + 2(m2 Gcc1)2 r2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='16) It is obvious from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='15) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='15) that the Ricci scalar and Kretschmann scalar take finite value for finite r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' The solution described by metric function (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='14) is not a regular solution as the singularity is not a coordinate one but a essential singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' This solution behaves like AdS black holes asymptotically [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Above solutions are already presented by Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' [19, 48] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='2 Equilibrium thermodynamics In order to discuss the equilibrium thermodynamics of this black hole, we first compute the black hole mass M as [46], M = m 8 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='17) where m can be estimated by vanishing metric function f(r)|r=r+ = 0 as m = −Λr2 + m2 Gcc1r − 2b2r2 (β − 1) + q2 − 2q2 ln � r 2ℓ (1 + β) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='18) The Hawking temperature using surface gravity is calculated by T = 1 4π � m2 Gcc1 − 2Λr+ − 4q2 r+(1 + β+) � , with β+ = β|r=r+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='19) The equilibrium Hawking-Bekenstein entropy for the BI BTZ black holes in massive gravity rainbow is calculated by S0 = πr+ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='20) – 5 – The electric potential Φ(r+) and conserved electric charge Q are calculated, respectively, by Φ(r+) = −q ln �r+ 2ℓ (1 + β+) � , Q = q 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='21) With the above calculated thermodynamical quantities at equilibrium, one can easily check the validity of first-law of thermodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' 3 Quantum corrected thermodynamics due to the LC entropy Now, we focus to the correction in entropy as confirmed by microstate counting in string theory as well as loop quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' The particular case of the LC entropy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='2) is given by [22] S(LC) = S0 + α ln S0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='1) where the expression of equilibrium area-law entropy S0 is given by equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='20), and α is the correction coefficient [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' This consideration modifies the thermodynamics for BI black holes in massive gravity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' From equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='2), the horizon radius can be expressed in terms of LC entropy as r+ = 2α π W(η), with η = 1 α exp � S(LC) α � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='2) where W(x) is the well-known Lambert function satisfying the equation W(x)eW (x) = x [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Now, plugging the value of r+ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='2) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='17), we obtain M(Q, S(LC)) = −αW(η) 4π2 � 2ΛαW(η) − m2 Gcc1π + 4b2αW(η) (βS(LC) − 1) − 2π2Q2 αW(η) � 1 − 2 ln �αW(η) πℓ (1 + βS(LC)) ��� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='3) where the definition βS(LC) = � 1 + π2Q2 b2α2 W −2(η) is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' The Logarithmic quantum corrected potential and temperature are calculated by Φ(LC) = �∂M ∂Q � S(LC) = Q � 1 − 2 ln �αW(η) πℓ (1 + βS(LC)) �� , T (LC) = � ∂M ∂S(LC) � Q = r+ � m2 Gcc1 − 2Λr+ − 4q2 r+(1+β+) � 4(πr+ + 2α) = T 1 + α S0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='4) The first-law of thermodynamics under the consideration of LC entropy is given by dM(Q, S(LC), λ(LC)) = T (LC)dS(LC) + Φ(LC)dQ = TdS(LC) + Φ(LC)dQ − αdλ(LC), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='5) where, dλ(LC) = T S0 dS0 = T(r+)dr+ r+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='6) – 6 – Indeed, λ(LC) can be a new thermodynamics parameter responsible for logarithmic correction term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Here, coefficient α can play the role of its conjugate variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Hence, the thermodynamic relations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='4) must be extended as Φ(LC) = �∂M ∂Q � S(LC),λ(LC) , T (LC) = � ∂M ∂S(LC) � Q,λ(LC) , α = − � ∂M ∂λ(LC) � Q,S(LC) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='7) Certainly, such relations will help to fix the value of correction coefficient for various horizon radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' With the help of equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='19) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='6), the value of λ(LC) is simplified to λ(LC) = 1 4π � mG2cc1 ln r+ + 2 � 2b2 − Λ � r+ − 4b � br+β+ − q ln � 2 r+ + 2b q β+ ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='8) On the other hand, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='2) suggests that S(LC) is a function of S0 and dS(LC) = � 1 + α S0 � dS0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' This eventually leads to dM(Q, S(LC), λ(LC)) = dM(Q, S0) = TdS0 + Φ(LC)dQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='9) which is another form of the first-law of thermodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' In order to study the effects of LC on the phase transition and stability of the BI massive black holes, one requires the signature of specific heat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' The specific heat of the BI black hole in massive gravity can be calculated from the following definition: C(LC) Q = T � ∂S(LC) ∂T � Q , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='10) This yields C(LC) Q = πβ+[m2 Gcc1 − 4b2r+(β+ − 1) − 2Λr+] 4[2b2(β+ − 1) − Λβ+] � 1 + 2α πr+ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='11) The vanishing numerator of above expression determines first-order phase transition point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' How- ever, vanishing numerator of specific heat shows the (divergent) second-order phase transition point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' The expression suggests that second-order phase transition occurs at the center of black hole due to correction term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' However, the first-order phase transition occurs at two points of horizon radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' This specific heat (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='11) can be expressed in terms of equilibrium specific heat C(0) Q as C(LC) Q = C(0) Q � 1 + α S0 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='12) – 7 – Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Specific heat C(LC) Q in terms of r+ for unit values of the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' where C(0) Q = πβ+[m2 Gcc1 − 4b2r+(β+ − 1) − 2Λr+] 4[2b2(β+ − 1) − Λβ+] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='13) is uncorrected specific heat corresponding to α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' 1 we can see typical behavior of C(LC) Q in terms of r+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Clearly, in the absence of loga- rithmic correction there is only one point of second-order phase transition located at r+ = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' It means that the black holes with horizon radius equal to R undergo second-order phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' The black holes with horizon radii greater than R are locally stable while those with horizon radii smaller than R are unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' The logarithmic correction with positive α (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='5 and α = 1) has no any effect on the stability of BI BTZ black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' When the logarithmic correc- tion is considered with negative α (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' α = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='5 and α = −1), in addition to the mentioned second-order phase transition, there is a first-order phase transition too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' If α = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='5 is chosen, the first-order phase transition occurs at r+ = R1 < R and it happens at r+ = R2 > R for α = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' In the case α = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='5, the black holes with horizon radii smaller than R1 and greater than R are stable wile in the case α = −1 those with horizon radii smaller than R and greater than R2 remain stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' The Helmholtz free energy is given by, F (LC) = − � S(LC)dT = − � (S0 + α ln S0) dT = F (0) + αδF (LC), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='14) where the equilibrium Helmholtz free energy is given by F (0) = − � S0dT = 1 8 � Λr2 + − 2q2 � 2 ln �β+ + 1 β+ − 1 � − 1 β+ + 1 �� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='15) – 8 – Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Helmholtz free energy F (LC) in terms of r+ for unit values of the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' and the correction term is given by δF (LC) = − � ln S0 dT, = r+ 2π � Λ (ln r+ − 1) � Λ − 2b2� − 2qb r+ ln � 2 r+ � 1 + br+ q β+ �� − 2b2β+ (ln r+ + 1) � + T ln � 2 π � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='16) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' 2, we can see typical behavior of F (LC) in terms of r+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' We can see that the logarithmic correction, depending on the correction coefficient, may increase or decrease value of the Helmholtz free energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' In the case of α = −1, Helmholtz free energy is completely positive including a positive minimum (see dashed red line of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Solid orange line of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' 2 represent the case of α = − 1 2 [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' The part of the logarithmic corrected thermodynamics presented here is already presented by the Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' However here we succeed to determine the correction coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' 4 Quantum corrected thermodynamics due to the EC entropy In the main part of this paper we would like to consider the exponential correction on the black hole entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' As we know, the entropy of black hole gets exponential correction when microstate counting is performed for quantum states residing on the horizon only [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Here, we consider the entropy perturbation due to exponential term only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' This is given by [54] S(EC) = S0 + ωe−S0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='1) – 9 – where ω is the correction coefficient [55, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' This leads to r+ = 2 π � S(EC) − W(ξ) � , with ξ = ωe−S(EC), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='2) where, W(ξ) is the well-known Lambert W function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Now, inserting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='2) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='17), we obtain mass in terms of EC entropy as M(Q, S(EC)) = W(ξ) − S(EC) 2π2 � Λ � S(EC) − W(ξ) � + 2b2 � S(EC) − W(ξ) � (ωS(EC) − 1) − π 2 m2 Gcc1 − π2Q2 S(EC) − W(ξ) � 1 − 2 ln � [S(EC) − W(ξ)] (1 + ωS(EC)) ℓ ��� ,(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='3) where ωS(EC) = � 1 + π2Q2 b2 [S(EC) − W(ξ)]−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' From the straightforward calculations, we obtain the following expressions: Φ(EC) = �∂M ∂Q � S(EC) = Q � 1 − 2 ln � [S(EC) − W(ξ)] (1 + ωS(EC)) ℓ �� , T (EC) = � ∂M ∂S(EC) � Q = T 1 − ωe−S0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='4) By considering EC entropy, the first-law of thermodynamics for this black hole is given by dM(Q, S(EC), λ(EC)) = T (EC)dS(EC) + Φ(EC)dQ = TdS(EC) + Φ(EC)dQ + ωdλ(EC), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='5) where dλ(EC) has following expression: dλ(EC) = Te−S0dS0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='6) Indeed, λ(EC) is introduced as a new thermodynamical variable responsible for exponential correction term with conjugate variable ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Hence, the thermodynamic relations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='4) can be extended as Φ(EC) = �∂M ∂Q � S(EC),λ(EC) , T (EC) = � ∂M ∂S(EC) � Q,λ(EC) , ω = − � ∂M ∂λ(EC) � Q,S(EC) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='7) This relation will be helpful in order to fix the value of EC parameter for various horizon radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Finally, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='6) simplified to λ(EC) = − 1 4π � m2 Gcc1 e−S0 + 4 π � 2b2 − Λ � (1 + S0) e−S0 + 8b2 π � ωS0e−S0S0dS0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='8) – 10 – From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='1), we note that S(EC) is a function of S0 and dS(EC) = � 1 − ωe−S0� dS0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' This, eventually, modifies the first-law of thermodynamics (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='5) as dM(Q, S(EC), λ(EC)) = dM(Q, S0) = TdS0 + Φ(EC)dQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='9) Starting from the following definition of the black hole specific heat C(EC) Q = T � ∂S(EC) ∂T � Q , = C(0) Q � 1 − ωe−S0� , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='10) where C(0) Q , as the un-corrected specific heat corresponding to ω = 0, is given by the equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' 3 we have shown typical behavior of the specific heat for different values of ω by letting mG = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' In the absence of correction we see that a second-order phase transition may happen at r+ = r and a first-order one at r+ = r1 > r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' The BI black holes with the horizon radii in the ranges r+ < r and r+ > r1 are stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Note that the difference between α = 0 and ω = 0 cases arson from different choices of mG, which reflects the impacts of massive gravitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' In the presence of exponential correction with negative ω the situation remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' When ω is taken positive (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' ω = 2 and ω = 3) stability of the black holes is affected by the exponential quantum correction such that an additional first-order phase transition can occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' In the case ω = 2 it appears at r+ = r0 < r and for ω = 3 at r+ = r′, with r < r′ < r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' The black holes with ω = 2 are stable for horizon radii in the ranges r0 < r+ < r and r+ > r1, also those with ω = 2 and horizon radii in the intervals r < r+ < r′ and r+ > r1 prefer thermal stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' The Helmholtz free energy corresponding to EC entropy is derived as F (EC) = − � S(EC)dT = − � � S0 + ωe−S0� dT = F (0) + ωδF (EC), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='11) in which F (0) has same form as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='15) and δF (EC) = − � e−S0dT = e−S0 π2 � Λ + 2b2� + b2 π � e−S0 β+ dr+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='12) With this expression, one can check the effects of exponential correction on free energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Assuming small b, one can obtain, F (EC) = r+ 8 + Q ln (r+) 3Q b2r2 + + ωe− πr+ 2 π2 � 2b2 + Λ − b3(πr+ + 2) Qπ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='13) Then, using the thermodynamics relation, P = − � ∂F (EC) ∂V � T , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='14) – 11 – Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Specific heat C(EC) Q in terms of r+ for mG = 2 and unit values of the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' one can obtain the pressure as the following expression, P = ωb2r3 + � (b2 + Λ 2 )Q − b3r+ 2 � e− πr+ 2 − πQ � (b2r2 + − 6)Q2 + b2r3 + 8 � 2πvQb2r4 + , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='15) where we assumed V ∝ r2 + (considered v as a proportionality constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Assuming Q = 4b π one can rewrite the equation of state as the following form, PV T = 1 − B V + f(T), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='16) where B = − 6v b2 , and f(T) is a temperature dependent function which we draw its typical behavior in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' We can see that f(T) vanishes at higher temperature and higher temperature means in- finitesimal radii where the exponential correction is dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Therefore equation of state (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content='16) at small black hole phase reflects the leading order virial expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' 5 Conclusion First of all, we have considered the three-dimensional massive gravity coupled with nonlinear electrodynamics and calculated the relate equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' By solving them, we obtained a class of BI BTZ black hole solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' In order to study the space-time singularities, we have estimated the Ricci and Kretschmann scalars which are singular at r = 0 and, take finite value – 12 – Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Typical behavior of f(T ) in terms of T for unit values of the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' for finite r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' This confirms that obtained black hole solution is not a regular solution as the singularity is not due to coordinate one but it is an essential singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Asymptotically, this solution behaves like AdS black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Furthermore, we have discussed the thermodynamics of such system in equilibrium and, espe- cially entropy, mass, Hawking temperature and electric potential have been calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' It is well-known that there exists some quantum corrections on the entropy of black holes [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' For instance, the logarithmic correction in entropy has been confirmed by micro-state counting in string theory as well as loop quantum gravity [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' However, the entropy of black hole gets ex- ponential correction when micro-state counting is performed for quantum states residing on the horizon only [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Here, we have first computed the quantum corrected thermodynamics for the BI BTZ black hole in massive gravity due to the LC entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' The effects of quantum correction due to LC entropy on the stability is also studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' The calculated specific heat corresponding to LC entropy suggests that in addition to a second-order phase transition there is first-order phase transition which is due to consideration of LC entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' However, the logarithmic cor- rection has no effect on the second-order phase transition points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' By this we mean that the number and location of second-order phase transition points remain unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' We have also found that logarithmic correction affects the Helmholtz free energy significantly for larger black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' The effects of exponential correction is also studied on the thermodynamics and stability of BI massive BTZ black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' We have shown that the BI black holes in the presence of mas- sive gravitons exhibit one second-order and one first-order phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' So the exponential correction may yield to a second first-order phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Moreover, stability properties of black holes is affected by the mass of graviton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' This fact can be seen through comparison of un-corrected specific heats as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' 1 and 3 for different values of mG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Also, the – 13 – exponential corrected thermodynamics of BI black hole in massive gravity yields to the leading order virial expansion which is not already reported by the ordinary BI black hole [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' References [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Khan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Upadhyay, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Ganai, Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' A 36 (2021) 2130023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' [27] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Pourhassan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Upadhyay, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Plus 136 (2021) 311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' [28] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Upadhyay, Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' 50 (2018) 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' [29] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Upadhyay, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' B 775 (2017) 130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' [30] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Upadhyay, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Pourhassan, Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' 013B03 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' – 14 – [31] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Khan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' A.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Infeld, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Lond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' 144 (1934) 425.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' [33] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Fradkin and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Tseytlin, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' B 163 (1985) 123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' [34] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} 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104032.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' [36] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Jusufi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Ov¨gun, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Banerjee and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Sakallı, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} 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–' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfov3Z/content/2301.01603v1.pdf'} diff --git a/idE2T4oBgHgl3EQfHwb9/content/tmp_files/2301.03673v1.pdf.txt b/idE2T4oBgHgl3EQfHwb9/content/tmp_files/2301.03673v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9896c6d4f401b686d1e78b6bef8007409f2a7306 --- /dev/null +++ b/idE2T4oBgHgl3EQfHwb9/content/tmp_files/2301.03673v1.pdf.txt @@ -0,0 +1,2596 @@ +Prototype Global Analysis of LISA Data with Multiple Source Types +Tyson B. Littenberg +NASA Marshall Space Flight Center, Huntsville, Alabama 35811, USA +Neil J. Cornish +eXtreme Gravity Institute, Department of Physics, +Montana State University, Bozeman, Montana 59717, USA +(Dated: January 11, 2023) +The novel data analysis challenges posed by the Laser Interferometer Space Antenna (LISA) +arise from the overwhelmingly large number of astrophysical sources in the measurement band and +the density with which they are found in the data. Robust detection and characterization of the +numerous gravitational wave sources in LISA data can not be done sequentially, but rather through +a simultaneous global fit of a data model containing the full suite of astrophysical and instrumental +features present in the data. While previous analyses have focused on individual source types in +isolation, here we present the first demonstration of a LISA global fit analysis containing combined +astrophysical populations. The prototype pipeline uses a blocked Metropolis Hastings algorithm to +alternatingly fit to a population of ultra compact galactic binaries, known “verification binaries” +already identified by electromagnetic observations, a population of massive black hole mergers, +and an instrument noise model. The Global LISA Analysis Software Suite (GLASS) is assembled +from independently developed samplers for the different model components. The modular design +enables flexibility to future development by defining standard interfaces for adding new, or updating +additional, components to the global fit without being overly prescriptive for how those modules +must be internally designed. The GLASS pipeline is demonstrated on data simulated for the LISA +Data Challenge 2b. Results of the analysis and a road-map for continued development are described +in detail. +I. +INTRODUCTION +The mHz band of the gravitational wave spectrum +is expected to contain an unprecedented abundance of +galactic, extra-galactic, and cosmological gravitational +wave (GW) sources. The Laser Interferometer Space An- +tenna (LISA) will survey the mHz GW band and provide +unique observational constraints on the formation and +evolution of compact binaries in the Milky Way, the ori- +gin and growth of massive black holes throughout cosmic +history, the dynamics of dense stellar environments in +galactic nuclei, the fundamental nature of gravity and +black holes, and more [1]. +The richness of the LISA +source catalog comes at the price of a more complicated +analysis framework than is required for currently operat- +ing GW observatories. While aspects of the methodology +developed for ground-based interferometers (many dis- +crete sources) [2] and pulsar timing (overlapping sources, +sophisticated noise modeling) [3] under-gird development +of LISA analysis pipelines, new strategies are needed to +account for the overwhelming number and density of GW +signals in the LISA data. +The fundamental challenge of LISA analysis stems +from the large number (O(104)) and long duration +(months to years) of detectable signals, resulting in non- +negligible overlaps in time and frequency between dis- +crete sources. As a result, analyses cannot treat sources +independently and sequentially work through a list of +candidate detections. Instead, the LISA analysis has to +be approached globally, simultaneously fitting complete +data models including all of the detectable GW sources +and the detector noise. The need for a “Global Fit” was +first described in 2005 [4], and has been identified as the +primary challenge to the LISA analysis since early in the +mission formulation [5]. This has lead to a coordinated +effort to develop capable algorithms well in advance of +mission operations [6]. +Global fit analyses are not unique to LISA, as there +are analogous methods used elsewhere in GW astron- +omy, and more broadly within astronomy and astro- +physics (e.g. +Gaia [7]). +For LIGO-Virgo analysis, +the BayesWave pipeline simultaneously models Gaussian +noise, non-Gaussian noise artifacts, and short-duration +GW transients [8]. PTA analyses use a global fit to si- +multaneously model a correlated, stochastic gravitational +wave background, a solar system ephemeris model, and +multiple noise sources for each pulsar in the array [9, 10]. +Some PTA analyses also perform a global fit for mul- +tiple source types, such as the signals from individual +black hole binaries and a stochastic confusion noise from +unresolved binaries [11], or perform a BayesWave-style +analysis to reconstruct un-modeled burst signals [12]. +Where the analogy ends is the scale of the LISA prob- +lem compared to elsewhere in GW astronomy, evident in +the number of sources that are part of the global anal- +ysis, the diversity of source types (SMBHBs, EMRIs, +UCBs, SGWBs, SOBHs, etc.), and data complications +from multi-year integration times (glitches, nonstation- +ary noise, gaps, etc.). +In this paper we present the first demonstration of a +LISA global fit analysis contending with multiple source +types. +We call the algorithm the Global LISA Anal- +ysis Software Suite, or GLASS. As the proving ground +arXiv:2301.03673v1 [gr-qc] 9 Jan 2023 + +2 +for GLASS we use the simulated data released in the +second round of the LISA Data Challenges (Challenge +LDC2a-v2) [13]. +The simulated data contain Gaussian +detector noise; a simulated population of Milky Way +ultra compact binaries (UCBs); 35 galactic UCBs al- +ready discovered by electromagnetic observations, the so- +called “verification binaries” (VGBs); and a population +of merging massive black hole binaries (MBHBs). +The philosophy of GLASS is to incorporate indepen- +dently developed, stochastic sampling algorithms de- +signed to address different facets of the full LISA anal- +ysis. GLASS is then effectively an overarching umbrella +that manages the interfaces, matches data formats, and +orchestrates how the different samplers will work together +to converge and adequately cover the target, high dimen- +sional, joint posterior distribution function. +GLASS uses a four-component model to fit the simu- +lated LDC2a-v2 data. The UCB, VGB, and MBHB mod- +els each employ analytic template waveforms to fit the +detectable sources, while the noise level, including the +unresolved astrophysical foreground from the galactic bi- +naries, is fit with a phenomenological model. The param- +eters of the four model components are optimized using +a blocked Metropolis Hastings (MH) algorithm. The test +data set spans one year of simulated LISA observations +however our analysis processes the data sequentially, an- +alyzing increasingly large strides of the “observed” time +series as we envision would be done during mission oper- +ations. Results from the analysis of the LDC2a-v2 data +are compared to the input populations to assess the per- +formance of the algorithm. +Figure 1 summarizes the LDC2a-v2 results by show- +ing the combined reconstructed model components rep- +resented as the amplitude spectral density of the time +delay interferometry (TDI) A channel [14]. +The orig- +inal data are shown in gray in the background of the +figure. +The purple lines are the posteriors of the re- +constructed UCB waveforms while orange are the recon- +structed VGBs. The magenta broadband curves are the +reconstructed massive black hole signals, and the light +blue curve is the noise model. Note that at low frequency +the credible 50 and 90% credible intervals are visible in +the black hole signals while otherwise the credible inter- +vals on the reconstructions are too narrow to be visible +in this plot. This figure represents the key result of this +study: We are demonstrating a prototype pipeline able to +simultaneously fit thousands of overlapping signals of dif- +ferent morphology and an a priori unknown noise level. +The remainder of this paper will provide a detailed de- +scription of the algorithm and a demonstration of the per- +formance. Section II describes the analysis architecture, +and how the different modules are integrated together +into a global fit pipeline. +Section III A describes the +noise model and sampling algorithm, adapted from the +BayesLine algorithm used for LIGO-Virgo noise model- +ing [15]. Section III B summarizes updates to the galac- +tic binary sampler GBMCMC first described in Ref [16], +FIG. 1: Median reconstructed global fit model components +for the full 12 month LDC2a-v2 data, shown as the ASD in +the TDI A channel. Gray is the residual, purple are the UCB +detections, orange or the fits to the known binaries, magenta +are the MBHB mergers, and light blue is the noise model. +while Section III C specifies the configuration changes for +GBMCMC to perform targeted analysis of known binaries. +Section III D describes how the MBHBMCMC massive black +hole sampler from Ref [17] was adapted for this work. +Section IV presents the evolving results from 1.5, 3, 6, +and 12 month analyses of the LDC2a-v2 data before we +conclude in Section V with a development road map to +improve GLASS’s capabilities in order to analyze increas- +ingly realistic LISA data. +II. +THE GLASS ARCHITECTURE +The central engine of GLASS is a blocked Markov Chain +Monte Carlo sampler [18]. +In the blocked sampling +scheme, a subset of the model parameters (a block) is +updated while holding all other parameters fixed. Differ- +ent blocks are updated independently in sequence, and +the process cycles until the sampler has converged. For +the LISA Global Fit problem, blocked MH samplers have +two advantages. First, they work well for high dimension +spaces when parameter correlations are confined to rela- +tively small and a priori identified sub spaces. Second, +they are naturally modular, turning the daunting task of +building an algorithm equal to the complexity of LISA +data into a well defined set of components that are de- +veloped independently and then integrated. +The blocked MH scheme in GLASS is hierarchical where +the top level blocks, which we will refer to as “modules,” +are the joint set of parameters for the different model +components i.e., blocks for the noise, VGB, UCB, and +MBHB parameter sets. The sampling within the VGB +and MBHB modules is further grouped into blocks by +individual sources, while the UCB module has one more +layer of hierarchy–where model parameters are grouped +by narrow-band frequency segments, and then by indi- + +10-18 +10-19 +1020 + 10-21 +10-22, +10-23 +10~24, +10~4 +10~3 +10~2 +f (a)3 +vidual sources within each segment. +Each module uses a customized parallel tempered +Markov Chain Monte Carlo sampler developed indepen- +dently of one another. The role of GLASS is to coordinate +which blocks are updating and to exchange state data be- +tween modules. The modules work on their own subset +of the data, use their own likelihood function, tempering +scheme, proposals, etc., and in principle could even use +a different representation of the data (e.g., time series, +frequency series, or wavelets). In the current implemen- +tation of GLASS each module is working in the frequency +domain. +Figure 2 is a schematic diagram for how modules op- +erate on different bandwidths of the data. Note that the +colors indicating each module will be consistent through- +out this paper. The noise model is fit over the full fre- +quency measurement band of the data (light blue). For +the LDC2a-v2 data we take that to range from ∼10−5 +to ∼30 mHz. Massive black hole mergers are broadband +signals where the maximum frequency is determined by +the total mass of the binary. The MBHB module band- +width is dynamically based on the source parameters, +but generally extends up to a few to O(10) mHz (ma- +genta). +The UCBs are narrow-band signals, generally +spanning ≲ 10 µHz, but are by far the most numerous +of the LISA sources, and will be found throughout the +measurement band of our analysis, though sparsely above +10 mHz. The UCB module consists of several instances +of the same sampler, each focusing on a band-limited +segment of data (purple). The bandwidth of each seg- +ment depends on the frequency, using smaller segments +where sources are most densely spaced. Finally the VGB +module is conducting a narrow-band targeted analysis for +individually known sources (orange). +Known Binaries +Ultra-compact Binaries +Massive Black Hole Mergers +Noise +f +... +... +FIG. 2: Schematic block diagram for how frequency domain +data are segmented by the different model components. The +noise module must cover the full frequency range (light +blue). The MBHB module is broadband, covering almost the +same width as the noise model (magenta). The UCB module +divides the data into narrow-band, overlapping, segments +(purple), while the VGB model targets only the frequency +range spanned by each individual known binary (orange). +To understand the interface between modules consult +the joint likelihood function for the global fit: +p(d|θ) = (2π)− N +2 det (C(θnoise))− 1 +2 e− 1 +2 (d−h(θMBHB)−h(θUCB)−h(θVGB))T C(θnoise)−1(d−h(θMBHB)−h(θUCB)−h(θVGB)) +(1) +where d is the data, N is the number of data points, +C is the noise covariance matrix, θ represents the full +parameter set, θi are the model parameters in the block +for module i, and h are the co-added detector responses +to the modeled sources in each module. +For example, +h(θMBHB) is really shorthand for � +n h(θn +MBHB) where n +is indexing all of the sources in the MBHB model. +Sticking +with +the +MBHB +example, +the +sampler +adopted for that model was developed assuming no other +sources in the data, and a known noise covariance ma- +trix. The internal likelihood for the kth component of the +MBHB module is then just +p(d|θk +MBH) = e− 1 +2( ¯d−h(θk +MBHB)) +T C−1( ¯d−h(θk +MBHB)) +(2) +where the normalization term is absent because the +sampler only considers the likelihood ratio between two +points in parameter space and the covariance matrix is in- +dependent of the (MBHB) parameters. To interface this +sampler with the rest of GLASS at the beginning of each +one of the MBHB blocks’ updates the covariance matrix +is replaced based on the current state of the noise model +θnoise and the “data” as seen by the MBHB sampler is the +residual after subtracting all other model components– +as far as the MBHB sampler for the kth MBHB is con- +cerned ¯d = d−h(θUCB)−h(θVGB)−� +n̸=k h(θn +MBHB). The +MBHB sampler is otherwise blissfully unaware of what +is happening in the rest of the global fit. It is the job of +GLASS to keep track of the current state of each module, +prepare the effective data and noise covariance matrix, +and refresh the likelihood (using the new effective data +and noise model) of the current state before a sampler +updates it’s block of parameters. An identical argument +applies to the other modules. +The individual samplers for the modules have been de- +veloped and published elsewhere and will be briefly sum- +marized in later sections before focusing on updates made +to each of them for the LISA global analysis. To under- +stand how the data is shared between modules a block +diagram of the workflow is shown in Fig. 3. +The dia- +gram is a simplified version of the true workflow, depict- +ing only three UCB and MBHB nodes each. +In prac- +tice GLASS uses several hundred UCB nodes and one +MBHB node per source in the model. The noise, VGB, +UCB, and MBHB model updates are executed by the +BayesLine, VBMCMC, GBMCMC, and MBHBMCMC blocks, re- +spectively. +GLASS uses the Message Passing Interface +(MPI) standard to exchange information between the dif- +ferent modules. For shorthand we will refer each MPI + +4 +process as a “node” of the analysis, though in practice +we use multiple MPI processes per node. +The BayesLine module uses a single node which also +serves as the root process responsible for the work shared +by all nodes and the overall orchestration of the analysis +(P0 in the flow chart). At start up the root node handles +data parsing, selection, and conditioning before broad- +casting the data and the initial state of each sampler to +all other worker nodes. During each iteration of the sam- +pling nodes P0 and P1 first update their parameter blocks +for the noise model and VGB model, respectively. At the +end of the noise and VGB module updates (each involv- +ing several internal MCMC steps for each model), the +VGB process sends the current state of the VGB model +in the frequency domain to the root process. The root +process then broadcasts the current state of the noise +model and VGB model to the UCB (P2-P4) and MBHB +(P5-P7) nodes. The UCB and MBHB processes then cre- +ate their respective residuals and update their block of +parameters. +Each UCB process is responsible for a narrow-band +segment of the data but care must be taken at the seg- +ment boundaries where individual sources can span the +interface. +Each node shares its current state with the +neighboring nodes (i.e. the adjacent frequency segments) +so that the receiving node can remove the state of the +neighboring model when forming the residual that will +effectively serve as the data for the current update. The +segments overlap in frequency and each node is responsi- +ble for fitting to the sources in its half of the overlapping +region. This overlap, which is set to be a factor of two +larger than the typical bandwidth of a source at that fre- +quency, ensures that the templates for sources located +near the boundaries are not artificially truncated. +To +preserve the correlations between sources that are close +to one another on either side of a boundary, the UCB +nodes alternate which block is updating and which is +waiting. For example, all of the odd-numbered processes +will update their parameters, exchange with their neigh- +bors, and then the even processes will update. +For the MBHB modules, additional pre-processing is +needed once their residuals are formed. The MBHB mod- +ule relies on the heterodyned likelihood described in [19– +21] and recomputes the coefficients based on the current +state of the sampler, as well as updating proposal dis- +tributions used within the sampler that use the informa- +tion matrix as an approximation to the covariance matrix +of the posterior. After that pre-processing each MBHB +module updates its parameter block in parallel with the +other MBHBs in the fit. +At the end of the UCB and +MBHB updates each module sends the current state of +its model in the frequency domain to the root process to +broadcast to all of the other nodes, and the entire cycle +repeats. After many iterations when the sampling has +finished each process performs a minimal level of post- +processing to prepare for the next stage of the pipeline +when the posterior samples are consolidated into a source +catalog. +Note that in the traditional blocked MH sampling +scheme only one block of parameters are updated at a +time while the others are held fixed whereas GLASS is up- +dating blocks of parameters in parallel. This was a choice +made to improve the computational efficiency of the algo- +rithm. The current bottlenecks for the analysis are the +heterodyne step for the MBHB model and the conver- +gence time for the highest frequency UCBs. Those two +aspects of the problem set the scale for the cost of each it- +eration and the number of iterations needed, respectively. +To maximize efficiency, we do enough MBHB parameter +updates to match the cost of updating the heterodyne. +The number of internal GBMCMC updates per cycle of the +full sampler are then dynamically adjusted to take ap- +proximately the same amount of computational time as +the MBHB models. The comparative costs of the noise +model and VBMCMC updates are significantly lower, similar +to the costs of the data sharing and common processing +that must get done before each iteration (e.g. writing +results to file, etc.). +This scheme was thus created to +maximize the duty cycle of individual nodes by mini- +mizing the amount of time nodes are blocked waiting +for other processes to finish their work. While techni- +cally violating the conditions needed to have the resulting +samples be representative of the target posterior distri- +bution function, the effects are only noticeable for blocks +that are correlated. Updating alternating UCB blocks +ensures that no correlated UCB parameters are being +altered in parallel as the data segments for each UCB +node are larger than the bandwidth of a single UCB sig- +nal. Similar arguments can be made between other blocks +being updated in parallel although a production analysis +on observational data requires more through testing for +confirmation, and/or a more conservative approach and +a higher computational cost. It is a trivial rearrangement +of where the MPI exchanges take place to revert to the +traditional serial update of all parameter blocks. +Each of the samplers use parallel tempering [22] to im- +prove the mixing of the chains. Parallel tempering is es- +pecially critical for promoting transitions between mod- +els in the trans-dimensional samplers. +The tempering +scheme is independently developed and tuned for the dif- +ferent modules, and only the zero temperature chain pa- +rameters or state are shared between different processes. +Each of the parallel tempering samplers is multi-threaded +ideally using a single CPU per chain. In practice there +is a trade space between the number of resources needed +for the analysis and the amount of time those resources +are needed. Processing of the LDC2a-v2 data was done +on Amazon World Service (AWS) cloud computing in- +frastructure which favors smaller-scale jobs running for +longer. As a result the LDC2a-v2 analysis used multi- +ple threads per CPU. The final configuration for the full +LDC2a-v2 analysis (12 months of simulated data) used +624 MPI tasks and 12 CPUs per task for a total of 7488 +CPUs. The run was deployed on 78 × 96 CPU nodes. +The noise model and verification binary modules used +one MPI task each. There were 15 MBHB mergers in + +5 +the data each run with a dedicated MPI task. The re- +maining 607 MPI tasks were dedicated to the UCB model +covering .03 to 23 mHz. +III. +DESCRIPTION OF THE INDIVIDUAL +SAMPLERS +The individual model components that are integrated +into the GLASS architecture are independently developed, +described, and published. Each is still under active de- +velopment so it is useful to overview each sampler with +emphasis on updates that have been made since the most +recent publications. +A. +Global Noise Model +For the noise model we use an adaptation of the +BayesLine algorithm originally developed for LIGO- +Virgo noise modeling [15]. The original BayesLine algo- +rithm fits the power spectral density (PSD) of the noise +Sn(f) independently in each detector. The LIGO-Virgo +version of the pipeline uses a two-component fit to phe- +nomenologically model the noise spectrum. +The main +component is a broadband noise spectrum that looks sim- +ilar to a sum of power laws, with steeply rising noise at +low frequency and a more gradual increase at high fre- +quency. +Modeling the actual LIGO-Virgo data with a +broken power law is not sufficiently flexible so BayesLine +uses a cubic spline interpolation where each spline con- +trol point i is parameterized by its frequency and PSD +level [f i, Si +n]. The location of the spline points, as well +as the total number, are free parameters sampled over +with a trans-dimensional MCMC. For the LIGO-Virgo +application BayesLine also includes a linear combina- +tion of Lorentzians to fit the narrowband features in the +spectrum due to calibration lines, the power supply, res- +onances of the mirror suspension system, etc. The sim- +ulated LISA data do not contain narrowband noise fea- +tures and so GLASS’s implementation of BayesLine does +not use the line model. Note, however, that there were +spectral lines in the LISA Pathfinder data and so future +version of the model will need such a feature [23]. There +are two important differences between the BayesLine im- +plementation integrated into GLASS and that which has +been used for LIGO-Virgo data. +First, the LISA noise spectrum spans the frequency +regime where finite arm length effects of the detector +response are in the measurement band, unlike ground- +based interferometers which operate entirely in the long +wavelength limit. The arm length manifests in the noise +spectrum as sharp features where the PSD, and in- +strument response, formally go to zero for signals with +wavelength that fit an integer number of cycles within +the detector arm. +Mathematically this is a result of +terms proportional to sin2(f/f ∗) appearing in the de- +tector response functions with the “transfer frequency” +f ∗ ≡ c/2πL where c is the speed of light and L ≈ 2.5 +Gm is the arm length of the LISA detector. The result- +ing spectrum is not well modeled by a spline interpola- +tion at high frequencies. However, the difficult features +for a spline interpolation to track are a purely geomet- +ric effect set by the size of the detector. We therefore +model the difference between a reference noise spectrum, +including the geometric effects, and the observed data +Smodeled +n += Sobserved +n +− Sreference +n +, i.e. we are fitting for +broadband differences between the reference noise level, +derived from the current best estimate of the LISA per- +formance, and the observed data. Where the reference +model is accurate the modeled PSD will be consistent +with zero. +Second, the interpolation between control points for +the GLASS application of the spline model employs Akima +splines [24] rather than the cubic splines used in the +LIGO-Virgo applications. +The Akima splines are less +prone to oscillations between control points by relaxing +the requirement of a continuity in the second derivative +of the interpolated curve. The tendency for cubic splines +to oscillate is exacerbated by the large dynamic range +and steeply changing spectrum at low frequency. Akima +splines perform better on the LISA spectrum and are +worth considering for LIGO-Virgo noise modeling as well. +B. +GBMCMC Updates +The UCB sampler is the GBMCMC pipeline described +in Ref. [16]. The GBMCMC application is the latest in a +long line of algorithms designed for the LISA galactic bi- +naries which partition the frequency domain data into +many narrow-band segments and uses model selection +to determine the number of detectable binaries in each +segment [25, 26]. The model selection method of choice +used by GBMCMC is a transdimensional, or reversible jump +Markov Chain Monte Carlo (RJMCMC). +Of the different samplers integrated by GLASS GBMCMC +has seen the most additional development. +The sam- +pler was updated to use multi-threading for the parallel +tempered chains, making a significant improvement in +the run time especially when leveraging the increasingly +large number of CPUs available per node. +The pipeline has also updated the way results from +previous runs are incorporated as proposal distributions +for subsequent analyses. As described in Ref [16], the +LISA data are processed in increasingly-long time epochs, +starting with the first 1.5 month segment of data and +re-processing each time the available data has doubled +(i.e., after 3, 6, and 12 months). +In the first version +of the GBMCMC pipeline multivariate Gaussian proposals +were built using the covariance matrix of the posterior +samples for each UCB in the catalog. In the latest version +the single Gaussian proposal was replaced by a Gaussian +Mixture Model (GMM) which is fit using the Expecta- +tion Maximization (EM) algorithm run on the posterior +samples for sources in the previous epoch’s catalog. Eval- + +6 +P0 +P4 +P3 +P2 +P0 +Read Data +P0 +P3 +P2 +P1 +P5 +P6 +P7 +MBHMCMC +Step +MBHMCMC +Step +MBHMCMC +Step +P5 +P6 +P7 +P4 +P5 +P6 +P2 +P3 +P4 +GBMCMC +Step +GBMCMC +Step +P4 +P3 +P2 +P5 +P6 +P7 +Post +Processing +Post +Processing +Post +Processing +Post +Processing +Post +Processing +Post +Processing +Post +Processing +MBH +Root/Noise +GBMCMC +P0 +P7 +Post +Processing +P1 +VBMCMC +P0 +BayesLine +Step +P0 +P1 +VBMCMC +Step +P1 +P0 +P4 +P3 +P2 +GBMCMC +Step +P2 +P3 +P4 +P5 +P6 +P7 +Heterodyne +Heterodyne +Heterodyne +P4 +P3 +P2 +FIG. 3: Block diagram of Global Fit architecture. Process P0 is the root process and runs the noise model sampler. +Processes P1 to P3 are for the UCB model. Processes P4 to P6 run the MBHB model. Gray is the blocked MH sampler. +Purple are the NMBHB independent MBHB sampling steps. Green are the NUCB coupled UCB processes, which exchange only +between adjacent segments. Orange is the Noise model which is run on the root processes. Data from all processes are shared +with, and broadcast from, root. In practice O(103) UCB processes are needed, and O(10) MBHB processes. +uating the GMM proposal is more computationally costly +than the single multivariate proposal, but we have found +it to be offset by the improvement in convergence time. +The GBMCMC sampler now also includes a basic “split- +merge” proposal for trans-dimensional steps, whereas the +original algorithm only used “birth-death” moves. +A +birth-death move chooses to either remove or add a fea- +ture to the model (in GBMCMC’s case, a source from or +to the fit). The split-merge proposal attempts to divide +a single feature into two, or combine a pair of features + +7 +into one. The current implementation of the the split- +merge proposal is naive, choosing to remove one source +and replace it by two drawn from the same distribution +as is used by the birth-death moves, or to remove two of +the current sources and replace them by a single draw. +In other words, the current split-merge proposal is really +two birth moves and one death move, or two death moves +and one birth move, respectively. Further development +of more efficient split-merge proposals will be a critical +area to improve the sampling. +Finally, in the previous applications of the GBMCMC sam- +pler the frequency segments were of equal bandwidth over +the entire observing band. +Because each segment was +analyzed independently the overall number of segments +(and therefore nodes) needed for the analysis was not +a limiting factor in its deployment. Within the GLASS +architecture when all of the processes are communicat- +ing via MPI we need to be more parsimonious about +the number of segments being analyzed. +To that end +we adopted an adaptive segment size depending on the +source density. At low frequency where the source density +is the highest the segments are more narrow, and at high +frequency where the source density is low (and the sig- +nals have larger bandwidth) the segments are wider. The +exact segmenting was fixed before the LDC2a-v2 analysis +was started and kept the same for each epoch’s analysis. +A more efficient approach would be to use the previous +epoch’s catalog to dynamically determine where to best +place the segment boundaries to both keep the source +density per segment near constant, and to avoid hav- +ing loud signals near segment boundaries as an insurance +policy, even though the “edge effects” of the segmenting +are already ameliorated by GLASS’s data sharing scheme +between UCB segments. +C. +VBMCMC Updates +The verification binary sampler (VBMCMC) in GLASS is +identical to GBMCMC but is run in a different configura- +tion. Whereas GBMCMC is performing a blind search for +UCBs in the LISA data, part of which includes a model +selection step to determine if a candidate source in the +data is detectable, VBMCMC is executing a targeted analy- +sis of binaries which have already been identified as LISA +sources by EM surveys [27]. +The VBMCMC sampler therefore uses a fixed-dimension +analysis with priors on the orbital period and sky loca- +tion of the binaries derived from the EM observations. +Because the sky localization from the EM observations +is orders of magnitude more precise than LISA will ever +achieve, the sky location parameters in VBMCMC are fixed +to the EM-observed values, effectively using delta func- +tions for the priors. +The same is true for the orbital +period of the binaries (converted to GW frequency for +the VBMCMC model). While it is possible that some of the +currently-known binaries’ orbital period measurements +are not as precise as what LISA will infer, the long tem- +poral baseline of EM observations should effectively pin +the orbital period for LISA observations assuming con- +tinued effort to periodically monitor the known binaries +prior to LISA’s launch. For the known binaries where +this is not the case, replacing the delta function prior on +GW frequency with a Gaussian distribution with width +from the EM uncertainties is a trivial change. +Using a fixed-dimension targeted search for the known +binaries instead of retroactively extracting the known bi- +naries from the full UCB source catalog allows for upper +limits to be set on binary parameters (most notably the +GW amplitude) in the event that some of the known +binaries are below the detection threshold at the time +of the global fit analysis, perhaps due to elevated lev- +els of the astrophysical foreground (confusion) noise, or +because they will require longer integration times with +LISA before becoming detectable. +The targeted VBMCMC analysis will also reduce contam- +ination of known binaries from other loud sources at sim- +ilar orbital periods, as many of the currently know bina- +ries are at frequencies where the galactic source density +is expected to be highest. +Analysis of known binaries +will be particularly vulnerable to source contamination +early in the LISA mission when the frequency resolution +and integrated signal to noise levels are still improving +at the same time as when UCB observations may play an +important role in the early phases of instrument charac- +terization. +D. +MBHB Updates +The +MBHB +module +uses +elements +of +the +LISA-Massive-Black-Hole-Binary +pipeline +origi- +nally developed for low-latency detection and parameter +estimation of massive black hole mergers with LISA [21]. +The pipeline starts with a pre-processing search phase +that uses short stretches of data (typically a few weeks), +treating the galactic foreground as a noise source, and us- +ing a maximized likelihood function to rapidly lock on to +any massive black hole binaries. The search is repeated +on each segment of data after subtracting the previously +found signals until no additional sources above a S/N +threshold are found. The rapid search is then followed +with a full MCMC exploration of each source, taken one +at a time, that refines the parameter estimates. These es- +timates are then used as the starting point for the MBHB +analysis in GLASS. +The same PTMCMC sampling routine is used in the +global fit, but now with the noise model replaced by the +spline model, and with the resolved UCBs subtracted. +Another key difference is that the model components are +updated in an alternating fashion, in contrast with the +low latency analysis where the noise model is fixed and +the MBHB signals are analyzed sequentially. +The MBHB block of parameters is updated as follows: +When it comes time to update a particular MBHB, the +sampler receives the current state of the residuals residual + +8 +constructed from the other model components. That is, +the original data with the current state of the combined +UCB and VGB models, as well as the other MBHB wave- +form models, subtracted. A key element of the MBHB +sampler is the use of heterodyning to accelerate the like- +lihood calculations [19–21]. The heterodyne is computed +using a reference waveform–in this case one based on the +parameters of the MBHB model at the end of the last +update, and the current state of the residual. The com- +putational cost of setting up the heterodyne is equal to a +few times the cost of a standard likelihood evaluation, so +to be cost effective it makes sense to perform hundreds +of iterations with the fast heterodyned likelihood before +moving on to the next block in the sequence of updates. +The current version of the MBHB sampler uses cus- +tomized implementation of the IMRPhenomD waveform +model which describes the dominant (2, 2) harmonic for +spin-aligned, quasi-circular binaries. In future versions +of the sampler the waveform model will be generalized to +include spin precession effects, sub-dominant waveform +harmonics, and eventually, orbital eccentricity. +Another limitation of the current implementation is +that the dimension of the MBHB model is fixed to what- +ever was found by the low latency search. The MBHB +model also needs to be trans-dimensional with the results +from the search phase being used to propose adding or +removing sources. +IV. +DEMONSTRATION +Having described the overall GLASS architecture we +now turn to a demonstration of the pipeline’s perfor- +mance on simulated LISA data. The test data set was +produced by the LISA Data Challenge (LDC) team and +is the first of the LDC data sets to contain a combination +of different source types [28]. The following demonstra- +tion of GLASS’s current capabilities uses the LDC2a-v2 +data which contain ∼ 30 million galactic UCBs, 37 VGBs, +15 MBHB mergers, and an unspecified instrument noise +level. The LDC2a-v2 data span one year of LISA obser- +vation time assuming a 100% duty cycle. In reality there +will be periodic and sporadic interruptions to the data +taking which will require further development of GLASS’s +noise model and likelihood functions. +For each LDC simulation there are “blind” and “train- +ing” data sets, where the training data contain the list +of signals (injections) used for the simulation. The blind +data are simulated using a different realization of the +same population that is found in the training data. For +this demonstration the training data are used, enabling +assessment of the pipeline performance through compar- +isons of the resulting source catalog to the injected sig- +nals. +The LDC2a-v2 data TDI channels are generated +assuming an equal arm interferometer with stationary +instrument noise such that the TDI “A” and “E” chan- +nels are noise-orthogonal [14]. The GLASS noise model +correspondingly uses an independent fit to the A and E +instrument noise levels. This simplifying assumption will +need to be relaxed for analysis of the observational data +but will only effect the computational cost of the analy- +sis by introducing correlations between TDI data streams +which result in non-zero off diagonal terms in the noise +covariance matrix at each frequency. The resulting like- +lihood evaluations require more operations to compute +(see Eq. 1) but will not effect the overall complexity of +the global fit. +FIG. 4: Amplitude spectral density (ASD) of TDI A +channel data analyzed by each block of the sampler. The +UCB segments are shown in alternating shades of purple. +Note the changing bandwidth of the UCB segments, which +are larger where the source-density is lower. VGB segments +are orange. The MBHB (magenta) and noise models (blue) +cover the full analysis band, with the noise model extending +to slightly higher and lower frequencies for margin. This +example uses the first 6 months of the LDC2a-v2 data. +Fig 4 shows the amplitude spectral density (ASD) of +the TDI A channel after the first six months of the +LDC2a-v2 data. Through the remainder of the paper, the +A channel will be used to visualize the data and/or sig- +nal models. The differences between the A and E channel +data are subtle and not informative at this level, though +they are crucial for the analysis to decompose the ob- +served signal into the the two GW polarization states. +The data shown in the figure are colored over the in- +tervals being analyzed by different model components. +The noise model (light blue) covers the full frequency +range. +The MBHB model (magenta) spans a similar +bandwidth, though there is additional padding for the +noise model to ensure that it extends beyond where the +MBHB signals are in band during the time they are ob- +servable. The UCB segments are shown in alternating +light and dark purple bands. +Though it is difficult to +discern from the figure, especially due to the frequency +axis being on a log scale, the width of the UCB segments +is frequency-dependent, roughly tuned to use narrower +segments where the source density is higher. Finally the +locations of each narrow-band segment for the targeted + +10-19 +10-20 + 10-21 +10-22 +10-23 +10-3 +10-2 +f (Hz)9 +VGB analyses are shown in orange. Throughout the re- +mainder of the paper the same color scheme will be used +to identify different model components: Blue for noise, +purple for UCBs, orange for VGBs, and magenta for MB- +HBs. +FIG. 5: Same as Fig. 1 but focused on a narrow frequency +band near 6 mHz. The known binary in this segment of data +(orange) is representative of how HM Cnc will appear in the +LISA data. +For the headline demonstration of GLASS at work, +Fig. 5 shows the reconstructed components of each part +of the data model. The figure is showing the same con- +tent as Fig. 1 but zoomed in to a narrow interval around +6 mHz containing one of the loudest currently known +sources, HM Cnc [29], shown in orange. Here we can see +all of the model components on display, with a densely- +packed collection of UCBs in addition to HM Cnc all +overlapping one another (purple), and the MBHB merg- +ers sweeping through the band (magenta). +The gray +curve depicting the residual after all model components +have been subtracted from the data is fit by noise model +shown in blue. Note that in this figure the uncertainty +in the reconstructions is thinner than the line widths in +the figure, as all of the sources in this interval have high +signal to noise ratio (S/N). +Broadening the aperture to the full analysis band of +the demonstration, Fig. 6 shows the original data’s ASD +which is dominated by the UCBs and is thus shown in +purple. Removing the resolved UCBs (and VGBs) leaves +the magenta residual containing a bump in the spectrum +from the combined signals of the MBHBs. The final (light +blue) residual is after all of the resolved GW signals in the +fit are subtracted from the data. The remaining bump +in the residual spanning ∼3 × 10−4 to ∼5 × 10−3 Hz is +due to the foreground of un-resolvable UCBs. +As described above the analysis is repeated on increas- +ingly long epochs of the full data set, starting with the +first 1.5 months of observations going up to the full year +of data. Analyses are conducted each time the data vol- +ume has doubled, resulting in analyses of 1.5, 3, 6, and +12 month segments. As the observing time increases the +FIG. 6: ASD of the data including all signals (purple), after +removing the fit to the resolvable UCBs leaving behind only +the MBHBs (magenta), and then the final residual after all +signals in the fit are removed (light blue). +FIG. 7: Number of UCB (top) and MBHB (bottom) +detections as a function of observation time. The UCB +detection number is the number of candidates from the +maximum a posteriori model after clustering samples by +waveform match and then selecting candidates with z > 0.5 +(lightest shade), z > 0.9 (medium shade), and those that +uniquely correspond to a source in the injected population +with a match of m > 0.9. See Sec. IV B for a full +explanation of the match. +number of detectable signals grows. +For UCBs, which +are continuous sources, this is due to the source building +signal power over time and the improving frequency res- +olution of the data. The MBHBs are transient sources +so longer observation times provide more opportunity to +catch a black hole merger in the act. Fig. 7 shows the +number of candidate detections in the source catalogs for +the UCB (purple, left) and MBHB (magenta, right) mod- +els as the observing time increases. The drop in the total +number of UCB detection candidates between 1.5 and 3 + +10-19 +10~20 +10-21 +10~22 +6.210 +6.216 +6.220 +6.226 +6.230 +6.236 +6.240 +6.246 +f (mHz)10-18 +10~19 +10~20 +410-21 +10~22 +10-23 +1024 +10-4 +10~3 +10~2 +f (Hz)8000 +6000 +EDJ +4000 +2000 +0 +15 +NE +10 +9 +12 +ntha10 +months is unexpected. However, the number of confident +detections that are clearly associated with an injected sig- +nal increases, as will be described in section IV B. Time- +dependent analyses of data with such short observing +times will be particularly sensitive to the initial orienta- +tion of the spacecraft constellation relative to the galactic +center, and the modeling of the time-varying noise due +to the galactic foreground [30]. While needing further +study, our assessment is that the initial conditions of the +LISA orbits and our admittedly incorrect assumption of +stationary noise lead to this counter-intuitive result. +Having summarized the GLASS performance on a data- +wide level, we now take a detailed look at the perfor- +mance of the individual model components by studying +the properties of the recovered source catalogs and com- +paring them to the input populations. +A. +Noise Model +The instrument noise model parameters used in GLASS +are not physically meaningful and so the primary diag- +nostics for the performance of the sampler are functional +tests. In each cycle of the GLASS blocked MH sampler, the +PSD model is fitting the residual after the current state +of the UCB, VGB, and MBHB models have been sub- +tracted from the data. That residual includes the instru- +ment noise as well as the unresolved galactic foreground, +referred to in the LISA literature as “confusion noise,” +which is expected to be the dominant source of residual +power between ∼10−1 and ∼3 mHz. Exactly where the +galactic foreground drops below the instrument noise de- +pends on details of the galactic population of compact +binaries [31], the performance of the LISA instrument, +and the observation time [32, 33]. +Fig. 8 shows the power spectrum of the 12 month data +A channel (dark gray), and the residual after removal +of a fair draw from the joint UCB+VGB+MBHB model +(light gray). The colored lines are the PSD fits from the +noise model for the 1.5, 6, and 12 month runs. The 3 +month result is omitted for clarity. +The black dashed +line is the PSD used for simulating the instrument noise. +In each of the PSD fits the spline model used ∼15 to +∼30 control points. +The results clearly show how the +prominent bump in the spectrum where the astrophysical +foreground dominates initially grows with time across the +band as the joint S/N of the galaxy increases, and then is +slowly reduced as the UCB model is able to resolve more +binaries, particularly at higher frequency. +Outside of +the interval dominated by the astrophysical foreground, +the modeled PSD matches the simulated levels. The 1.5 +month PSD fit is truncated at low frequency because the +bandwidth of the noise fit is dynamically set based on +the signal content of GLASS which does not include any +MBHB merger signals in the first month of the LDC2a-v2 +data, alleviating the need to model the PSD below ∼0.3 +mHz. +A more quantitative assessment of the PSD model +FIG. 8: Median inferred noise PSD for three (purple), six +(green), and 12e (orange) month observing times. The black +dashed line is the true PSD used when simulating the data. +For reference, the dark gray is the power spectrum of the 12 +month data, and the darker gray is the residual after +removal of the UCB, VGB, and MBHB models. The +difference between the inferred and true PSD between +∼ 2 × 10−4 and +∼ 6 × 10−3 Hz is due to the unresolved +galactic foreground, or “confusion noise.” +is possible by testing the whitened data +˜w(f) +≡ +˜d(f)/ +� +Sn(f) where the tilde denotes a Fourier trans- +form, d is the data, and Sn(f) is the PSD. The PSD is +proportional to the frequency-dependant variance of the +noise and therefore the whitened data should be consis- +tent with a zero mean unit variance normal distribution +N[0, 1]. +Fig. 9 shows histograms of the combined real +and imaginary components of the Fourier transformed +whitened residuals for the 1.5 (left, purple), 6 (middle, +orange) and 12 (right, pink) data. Displayed above each +panel is the mean and standard deviation of the whitened +residuals, in agreement with the expected results. While +the performance of the noise model passes the tests pre- +sented here we know that the model is incomplete and +demands further development to meet the challenges of +the real observing data. The primary limitation of the +current model is the implicit assumption that the noise is +stationary. In practice the LISA noise will have a time- +varying PSD due to secular and random fluctuations of +the instrument performance, as well as the cyclostation- +ary modulations of the galactic foreground imparted by +LISA’s orbit [30, 34]. Generalizing the noise model us- +ing time-frequency methods [35] is an immediate priority +for future development of GLASS, and has ripple effects +through the rest of the model components. +B. +UCB Catalog +The UCB catalog contains ∼2000 candidate detections +after the first 1.5 months of observing, climbing to ∼8500 +by the end of the 1 year LDC2a-v2 data set. Fig. 10 shows + +10-36 +10-38 +TDI A Channel Sn(f) +10-42 +44 +10-46 +10-48 +10-4 +10-3 +10-2 +f (Hz)11 +(a) 1.5 mo +(b) 6 mo +(c) 12 mo +FIG. 9: Distribution of the whitened data +˜w(f) = ˜d(f)/ +� +Sn(f) for the 1 month [left], 6 month +[middle] and 12 month [right] whitened residual data. If the +PSD model is functioning correctly the whitened data +should be distributed as a zero mean unit variance Gaussian. +The mean and standard deviation computed from the +whitened data are printed above each panel. +FIG. 10: Scatter plot of frequency f and GW amplitude A +of UCBs in the 12 month source catalog. The black line is +an example LISA sensitivity curve. +a scatter plot of the point-estimate frequency and GW +amplitude parameters (f, A) of the recovered sources in +the catalog after analysis of the full 12 month LDC2a-v2 +data. The black line is a representative LISA sensitivity +curve, so that the S/N of each source is proportional to +its height above the curve. The “cavity” of sources above +the curve at low frequency is due to the foreground of +unresolved galactic binaries becoming the dominant noise +source. +Distilling the output of the GBMCMC sampler to a dis- +crete list of catalog sources is a nuanced and lossy pro- +cess, a detailed description of which is found in Ref. [16]. +To summarize: +In each frequency segment the maxi- +mum a posteriori (MAP) number of source templates +used to fit the data (i.e., the number of templates used +most frequently in the RJMCMC sampler) is selected +FIG. 11: LDC0100498745 parameter estimation over time. +Green, orange, and pink contours correspond to the +measurement after 3, 6, and 12 months of observing with +LISA. The contours mark the 1 and 2σ credible intervals. +The sampling parameters from GBMCMC are re-parameterized +into orbital period and derivative (P, ˙P) [top left]; amplitude +and inclination (A, ι) [top right]; galactic latitude and +longitude (l, b) [bottom left]; chirp mass and luminosity +distance (M, DL) [bottom right]. The black dashed lines +show the injected parameter values. +as the reference model. The posterior samples from that +model are clustered into discrete catalog entries using the +match m ≡ (hi|hj)/ +� +(hi|hi)(hj|hj) between the wave- +forms computed from the chain samples at step i and +j where (·|·) is the standard noise-weighted inner prod- +uct. The threshold for considering a chain sample as a +member of a cluster is m > 0.8. +The fraction of the +total number of steps in the chain that have a sample +in a particular entry is interpreted as a detection confi- +dence z. The threshold for inclusion in the final catalog +is z > 0.5 i.e., that a catalog entry has a sample in more +than half of the total number of chain steps in the MAP +model. This is not a strict criteria, roughly equating to +sources with a Bayesian odds ratio > 1 being included +in the model. We therefore use an additional threshold +of z > 0.9 for catalog sources to be considered confident +detections. Neither the match requirement for inclusion +as a sample belonging to a catalog entry, or the frac- +tion of samples from the chain that an entry contains, +are extensively tested or optimized through large scale +injection studies. Such critical work must be thoroughly +undertaken in advance of using GLASS or anything like it +for production analyses. +As an example of the content contained for a single +UCB, Fig. 11 shows a set of marginalized 2-dimensional +posterior distributions for a high S/N binary found near +10 mHz. UCBs in the GLASS catalog are identified by +their median frequency, so in the 12 month catalog this +binary was labeled LDC0100498745. See [16] for a dis- + +w = 0.00 ± 0.98 +ww = 0.00 ± 0.99 +w1020, +10~21, +1023 , +10-4 +10~3 +10~2 +f []P [s/s] [×10-10] +P [s] [×10-4 + 1.99007×102A「×10-22 +t [deg[deg] +10.5 +[deg.03 months +06 months +12 months +DL [kpc] +M [Mo]w = 0.00 ± 0.9912 +cussion and demonstration of how UCB candidates are +traced through versions of the source catalogs from earlier +analysis epochs (i.e. tracking how this particular source +was labeled in the 6 month catalog, etc.). Shaded con- +tours are the 1 and 2σ credible intervals, and the colors +correspond to observing time with blue green, orange, +and pink representing 3, 6, and 12 months respectively. +As with the color-coded source types, throughout the pa- +per these colors (along with light-purple for 1 month) will +consistently represent the observing times in subsequent +figures. The posteriors are represented in a different pa- +rameterization than is used in the sampling, to better +match the observables customarily used by the EM ob- +serving community. The top left panel shows the orbital +period P and first derivative +˙P of the binary system. +Note how the measurement precision of ˙P increases more +rapidly than other parameters. This is because the or- +bital evolution of the binary enters the phase as a T 2- +dependent term, so the information accumulates more +rapidly than the typical +√ +T scaling due to the increasing +S/N of a continuous source. The top right panel shows +the gravitational wave amplitude and the binary’s or- +bital inclination in degrees. The bottom left panel is the +sky location in galactic coordinates, with l as the galac- +tic latitude and b the galactic longitude. +The bottom +right panel are the chirp mass M and luminosity dis- +tance DL parameters derived from the GW observables +assuming the orbital evolution is purely driven by emis- +sion of gravitational waves. The horizontal and vertical +lines mark the parameter values for the injected signal, +i.e. the “right answer” when we have the luxury of know- +ing the true source parameters. +One intriguing opportunity afforded by the LISA UCB +catalog is to map the Milky Way’s stellar remnant popu- +lation. Fig. 12 shows the map of the UCB sky in galactic +coordinates after 1.5, 3, 6, and 12 months from top left +to bottom right. The maps are constructed by combining +the posterior samples from all of the sources in the UCB +catalog. After only 1.5 months of observing large scale +galactic features like the bulge and disk are evident in the +maps. The resolution of the image continues to improve +as the observing time increases, revealing a remarkably +clear view of the galactic disk and bulge with hundreds +of well-localized sources. The quality of the image will +steadily improve over the LISA mission life time, and will +include distance information from the chirping binaries, +enabling three-dimensional inferences on the spatial dis- +tribution of binaries throughout the galaxy [36]. +With the benefit of knowing the input source popula- +tion, the observed UCB catalog is compared to the in- +jected binaries to study the detection efficiency of the +analysis. The primary metric for assessing the the qual- +ity of the inferred source catalog is the maximum match +m between the waveform computed from the point es- +timate source parameters in the catalog and the wave- +forms from the injected population. For computational +efficiency the match is only computed between the cata- +log waveform and an injected waveform if their frequency +parameters are within 10 frequency bins of one another. +Fig. 13 shows the distribution of the match values be- +tween the sources in the catalog and those in the input +population. For inclusion in the sample we select catalog +candidates with detection confidence z > 0.9 instead of +the z > 0.5 criteria for inclusion in the catalog. The top +panel shows the cumulative distribution function of the +matches. The vertical axis is thus interpreted as the frac- +tion of sources in the catalog with match below m. The +bottom panel is the un-normalized survival function, i.e. +the total number of sources in the catalog with match +greater than m. If we consider m > 0.9 as a criteria for +an unambiguous mapping between a source in the catalog +and an injection then ∼75% (∼80%) of the confidently +identified (z > 0.9) binaries in the 12 (6) month catalog +exceed the criteria equating to ∼5000 (∼4500) sources. +The most striking feature in the results is a population +of sources with matches below ∼0.9 that emerges in the +12 month catalog. Previous analyses of simulated galax- +ies in LISA data have not shown a similarly-populated +low match tail [25, 26], instead finding ≳90% of sources +with m > 0.9. The comparatively high rate of low-match +sources in the GLASS UCB catalog demands further in- +vestigation and discussion. +To begin, compare the matches for each search over +different projections of the full parameter space shown in +Fig. 14 displaying the frequency-amplitude plane on the +left and the sky location on the right. Here the sky loca- +tion is displayed using the sampling parameters (ecliptic +coordinates) rather than galactic coordinates as in Fig 12. +The source model in GBMCMC is parameterized using eclip- +tic coordinates to minimize covariances between parame- +ters, and thereby make it easier to sample the posterior. +In this figure each point in the scatter plot is colored by +the waveform mismatch defined as 1 − m, and the color +map uses a log scale. Thus cool colors are good matches, +and warm colors are poor matches. +The majority of low match sources are found in the +frequency interval between 1 and 6 mHz where the un- +resolved UCBs are the dominant source of noise. One +significant difference between the GLASS analysis and pre- +vious UCB searches is the global spline noise model. In +references [25, 26] each narrow-band frequency segment +independently modeled the noise level effectively using a +O(103) parameter piece-wise (and discontinuous) fit to +the instrument noise. +The GLASS noise model is con- +strained to be a continuous function, and is using model +selection to determine the most parsimonious number of +parameters, naturally making the noise model less flexi- +ble. It holds together that the differences between a fixed +(and high dimensional) piece-wise fit and the parsimo- +nious spline model have a larger effect in the foreground- +dominated part of the spectrum where there is stronger +coupling between the UCB and noise models. +Another, and perhaps more impactful difference, is a +difference in the prior used for the GW source parameters +between the LDC2a-v2 analysis and previous demonstra- +tions. Previous incarnations of the GBMCMC sampler, and + +13 +FIG. 12: Maps of the source sky locations in galactic coordinates from the UCB catalogs after 1.5 [top left], 3 [top right], 6 +[bottom left] and 12 [bottom right] months of observing, showing the increasingly clear reconstruction of the Milky Way disk +and bulge structures. +it’s ancestors, have used a prior on the sky location pa- +rameters derived by assuming the sources followed the +spatial distribution of the galaxy. +While that prior is +still an option in GLASS, it was intentionally not used for +the LDC2a-v2 analysis in favor of a uniform prior on the +sky location parameters. The choice to not use a galaxy +prior was motivated by the idea of eventually performing +a hierarchichal analysis where the posterior samples of +the binaries in the catalog are used to constrain models +for the spatial distribution of sources in the galaxy [36]. +Hierarchichal analyses are complicated by non-trivial pri- +ors on the posterior samples and so the choice was made +to produce samples for the UCBs in the most accessi- +ble form possible. The expectation was that this choice +would produce larger uncertainties in the position recon- +struction of individual sources, but an unintended con- +sequence is the effect it had on the detection efficiency. +The right hand side of Fig. 14 clearly shows sources with +high mismatch are preferentially located outside of the +galactic plane, whereas high match sources follow the ex- +pected “U” shape of the galaxy in ecliptic coordinates. +Mismatching sources in the catalog generally arise from +two circumstances: Either a source template is fitting to +blended contributions from multiple injections or multi- +ple source templates are being used to fit a single injec- +tion. The former (one-to-many) can be the most parsimo- +nious solution (having the highest Bayesian evidence), or +it could be due to modeling issues either from the noise +or sky location prior while the latter (many-to-one) is +clearly a problem with sampler convergence. +In the regime where a few templates are fitting a larger +number of signals, clear attribution for why this was the +preferred configuration of the model is difficult to assign. +However, it is true that from a strict model selection +point of view if fewer templates can adequately fit fea- +tures in the data, i.e. a larger combination of sources, +the parsimonious solution is favored. Additional infor- +mation, such as a prior that favors sources in the galac- +tic plane, is needed to help further disentangle the over- +lapping sources. If the sky location prior and/or noise +model were the predominant cause of the poor matches, +why would it only appear in the 12 month analysis? One +possible explanation is that the recovered source catalogs +from shorter integration times are going to be dominated +by the loudest, most isolated, binaries in the population +and as the observing time increases the search is able fit + +60° +30° +a0 +1800 +135° +:06- +-45° +o0 +45° +:06 +135° +180° +30° +30° +60°60° +30° +30° +a0 +18 +135° +:06- +45° +45° +:06 +1359 +1800 +30° +30° +60°60° +30° +30° +a0 +18 +-135° +-90° +-45° +45° +006 +135° +1800 +30° +30° +60°60° +30° +30° +a0 +18° +-135° +45° +450 +900 +135° +180° +30° +30° +60°14 +FIG. 13: Top: Cumulative distribution function of matches. +Bottom: Un-normalized survival function of the match. +Purple, green, orange, and pink curves are for 1.5, 3, 6, and +12 month observing times respectively. Results are from +confident (z > 0.9) detections from the source catalog. +features in the data with lower intrinsic GW amplitude +where the injected source density increases. +Addition- +ally, the uniform prior on the sky location of binaries is +worse for longer observation times. With short-duration +observations the LISA UCB catalog will be dominated +by near-by sources, especially at lower frequency, where +a uniform distribution on the sky, while still not accurate, +is closer to the observed population than at later times +when the UCB catalog will sample the full galaxy. The +issues of source confusion and the dependence on priors +for what constitutes a “detection” are many, nuanced, +and require dedicated study. Another possibility is that +the jump from 6 months to 12 months was too big a step +for the Gaussian mixture model proposal which uses the +posteriors from the shorter time span analysis as a pro- +posal for the longer time span analysis. +Fig. 15 shows a pair of examples suspected of exhibit- +ing the parsimony “failure” mode. +Both show results +from a subset of a single UCB analysis segment from the +12 month data roughly bracketing the frequency range +where the low match sources are most prevalent. The +top panels show the ASD of the simulated data after the +MBHB signals have been removed (gray), the injected +UCB model (black), and the posterior distribution for +the recovered UCB model (purple). The middle panels +show the injected source parameters in the frequency- +amplitude plane (open circles), the point estimate param- +eters from the UCB catalog entries (filled purple circles), +unfiltered chain samples from GBMCMC (light purple), and +the posterior samples colored differently for each indi- +vidual source in the catalog. The bottom panel shows +the match for each posterior sample in the same colors +as the panel above, and open circles for the point esti- +mate match used to construct the curves in Fig. 13. In +these examples the catalog entries with poor matches are +typically found in parts of the frequency segment where +there are numerous unresolved injections. The posterior +samples do not obviously favor one injected value over +the other, yet the overall fit follows the injected signal +(top panel). +An example of the overfitting problem, where multiple +templates are fitting one source, is shown in fig. 16 using +the same format as +15. In this segment there are six +densely-packed injections, two of which are well-fit in the +catalog. The remaining four were fit by 13 templates in +the 12 month analysis–a clear convergence error. Exam- +ples like figures 15 and 16 will drive future development +of the GBMCMC pipeline. +To test our conjecture about the root cause of the high +rate of low match sources in the catalog, we reanalyzed +five UCB segments from the LDC2a-v2 data with the +MBHB injections removed, and used the GBMCMC sam- +pler alone. The frequency segments were chosen to be +evenly spaced between 1 and 6 mHz to cover the regime +where the low match catalog entries were most common. +Three different configurations were used to measure the +effect of the different suspects for the low match popu- +lation. The first uses proposals built from the 6 month +GLASS run, and the flat sky prior, just like the produc- +tion LDC2a-v2 results. The only difference between the +first example and the global analysis (ignoring covari- +ances with the MBHB model) is the noise model, which in +GBMCMC is parameterized as a constant over the frequency +interval of the segment. Assuming a constant PSD over +the each analysis segment effectively amounts to a sig- +nificantly more flexible noise model compared to what is +used in GLASS. The second configuration is the same as +the first, but with proposals built from a 9 month GBMCMC +analysis of the segments to test whether the low match +sources were due to convergence problems stemming from +the time steps between analyses being too large, reducing +the efficiency of the Gaussian mixture model proposals. +The final configuration reverts to the 6 month proposals +but includes the galaxy prior as described in [16]. +Fig. 17 shows the same type of results as fig. 13 but +only for the six test segments, and comparing the differ- +ent test configurations with the global analysis. Of the +different configurations tested, using the constant PSD +model slightly improves the purity of the catalog, but +similarly reduces the total number of detections in the +catalog. Including an intermediate 9 month analysis re- +sults in a higher number of detections and an improved +match distribution, meaning that the overall convergence +of the UCB model is heavily dependent on the efficiency +of the proposals. +Including the galaxy prior has the +largest influence on the results, both improving the cat- +alog purity as well as the total number of detections. +While a full-scale study is needed to conclusively assess +the performance of the different configurations, these re- +sults are suggestive that using a model for the galaxy +in the analysis and shorter time steps between global fit +processing will improve the quality of the UCB source +catalog. +The sensitivity to the galaxy prior also rein- +forces the fact that modeling choices have a significant + +match +1.0 +6'0 +0.8 +V +20 +count +0.6 +0.6 +0.4 +fraction +0.3 +0.2 +0.1 +0.0 +6000 +A 4000 +counfs +total +2000 +0+ +0.0 +0.1 +0.2 +0.3 +0.4 +0.6 +0.6 +0.7 +0.8 +6'0 +1.0 +match15 +FIG. 14: Scatter plot of point estimates for candidate sources in the UCB catalog after 12 months of observing colored by the +minimum mismatch between the catalog waveform and the injected waveforms. Left: GW frequency-amplitude plane showing +that most of the high mismatch sources are in the confusion noise regime below ∼6 mHz. Right: Sky location parameters in +ecliptic coordinates, revealing that the high mismatch sources are preferentially located out of the galactic plane. +FIG. 15: Investigation of possible causes for the low match population in the UCB catalog. Left and right panels focus on +different frequency intervals towards the low (∼1.8 mHz) and high (∼4 mHz) end of the region where the false alarms were +most frequent. The top panel shows ASD of the LDC2a-v2 data with MBHBs removed (gray), input UCB signal (black) and +the joint posterior for reconstruction from the GBMCMC sampler (purple). The middle panels are a scatter plot of the UCB +frequency-amplitude parameters for the injections (open circles), point estimate catalog entries (filled circles), chain samples +(light purple dots), and posterior samples for the individual sources (multi-color dots). The bottom panel shows the +maximum match between each posterior sample and an injected waveform (same colors as middle panel) and the match from +the catalog point estimate used in the summary plots like fig. 13. The low-match sources are consistent with fitting +combinations of injections. +impact on the resulting inferences, especially near the +detection threshold. +C. +VGB Catalog +Analysis and interpretation of the VGB catalog is more +straightforward due to the model using informative priors +derived from EM observations. Because the sampler is as- +suming a single source at known orbital period and sky +location, the most relevant parameters to be measured +by LISA are the GW amplitude and binary inclination. +In the event that the LISA observation time is not long +enough for the source to be “detectable,” the posterior +samples are useful for setting inclination-dependent up- +per limits on the amplitude, and therefore the combined + +1.00 +100 +0.75 +1021 +0.60 +10-1 +0.26 +mismatch +1022 +0.00 +0.26 + 10-2 +0.60 +10~23 +0.76 +1.00 +103 +10-3 +10~2 +0 +2 +3 +f (Hz)10~20, +10~21 +10~22 + 10-22, +10-23, +1.0 +match +0.6 +0.0. +1.802 +1.803 +1.804 +1.806 +1.806 +1.807 +1.808 +1.809 +f (mHz)10-20 +10-21 +10~22 +： +10-23 +1.0 +match +0.6 +O +0.0. +6f6'E +3.960 +3.961 +3.962 +3.963 +3.954 +f (mHz)16 +FIG. 16: Same as fig. 15 for an example where multiple +templates were fitting a smaller number of injections, i.e. a +clear example of a convergence failure for GBMCMC. +FIG. 17: Same as fig. 13 but for different test configurations +of GBMCMC on six frequency segments between 1 and 6 mHz +to explore possible causes for the high fraction of low match +sources in the GLASS UCB catalog. The dark blue curves are +the GLASS results for the test segments. Light blue use the +flat sky prior but a constant PSD model. Dark orange are +the same configuration as light blue but include proposals +generated from a 9 month run. Light orange uses the +constant PSD model and a prior that prefers sources to be +located in the galactic plane. Results are from confident +(z > 0.9) detections. In this limited test, using the galaxy +prior improves both the catalog purity and the number of +high-match sources in the catalog and shows that smaller +time steps between global fit runs are advantageous. +chirp mass of, and distance to, the binary. +Fig. 18 shows four representative examples from the +full set of known binaries comparing inferences between +3, 6, and 12 month observations (green, orange, and ma- +genta). +The two dimensional posteriors present the 1 +and 2σ contours, and the dashed black lines mark the in- +jected parameter values. The top two panels show results +(a) AM CVn +(b) HM Cnc +(c) UCXB 4U1820-30 +(d) CX0GBS J1751 +FIG. 18: Variety of results from targeted analysis of known +binaries in the LDC2a-v2 data showing measurement of the +GW amplitude and binary inclination as a function of +observing time, comparing 3 (green), 6 (orange), and 12 +(magenta) month observations. (a) AM CVn is a +straightforward example of a strong LISA source properly +identified early in the observing campaign with inferences +steadily improving over time. (b) HM Cnc shows similar +behavior as AM CVn but at higher S/N. (c) UCXB +4U182030 shows how a binary will transition from being +undetected, with the analysis providing upper limits on the +amplitude (green) to a point where the binary parameters +will be constrained (orange, magenta). (d) CX0GBS J1751 +is an example of improving upper limits with observation +time. This binary will require longer observing times to be +constrained. +for AM CVn and HM Cnc–two of the canonical known +binaries that are identifiable early in the LISA observa- +tions. +The bottom left panel is for the ultra compact +X-ray binary UCXB 4U1820-30 which transitions from a +regime where upper limits are set after 3 months of ob- +serving, to a constraint in the 6 and 12 month catalogs, +indicated by the open contours in green to the closed con- +tours in orange and magenta. Finally, source CX0GBS +J1751 remains undetectable by GLASS after 12 months of +observing but note that the upper limit inferred for the +amplitude decreases over time. +D. +MBHB Catalog +The final parts of the GLASS analysis to investigate are +the MBHB results. The most interesting single exam- +ple is the first MBHB to appear in the LDC2a-v2 data, +which merges during the second month of the simulated +observations. The simulated source also happens to be +one of highest S/N binaries in the population and is ob- + +10~20 +10~22, + 10~23 +10-24 +1.0 +. +match +0.6 +0.0. +6.230 +6.231 +6.232 +6.233 +6.234 +6.236 +f (mHz)match +1.0 +6'0 +GLASS +0.8 +GBMCMC, flat sky prior +0.7 +0.6 +GBMCMC, flat sky prior, 09mo step +0.6 +GBMCMC, galaxy prior +0.4 +fraction +0.3 +0.2 +0.1 +0.0 +match +100 +76 +A +counfs +60 +fotal +6 +0 +0.0 +0.1 +0.2 +0.3 +0.4 +0.6 +0.6 +0.7 +0.8 +6'0 +1.0 +match03 months +06 months +12 months +A「×10-22 +t (deg)03 months +06 months +12 months +0.90 +A「×10-22] +0.75 +t (deg)03 montHs +06 months +12 monthis +A「×10-23 +(deg)03 months +06 months +12 months +A[×10-22 +(deg)17 +FIG. 19: Marginalized posterior distributions for the mass +m and dimensionless spin χ parameters of the first MBHB +to merge in the LDC2a-v2 data, during the second month of +the simulated data. The dashed lines mark the parameter +values for the simulated signal. Note the parameter +estimation improves after the signal has left the LISA band +because the galactic foreground decreases as the UCB model +resolves more binaries. +servable in three of the different analysis epochs used for +this demonstration of GLASS. Fig. 19 shows the posterior +distribution function for the intrinsic source parameters: +m1 and m2 are the masses of the black holes in the bi- +nary while χ1 and χ2 are their respective dimensionless +spin parameters. Recall that the data and MBHB model +both currently assume the binaries have BH spin aligned +with the orbital angular momentum vector–an assump- +tion that is not valid in nature but made out of conve- +nience at this stage of development for simulations and +pipelines. +As with other examples, the color indicates +observing time and the dashed lines mark the injection +values. +What is remarkable about fig. 19 are the changes in the +posteriors over the observing time even though the binary +merged in month 2 of the simulated data. The subtle +reduction in the width of the posteriors is due to the +improved foreground subtraction from the UCB model, +which effectively lowers the noise level, and thereby in- +creases the S/N, of the MBHB mergers. Because of the +global nature of LISA analysis, inferences from transient +sources will continue to improve long after they have left +the measurement band. +Moving on to the full MBHB population, fig. 20 shows +the posterior distribution function for the mass parame- +ters from each of the 15 MBHBs injected into, and recov- +ered from, the LDC2a-v2 data. The MBHBs are labeled +in the GLASS catalog by the merger time (in seconds) +relative to the start of observations. The input popula- +tion covers a wide range off masses and the posteriors +are generally well-constrained due to the large number of +in-spiral cycles combined with the strength of the merger +signal. To compare against the true values from the sim- +ulations, the right-hand panel shows the same posteriors +but shifted by the injected mass values, so the point (0,0) +marks the truth. For visibility, only the 1σ contours are +shown on the right hand side. All but one of the binaries +contain the truth value inside of the 2σ contour. The +one source whose injected value is outside of the bulk of +the GLASS posterior is the lowest mass binary in the in- +put population. The same bias is seen in other prototype +analyses and is suspected of being the results of artifacts +in the LDC2a-v2 data from the waveform simulation pro- +cess [37]. +For the final demonstration of the MBHB catalog, +fig. 21 displays the sky map for each MBHB source in the +catalog, in order of merger time from top left to bottom +right, after the full 12 month analysis. The variety found +in the MBHB sky maps is the result of the relative im- +portance for each event of the three different “channels” +of localization information for these signals. +The first +channel is through the GW phase which is frequency- +modulated by the orbital motion of the spacecraft. The +second channel of localization information comes from +the relative arrival time of the GW signal’s wave-fronts +at the different spacecraft. The third, and least infor- +mative, is from the non-uniform detector response over +the sky which is encoded in the relative amplitudes of +the GW signal in the different TDI channels. Sky maps +that contain many modes of high probability are typi- +cally from higher mass binaries that are shorter lived in +the LISA data, missing out on the Doppler modulations +induced by the orbital motion of the detector and not +reaching high enough frequency to differentiate the sig- +nal arrival times at the different spacecraft. Lower mass +MBHB signals get the best of both worlds, as they are +in the measurement band for long enough to have clearly +detectable frequency modulation and they reach short +enough wavelengths to benefit from the time of arrival +measurements. +Note that even for the high mass and +short-lived binaries additional information from spin pre- +cession and, more importantly, higher harmonics of the +waveform will help break degeneracies [38]. See Ref. [39] +for a more comprehensive discussion and demonstration +of MBHB parameter estimation with LISA. +V. +DISCUSSION AND FUTURE WORK +The global analysis demonstrated here is one impor- +tant step towards a fully functioning pipeline ready for +LISA observational data but there is still a long way to +go. Obviously missing are the other anticipated source +types, though the GLASS architecture is designed to seam- +lessly accommodate additional modules. +Extending to +other source types is generally expected to be a small +perturbation relative to the overall scale of the analysis +which is set by the UCBs. +Another obvious direction of development is to reduce +the overall computational cost of the analysis. The cur- +rent version of GLASS used O(103) CPUs for O(5) days + +03 months +06 months +12 months +m2 (Mo)[×105] +×106 +m2 (M +×105 +X218 +FIG. 20: Mass posteriors for the entire observed MBHB population in the 12 month LDC2a-v2 data. Left: 1 and 2σ contours +for the inferred masses. Right: 1σ contours for the binaries shifted by the true values for each simulated source, such that +(0,0) marks the injected parameter location. +to process the 12 month data, and those processing times +will increase roughly linearly as the observation time +grows. +The current algorithm will become uncomfort- +ably expensive for multi-year data sets. +Reducing the +computational cost is of crucial importance for further +development because the main source of stress on the +analysis methods provided by LISA data is the scale. +Optimal development of the global analysis requires fre- +quent processing of full-scale data sets. The two prongs +for reducing the computational cost of the analysis are +by lowering the cost of each likelihood evaluation with +accelerated compuatational techniques (e.g. +Ref [40]), +and by reducing the total number of likelihood evalua- +tions by developing a more efficient sampling algorithm +through further development of data- and domain-driven +proposal distributions. +Beyond implementation improvements, there are a +number of model assumptions, reflected in the simplic- +ity of the likelihood function, that will need to be re- +laxed to properly handle observational data. Generally +speaking, it is the assumption of stationary noise that +is most problematic, though there are different sources +of non-stationarity that each deserve their own strategy. +As mentioned earlier, there will be periodic (due to the +galactic foreground) and secular (due to the instrument) +changes in the instrument noise level which introduce +non-zero off-diagonal elements of the frequency-domain +noise covariance matrix. We will mitigate these effects by +moving away from conducting the analysis in the Fourier +domain, in favor of a discrete wavelet basis, which still +yields a diagonal noise covariance matrix if the stationary +timescales are longer than the duration of the wavelet ba- +sis functions. Descriptions of waveform and noise models +in the wavelet domain are found in Refs [30, 35, 41] and +are scheduled to be integrated into GLASS. In the wavelet +domain the noise model will be a function of both time +and frequency to track the slow drift in the instrument +and foreground noise levels. +The time-frequency approach renders the heterodyn- +ing currently used for both the UCB and MBHB like- +lihood calculations redundant. Wavelet decompositions +incorporate a natural compression of GW signals since +the likelihood only integral only changes along the signal +track f(t), which has length ∼ +√ +N for a data set with +N data points [35, 41]. Wavelet domain likelihoods are +typically faster than their heterodyned analogs without +requiring a reference waveform or any pre-computation +step. +Faster likelihood functions allow for more rapid +convergence and better mixing between blocks of the MH +sampler for the same computational cost. The wavelet +domain is also better suited to handling data gaps than +frequency domain analyses. In the wavelet domain the +basis functions are finite duration with built-in window +functions that naturally suppress spectral leakage caused +by gaps in the data. Fourier methods require additional +data conditioning to deal with gaps, either through win- +dowing or data augmentation [42]. +Another modeling limitation of the current analysis is +that it treats the unresolved galactic signals as noise, +when in reality they are better described as a cyclo- +stationary stochastic signal. Going forward we will ex- +plore using a physically parameterized instrument noise +model [43], while treating the unresolved galactic bina- +ries as a separate, time-varying, stochastic signal [30, 43]. +Such a treatment requires that we include the (approx- +imately, at low frequency) noise-only TDI “T” channel, +and not just the A and E channels as is done in the cur- +rent version. Modeling of the galactic confusion will be +further improved by coupling the resolved UCB popula- +tion in the global fit to a physical model of the foreground +via parameterized priors for the spatial distribution of bi- +naries [36] and the overall number density of sources as +a function of frequency. +A source of non-stationary noise not well suited for +modeling with the power spectral density (or time- +frequency equivalent) are short duration noise transients + +MBH004799206 +MBH008746626 +MBH011167688 +0 +MBH011258537 +MBH011527103 +MBH011971300 +MBH013616801 +MBH016532115 +3.2 +MBH017244430 +(Mo)[×106] +MBH018605271 +MBH020426287 +MBH022228030 +MBH023440117 +MBH024408799 +MBH029515074 +5 +mi(Mo)[x106]MBH004799206 +MBH008746626 +MBH011167688 +MBH011258537 +MBH011527103 +MBH011971300 +MBH013616801 +MBH016532115 +MBH017244430 +MBH018605271 +MBH020426287 +MBH022228030 +7B023440117 +MBH024408799 +MBH02951507419 +FIG. 21: Marginalized distributions of MBHB sky location in ecliptic coordinates. The sky maps are ordered by merger time +from top left to bottom right. The different morphologies of the sky maps are due to the different maximum frequency and +duration of the signals, with lower mass binaries reaching high frequency and spending more time in band leading to precise +sky localization. Degeneracies in the sky location improve when higher harmonics are included in the waveforms. +or “glitches.” The path to incorporating a glitch model +(and, by corollary, a model for generic GW transients) +into GLASS has already been paved by similar work done +for LIGO-Virgo data [8] and theoretical demonstrations +using simulated LISA data [44]. There is already LDC +data that contain a simulated glitch population informed +by the LISA Pathfinder observations [45] and the in- +corporation of a transient noise module into the GLASS +framework is a near-term priority. +One final currently planned development direction for +GLASS is in the instrument model itself, enabling the +global analysis to start with lower level data products +than the TDI channels currently used. A data-driven ap- +proach to perform self-calibration coupled with the global +fit naturally propagates uncertainties at each stage of +the signal processing chain to the astrophysical inferences +made with the GW signals. Such capabilities may prove +to be important for science investigations where control + +60° +30° +30° +a0 +a0 +18 +0 +30° +30° +-60°60° +30° +30° +a0 +18 +30° +-60°60° +30° +30° +a0 +a0 +18 +30° +30° +60°60° +30° +30° +a0 +a0 +18 +30° +30° +60°60° +30° +30° +a0 +a0 +18 +30° +-60°60° +30° +30° +a0 +18 +30° +30° +60°60° +30° +30° +a0 +18 +30° +30° +-60°60° +30° +30° +a0 +18 +30° +30° +60°60° +30° +30° +a0 +a0 +18 +30° +-60°60° +30° +300 +a0 +18 +30° +30° +-60°60° +30° +30° +a0 +a0 +18 +30° +30° +60°60° +30° +30° +a0 +a0 +18 +30° +30° +60°60° +30° +30° +a0 +18 +30° +30° +60°60° +30° +30° +a0 +a0 +18 +30° +30° +60°60° +30° +30° +a0 +18 +30° +-60°20 +of systematic errors are vital, the most obvious example +of which would be testing the nature of gravity with high +S/N MBHBs or EMRIs. Already-demonstrated examples +of self-calibration methods for LISA include employing +UCBs as phase standards [46] and using the phasemeter +data to infer the light travel time between spacecraft for +cancellation of laser frequency noise and construction of +the TDI interferometer combinations [47, 48]. Another +possible capability to explore is the removal of noise in +the inter- and intra-spacecraft interferometer measure- +ments caused by angular jitter of the test masses mas- +querading as distance fluctuations–the so-called “tilt to +length coupling” inherent in the LISA measurement [49]. +An instrument module in the GLASS is valuable for quan- +titatively understanding and, if necessary, mitigating the +affect of calibration uncertainties an astrophysical infer- +ences. +As is clear from the long list of future work, the GLASS +architecture described in this paper does not represent a +finished design but instead is the scaffolding upon which +further development will be built. +Nevertheless, our +demonstrated results are an important way-point on the +path towards a fully functional pipeline ready for LISA +observational data. +VI. +ACKNOWLEDGEMENTS +Software: Results presented here used v2.0 of ldasoft, +a public C library which includes the noise, UCB, +VGB, and global fit samplers. +The MBHB sampler is +managed independently at LISA-Massive-Black-Hole. +Postprocessing and visualization tools for the source cat- +alogs are available the python package lisacattools +which in turn depends on numpy [50], pandas [51, +52], matplotlib [53], astropy [54], seaborn [55], and +ChainConsumer [56]. +The authors thank K. 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Zonca, and Astropy Project Contributors, The As- +tropy Project: Sustaining and Growing a Community- +oriented Open-source Project and the Latest Major Re- +lease (v5.0) of the Core Package, Astrophys. J. 935, 167 +(2022), arXiv:2206.14220 [astro-ph.IM]. +[55] M. L. Waskom, seaborn: statistical data visualization, +Journal of Open Source Software 6, 3021 (2021). +[56] S. R. Hinton, ChainConsumer, The Journal of Open +Source Software 1, 00045 (2016). + diff --git a/idE2T4oBgHgl3EQfHwb9/content/tmp_files/load_file.txt b/idE2T4oBgHgl3EQfHwb9/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9838fa345c31145a822bfc7eaf9eab91fc0f9204 --- /dev/null +++ b/idE2T4oBgHgl3EQfHwb9/content/tmp_files/load_file.txt @@ -0,0 +1,2081 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf,len=2080 +page_content='Prototype Global Analysis of LISA Data with Multiple Source Types Tyson B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Littenberg NASA Marshall Space Flight Center, Huntsville, Alabama 35811, USA Neil J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Cornish eXtreme Gravity Institute, Department of Physics, Montana State University, Bozeman, Montana 59717, USA (Dated: January 11, 2023) The novel data analysis challenges posed by the Laser Interferometer Space Antenna (LISA) arise from the overwhelmingly large number of astrophysical sources in the measurement band and the density with which they are found in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Robust detection and characterization of the numerous gravitational wave sources in LISA data can not be done sequentially, but rather through a simultaneous global fit of a data model containing the full suite of astrophysical and instrumental features present in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' While previous analyses have focused on individual source types in isolation, here we present the first demonstration of a LISA global fit analysis containing combined astrophysical populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The prototype pipeline uses a blocked Metropolis Hastings algorithm to alternatingly fit to a population of ultra compact galactic binaries, known “verification binaries” already identified by electromagnetic observations, a population of massive black hole mergers, and an instrument noise model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The Global LISA Analysis Software Suite (GLASS) is assembled from independently developed samplers for the different model components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The modular design enables flexibility to future development by defining standard interfaces for adding new, or updating additional, components to the global fit without being overly prescriptive for how those modules must be internally designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The GLASS pipeline is demonstrated on data simulated for the LISA Data Challenge 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Results of the analysis and a road-map for continued development are described in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' INTRODUCTION The mHz band of the gravitational wave spectrum is expected to contain an unprecedented abundance of galactic, extra-galactic, and cosmological gravitational wave (GW) sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The Laser Interferometer Space An- tenna (LISA) will survey the mHz GW band and provide unique observational constraints on the formation and evolution of compact binaries in the Milky Way, the ori- gin and growth of massive black holes throughout cosmic history, the dynamics of dense stellar environments in galactic nuclei, the fundamental nature of gravity and black holes, and more [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The richness of the LISA source catalog comes at the price of a more complicated analysis framework than is required for currently operat- ing GW observatories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' While aspects of the methodology developed for ground-based interferometers (many dis- crete sources) [2] and pulsar timing (overlapping sources, sophisticated noise modeling) [3] under-gird development of LISA analysis pipelines, new strategies are needed to account for the overwhelming number and density of GW signals in the LISA data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The fundamental challenge of LISA analysis stems from the large number (O(104)) and long duration (months to years) of detectable signals, resulting in non- negligible overlaps in time and frequency between dis- crete sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' As a result, analyses cannot treat sources independently and sequentially work through a list of candidate detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Instead, the LISA analysis has to be approached globally, simultaneously fitting complete data models including all of the detectable GW sources and the detector noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The need for a “Global Fit” was first described in 2005 [4], and has been identified as the primary challenge to the LISA analysis since early in the mission formulation [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' This has lead to a coordinated effort to develop capable algorithms well in advance of mission operations [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Global fit analyses are not unique to LISA, as there are analogous methods used elsewhere in GW astron- omy, and more broadly within astronomy and astro- physics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Gaia [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' For LIGO-Virgo analysis, the BayesWave pipeline simultaneously models Gaussian noise, non-Gaussian noise artifacts, and short-duration GW transients [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' PTA analyses use a global fit to si- multaneously model a correlated, stochastic gravitational wave background, a solar system ephemeris model, and multiple noise sources for each pulsar in the array [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Some PTA analyses also perform a global fit for mul- tiple source types, such as the signals from individual black hole binaries and a stochastic confusion noise from unresolved binaries [11], or perform a BayesWave-style analysis to reconstruct un-modeled burst signals [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Where the analogy ends is the scale of the LISA prob- lem compared to elsewhere in GW astronomy, evident in the number of sources that are part of the global anal- ysis, the diversity of source types (SMBHBs, EMRIs, UCBs, SGWBs, SOBHs, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' ), and data complications from multi-year integration times (glitches, nonstation- ary noise, gaps, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' In this paper we present the first demonstration of a LISA global fit analysis contending with multiple source types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' We call the algorithm the Global LISA Anal- ysis Software Suite, or GLASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' As the proving ground arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='03673v1 [gr-qc] 9 Jan 2023 2 for GLASS we use the simulated data released in the second round of the LISA Data Challenges (Challenge LDC2a-v2) [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The simulated data contain Gaussian detector noise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' a simulated population of Milky Way ultra compact binaries (UCBs);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 35 galactic UCBs al- ready discovered by electromagnetic observations, the so- called “verification binaries” (VGBs);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' and a population of merging massive black hole binaries (MBHBs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The philosophy of GLASS is to incorporate indepen- dently developed, stochastic sampling algorithms de- signed to address different facets of the full LISA anal- ysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' GLASS is then effectively an overarching umbrella that manages the interfaces, matches data formats, and orchestrates how the different samplers will work together to converge and adequately cover the target, high dimen- sional, joint posterior distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' GLASS uses a four-component model to fit the simu- lated LDC2a-v2 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The UCB, VGB, and MBHB mod- els each employ analytic template waveforms to fit the detectable sources, while the noise level, including the unresolved astrophysical foreground from the galactic bi- naries, is fit with a phenomenological model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The param- eters of the four model components are optimized using a blocked Metropolis Hastings (MH) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The test data set spans one year of simulated LISA observations however our analysis processes the data sequentially, an- alyzing increasingly large strides of the “observed” time series as we envision would be done during mission oper- ations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Results from the analysis of the LDC2a-v2 data are compared to the input populations to assess the per- formance of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Figure 1 summarizes the LDC2a-v2 results by show- ing the combined reconstructed model components rep- resented as the amplitude spectral density of the time delay interferometry (TDI) A channel [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The orig- inal data are shown in gray in the background of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The purple lines are the posteriors of the re- constructed UCB waveforms while orange are the recon- structed VGBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The magenta broadband curves are the reconstructed massive black hole signals, and the light blue curve is the noise model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Note that at low frequency the credible 50 and 90% credible intervals are visible in the black hole signals while otherwise the credible inter- vals on the reconstructions are too narrow to be visible in this plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' This figure represents the key result of this study: We are demonstrating a prototype pipeline able to simultaneously fit thousands of overlapping signals of dif- ferent morphology and an a priori unknown noise level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The remainder of this paper will provide a detailed de- scription of the algorithm and a demonstration of the per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Section II describes the analysis architecture, and how the different modules are integrated together into a global fit pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Section III A describes the noise model and sampling algorithm, adapted from the BayesLine algorithm used for LIGO-Virgo noise model- ing [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Section III B summarizes updates to the galac- tic binary sampler GBMCMC first described in Ref [16], FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 1: Median reconstructed global fit model components for the full 12 month LDC2a-v2 data, shown as the ASD in the TDI A channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Gray is the residual, purple are the UCB detections, orange or the fits to the known binaries, magenta are the MBHB mergers, and light blue is the noise model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' while Section III C specifies the configuration changes for GBMCMC to perform targeted analysis of known binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Section III D describes how the MBHBMCMC massive black hole sampler from Ref [17] was adapted for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Section IV presents the evolving results from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='5, 3, 6, and 12 month analyses of the LDC2a-v2 data before we conclude in Section V with a development road map to improve GLASS’s capabilities in order to analyze increas- ingly realistic LISA data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' THE GLASS ARCHITECTURE The central engine of GLASS is a blocked Markov Chain Monte Carlo sampler [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' In the blocked sampling scheme, a subset of the model parameters (a block) is updated while holding all other parameters fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Differ- ent blocks are updated independently in sequence, and the process cycles until the sampler has converged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' For the LISA Global Fit problem, blocked MH samplers have two advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' First, they work well for high dimension spaces when parameter correlations are confined to rela- tively small and a priori identified sub spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Second, they are naturally modular, turning the daunting task of building an algorithm equal to the complexity of LISA data into a well defined set of components that are de- veloped independently and then integrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The blocked MH scheme in GLASS is hierarchical where the top level blocks, which we will refer to as “modules,” are the joint set of parameters for the different model components i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=', blocks for the noise, VGB, UCB, and MBHB parameter sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The sampling within the VGB and MBHB modules is further grouped into blocks by individual sources, while the UCB module has one more layer of hierarchy–where model parameters are grouped by narrow-band frequency segments, and then by indi- 10-18 10-19 1020 10-21 10-22, 10-23 10~24, 10~4 10~3 10~2 f (a)3 vidual sources within each segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Each module uses a customized parallel tempered Markov Chain Monte Carlo sampler developed indepen- dently of one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The role of GLASS is to coordinate which blocks are updating and to exchange state data be- tween modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The modules work on their own subset of the data, use their own likelihood function, tempering scheme, proposals, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=', and in principle could even use a different representation of the data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=', time series, frequency series, or wavelets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' In the current implemen- tation of GLASS each module is working in the frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Figure 2 is a schematic diagram for how modules op- erate on different bandwidths of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Note that the colors indicating each module will be consistent through- out this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The noise model is fit over the full fre- quency measurement band of the data (light blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' For the LDC2a-v2 data we take that to range from ∼10−5 to ∼30 mHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Massive black hole mergers are broadband signals where the maximum frequency is determined by the total mass of the binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The MBHB module band- width is dynamically based on the source parameters, but generally extends up to a few to O(10) mHz (ma- genta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The UCBs are narrow-band signals, generally spanning ≲ 10 µHz, but are by far the most numerous of the LISA sources, and will be found throughout the measurement band of our analysis, though sparsely above 10 mHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The UCB module consists of several instances of the same sampler, each focusing on a band-limited segment of data (purple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The bandwidth of each seg- ment depends on the frequency, using smaller segments where sources are most densely spaced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Finally the VGB module is conducting a narrow-band targeted analysis for individually known sources (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Known Binaries Ultra-compact Binaries Massive Black Hole Mergers Noise f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 2: Schematic block diagram for how frequency domain data are segmented by the different model components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The noise module must cover the full frequency range (light blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The MBHB module is broadband, covering almost the same width as the noise model (magenta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The UCB module divides the data into narrow-band, overlapping, segments (purple), while the VGB model targets only the frequency range spanned by each individual known binary (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' To understand the interface between modules consult the joint likelihood function for the global fit: p(d|θ) = (2π)− N 2 det (C(θnoise))− 1 2 e− 1 2 (d−h(θMBHB)−h(θUCB)−h(θVGB))T C(θnoise)−1(d−h(θMBHB)−h(θUCB)−h(θVGB)) (1) where d is the data, N is the number of data points, C is the noise covariance matrix, θ represents the full parameter set, θi are the model parameters in the block for module i, and h are the co-added detector responses to the modeled sources in each module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' For example, h(θMBHB) is really shorthand for � n h(θn MBHB) where n is indexing all of the sources in the MBHB model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Sticking with the MBHB example, the sampler adopted for that model was developed assuming no other sources in the data, and a known noise covariance ma- trix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The internal likelihood for the kth component of the MBHB module is then just p(d|θk MBH) = e− 1 2( ¯d−h(θk MBHB)) T C−1( ¯d−h(θk MBHB)) (2) where the normalization term is absent because the sampler only considers the likelihood ratio between two points in parameter space and the covariance matrix is in- dependent of the (MBHB) parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' To interface this sampler with the rest of GLASS at the beginning of each one of the MBHB blocks’ updates the covariance matrix is replaced based on the current state of the noise model θnoise and the “data” as seen by the MBHB sampler is the residual after subtracting all other model components– as far as the MBHB sampler for the kth MBHB is con- cerned ¯d = d−h(θUCB)−h(θVGB)−� n̸=k h(θn MBHB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The MBHB sampler is otherwise blissfully unaware of what is happening in the rest of the global fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' It is the job of GLASS to keep track of the current state of each module, prepare the effective data and noise covariance matrix, and refresh the likelihood (using the new effective data and noise model) of the current state before a sampler updates it’s block of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' An identical argument applies to the other modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The individual samplers for the modules have been de- veloped and published elsewhere and will be briefly sum- marized in later sections before focusing on updates made to each of them for the LISA global analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' To under- stand how the data is shared between modules a block diagram of the workflow is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The dia- gram is a simplified version of the true workflow, depict- ing only three UCB and MBHB nodes each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' In prac- tice GLASS uses several hundred UCB nodes and one MBHB node per source in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The noise, VGB, UCB, and MBHB model updates are executed by the BayesLine, VBMCMC, GBMCMC, and MBHBMCMC blocks, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' GLASS uses the Message Passing Interface (MPI) standard to exchange information between the dif- ferent modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' For shorthand we will refer each MPI 4 process as a “node” of the analysis, though in practice we use multiple MPI processes per node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The BayesLine module uses a single node which also serves as the root process responsible for the work shared by all nodes and the overall orchestration of the analysis (P0 in the flow chart).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' At start up the root node handles data parsing, selection, and conditioning before broad- casting the data and the initial state of each sampler to all other worker nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' During each iteration of the sam- pling nodes P0 and P1 first update their parameter blocks for the noise model and VGB model, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' At the end of the noise and VGB module updates (each involv- ing several internal MCMC steps for each model), the VGB process sends the current state of the VGB model in the frequency domain to the root process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The root process then broadcasts the current state of the noise model and VGB model to the UCB (P2-P4) and MBHB (P5-P7) nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The UCB and MBHB processes then cre- ate their respective residuals and update their block of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Each UCB process is responsible for a narrow-band segment of the data but care must be taken at the seg- ment boundaries where individual sources can span the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Each node shares its current state with the neighboring nodes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' the adjacent frequency segments) so that the receiving node can remove the state of the neighboring model when forming the residual that will effectively serve as the data for the current update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The segments overlap in frequency and each node is responsi- ble for fitting to the sources in its half of the overlapping region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' This overlap, which is set to be a factor of two larger than the typical bandwidth of a source at that fre- quency, ensures that the templates for sources located near the boundaries are not artificially truncated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' To preserve the correlations between sources that are close to one another on either side of a boundary, the UCB nodes alternate which block is updating and which is waiting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' For example, all of the odd-numbered processes will update their parameters, exchange with their neigh- bors, and then the even processes will update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' For the MBHB modules, additional pre-processing is needed once their residuals are formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The MBHB mod- ule relies on the heterodyned likelihood described in [19– 21] and recomputes the coefficients based on the current state of the sampler, as well as updating proposal dis- tributions used within the sampler that use the informa- tion matrix as an approximation to the covariance matrix of the posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' After that pre-processing each MBHB module updates its parameter block in parallel with the other MBHBs in the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' At the end of the UCB and MBHB updates each module sends the current state of its model in the frequency domain to the root process to broadcast to all of the other nodes, and the entire cycle repeats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' After many iterations when the sampling has finished each process performs a minimal level of post- processing to prepare for the next stage of the pipeline when the posterior samples are consolidated into a source catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Note that in the traditional blocked MH sampling scheme only one block of parameters are updated at a time while the others are held fixed whereas GLASS is up- dating blocks of parameters in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' This was a choice made to improve the computational efficiency of the algo- rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The current bottlenecks for the analysis are the heterodyne step for the MBHB model and the conver- gence time for the highest frequency UCBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Those two aspects of the problem set the scale for the cost of each it- eration and the number of iterations needed, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' To maximize efficiency, we do enough MBHB parameter updates to match the cost of updating the heterodyne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The number of internal GBMCMC updates per cycle of the full sampler are then dynamically adjusted to take ap- proximately the same amount of computational time as the MBHB models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The comparative costs of the noise model and VBMCMC updates are significantly lower, similar to the costs of the data sharing and common processing that must get done before each iteration (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' writing results to file, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' This scheme was thus created to maximize the duty cycle of individual nodes by mini- mizing the amount of time nodes are blocked waiting for other processes to finish their work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' While techni- cally violating the conditions needed to have the resulting samples be representative of the target posterior distri- bution function, the effects are only noticeable for blocks that are correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Updating alternating UCB blocks ensures that no correlated UCB parameters are being altered in parallel as the data segments for each UCB node are larger than the bandwidth of a single UCB sig- nal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Similar arguments can be made between other blocks being updated in parallel although a production analysis on observational data requires more through testing for confirmation, and/or a more conservative approach and a higher computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' It is a trivial rearrangement of where the MPI exchanges take place to revert to the traditional serial update of all parameter blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Each of the samplers use parallel tempering [22] to im- prove the mixing of the chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Parallel tempering is es- pecially critical for promoting transitions between mod- els in the trans-dimensional samplers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The tempering scheme is independently developed and tuned for the dif- ferent modules, and only the zero temperature chain pa- rameters or state are shared between different processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Each of the parallel tempering samplers is multi-threaded ideally using a single CPU per chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' In practice there is a trade space between the number of resources needed for the analysis and the amount of time those resources are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Processing of the LDC2a-v2 data was done on Amazon World Service (AWS) cloud computing in- frastructure which favors smaller-scale jobs running for longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' As a result the LDC2a-v2 analysis used multi- ple threads per CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The final configuration for the full LDC2a-v2 analysis (12 months of simulated data) used 624 MPI tasks and 12 CPUs per task for a total of 7488 CPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The run was deployed on 78 × 96 CPU nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The noise model and verification binary modules used one MPI task each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' There were 15 MBHB mergers in 5 the data each run with a dedicated MPI task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The re- maining 607 MPI tasks were dedicated to the UCB model covering .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='03 to 23 mHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' DESCRIPTION OF THE INDIVIDUAL SAMPLERS The individual model components that are integrated into the GLASS architecture are independently developed, described, and published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Each is still under active de- velopment so it is useful to overview each sampler with emphasis on updates that have been made since the most recent publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Global Noise Model For the noise model we use an adaptation of the BayesLine algorithm originally developed for LIGO- Virgo noise modeling [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The original BayesLine algo- rithm fits the power spectral density (PSD) of the noise Sn(f) independently in each detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The LIGO-Virgo version of the pipeline uses a two-component fit to phe- nomenologically model the noise spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The main component is a broadband noise spectrum that looks sim- ilar to a sum of power laws, with steeply rising noise at low frequency and a more gradual increase at high fre- quency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Modeling the actual LIGO-Virgo data with a broken power law is not sufficiently flexible so BayesLine uses a cubic spline interpolation where each spline con- trol point i is parameterized by its frequency and PSD level [f i, Si n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The location of the spline points, as well as the total number, are free parameters sampled over with a trans-dimensional MCMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' For the LIGO-Virgo application BayesLine also includes a linear combina- tion of Lorentzians to fit the narrowband features in the spectrum due to calibration lines, the power supply, res- onances of the mirror suspension system, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The sim- ulated LISA data do not contain narrowband noise fea- tures and so GLASS’s implementation of BayesLine does not use the line model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Note, however, that there were spectral lines in the LISA Pathfinder data and so future version of the model will need such a feature [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' There are two important differences between the BayesLine im- plementation integrated into GLASS and that which has been used for LIGO-Virgo data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' First, the LISA noise spectrum spans the frequency regime where finite arm length effects of the detector response are in the measurement band, unlike ground- based interferometers which operate entirely in the long wavelength limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The arm length manifests in the noise spectrum as sharp features where the PSD, and in- strument response, formally go to zero for signals with wavelength that fit an integer number of cycles within the detector arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Mathematically this is a result of terms proportional to sin2(f/f ∗) appearing in the de- tector response functions with the “transfer frequency” f ∗ ≡ c/2πL where c is the speed of light and L ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='5 Gm is the arm length of the LISA detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The result- ing spectrum is not well modeled by a spline interpola- tion at high frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' However, the difficult features for a spline interpolation to track are a purely geomet- ric effect set by the size of the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' We therefore model the difference between a reference noise spectrum, including the geometric effects, and the observed data Smodeled n = Sobserved n − Sreference n , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' we are fitting for broadband differences between the reference noise level, derived from the current best estimate of the LISA per- formance, and the observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Where the reference model is accurate the modeled PSD will be consistent with zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Second, the interpolation between control points for the GLASS application of the spline model employs Akima splines [24] rather than the cubic splines used in the LIGO-Virgo applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The Akima splines are less prone to oscillations between control points by relaxing the requirement of a continuity in the second derivative of the interpolated curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The tendency for cubic splines to oscillate is exacerbated by the large dynamic range and steeply changing spectrum at low frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Akima splines perform better on the LISA spectrum and are worth considering for LIGO-Virgo noise modeling as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' GBMCMC Updates The UCB sampler is the GBMCMC pipeline described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The GBMCMC application is the latest in a long line of algorithms designed for the LISA galactic bi- naries which partition the frequency domain data into many narrow-band segments and uses model selection to determine the number of detectable binaries in each segment [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The model selection method of choice used by GBMCMC is a transdimensional, or reversible jump Markov Chain Monte Carlo (RJMCMC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Of the different samplers integrated by GLASS GBMCMC has seen the most additional development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The sam- pler was updated to use multi-threading for the parallel tempered chains, making a significant improvement in the run time especially when leveraging the increasingly large number of CPUs available per node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The pipeline has also updated the way results from previous runs are incorporated as proposal distributions for subsequent analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' As described in Ref [16], the LISA data are processed in increasingly-long time epochs, starting with the first 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='5 month segment of data and re-processing each time the available data has doubled (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=', after 3, 6, and 12 months).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' In the first version of the GBMCMC pipeline multivariate Gaussian proposals were built using the covariance matrix of the posterior samples for each UCB in the catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' In the latest version the single Gaussian proposal was replaced by a Gaussian Mixture Model (GMM) which is fit using the Expecta- tion Maximization (EM) algorithm run on the posterior samples for sources in the previous epoch’s catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Eval- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='P0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='P4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='P3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='P2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='P0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='Read Data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='P0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='P3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='P2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='P1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='P5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='P6 ' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Process P0 is the root process and runs the noise model sampler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Processes P1 to P3 are for the UCB model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Processes P4 to P6 run the MBHB model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Gray is the blocked MH sampler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Purple are the NMBHB independent MBHB sampling steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Green are the NUCB coupled UCB processes, which exchange only between adjacent segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Orange is the Noise model which is run on the root processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Data from all processes are shared with, and broadcast from, root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' In practice O(103) UCB processes are needed, and O(10) MBHB processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' uating the GMM proposal is more computationally costly than the single multivariate proposal, but we have found it to be offset by the improvement in convergence time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The GBMCMC sampler now also includes a basic “split- merge” proposal for trans-dimensional steps, whereas the original algorithm only used “birth-death” moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' A birth-death move chooses to either remove or add a fea- ture to the model (in GBMCMC’s case, a source from or to the fit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The split-merge proposal attempts to divide a single feature into two, or combine a pair of features 7 into one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The current implementation of the the split- merge proposal is naive, choosing to remove one source and replace it by two drawn from the same distribution as is used by the birth-death moves, or to remove two of the current sources and replace them by a single draw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' In other words, the current split-merge proposal is really two birth moves and one death move, or two death moves and one birth move, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Further development of more efficient split-merge proposals will be a critical area to improve the sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Finally, in the previous applications of the GBMCMC sam- pler the frequency segments were of equal bandwidth over the entire observing band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Because each segment was analyzed independently the overall number of segments (and therefore nodes) needed for the analysis was not a limiting factor in its deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Within the GLASS architecture when all of the processes are communicat- ing via MPI we need to be more parsimonious about the number of segments being analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' To that end we adopted an adaptive segment size depending on the source density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' At low frequency where the source density is the highest the segments are more narrow, and at high frequency where the source density is low (and the sig- nals have larger bandwidth) the segments are wider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The exact segmenting was fixed before the LDC2a-v2 analysis was started and kept the same for each epoch’s analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' A more efficient approach would be to use the previous epoch’s catalog to dynamically determine where to best place the segment boundaries to both keep the source density per segment near constant, and to avoid hav- ing loud signals near segment boundaries as an insurance policy, even though the “edge effects” of the segmenting are already ameliorated by GLASS’s data sharing scheme between UCB segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' VBMCMC Updates The verification binary sampler (VBMCMC) in GLASS is identical to GBMCMC but is run in a different configura- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Whereas GBMCMC is performing a blind search for UCBs in the LISA data, part of which includes a model selection step to determine if a candidate source in the data is detectable, VBMCMC is executing a targeted analy- sis of binaries which have already been identified as LISA sources by EM surveys [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The VBMCMC sampler therefore uses a fixed-dimension analysis with priors on the orbital period and sky loca- tion of the binaries derived from the EM observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Because the sky localization from the EM observations is orders of magnitude more precise than LISA will ever achieve, the sky location parameters in VBMCMC are fixed to the EM-observed values, effectively using delta func- tions for the priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The same is true for the orbital period of the binaries (converted to GW frequency for the VBMCMC model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' While it is possible that some of the currently-known binaries’ orbital period measurements are not as precise as what LISA will infer, the long tem- poral baseline of EM observations should effectively pin the orbital period for LISA observations assuming con- tinued effort to periodically monitor the known binaries prior to LISA’s launch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' For the known binaries where this is not the case, replacing the delta function prior on GW frequency with a Gaussian distribution with width from the EM uncertainties is a trivial change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Using a fixed-dimension targeted search for the known binaries instead of retroactively extracting the known bi- naries from the full UCB source catalog allows for upper limits to be set on binary parameters (most notably the GW amplitude) in the event that some of the known binaries are below the detection threshold at the time of the global fit analysis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' perhaps due to elevated lev- els of the astrophysical foreground (confusion) noise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' or because they will require longer integration times with LISA before becoming detectable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The targeted VBMCMC analysis will also reduce contam- ination of known binaries from other loud sources at sim- ilar orbital periods, as many of the currently know bina- ries are at frequencies where the galactic source density is expected to be highest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Analysis of known binaries will be particularly vulnerable to source contamination early in the LISA mission when the frequency resolution and integrated signal to noise levels are still improving at the same time as when UCB observations may play an important role in the early phases of instrument charac- terization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' MBHB Updates The MBHB module uses elements of the LISA-Massive-Black-Hole-Binary pipeline origi- nally developed for low-latency detection and parameter estimation of massive black hole mergers with LISA [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The pipeline starts with a pre-processing search phase that uses short stretches of data (typically a few weeks), treating the galactic foreground as a noise source, and us- ing a maximized likelihood function to rapidly lock on to any massive black hole binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The search is repeated on each segment of data after subtracting the previously found signals until no additional sources above a S/N threshold are found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The rapid search is then followed with a full MCMC exploration of each source, taken one at a time, that refines the parameter estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' These es- timates are then used as the starting point for the MBHB analysis in GLASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The same PTMCMC sampling routine is used in the global fit, but now with the noise model replaced by the spline model, and with the resolved UCBs subtracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Another key difference is that the model components are updated in an alternating fashion, in contrast with the low latency analysis where the noise model is fixed and the MBHB signals are analyzed sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The MBHB block of parameters is updated as follows: When it comes time to update a particular MBHB, the sampler receives the current state of the residuals residual 8 constructed from the other model components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' That is, the original data with the current state of the combined UCB and VGB models, as well as the other MBHB wave- form models, subtracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' A key element of the MBHB sampler is the use of heterodyning to accelerate the like- lihood calculations [19–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The heterodyne is computed using a reference waveform–in this case one based on the parameters of the MBHB model at the end of the last update, and the current state of the residual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The com- putational cost of setting up the heterodyne is equal to a few times the cost of a standard likelihood evaluation, so to be cost effective it makes sense to perform hundreds of iterations with the fast heterodyned likelihood before moving on to the next block in the sequence of updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The current version of the MBHB sampler uses cus- tomized implementation of the IMRPhenomD waveform model which describes the dominant (2, 2) harmonic for spin-aligned, quasi-circular binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' In future versions of the sampler the waveform model will be generalized to include spin precession effects, sub-dominant waveform harmonics, and eventually, orbital eccentricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Another limitation of the current implementation is that the dimension of the MBHB model is fixed to what- ever was found by the low latency search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The MBHB model also needs to be trans-dimensional with the results from the search phase being used to propose adding or removing sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' DEMONSTRATION Having described the overall GLASS architecture we now turn to a demonstration of the pipeline’s perfor- mance on simulated LISA data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The test data set was produced by the LISA Data Challenge (LDC) team and is the first of the LDC data sets to contain a combination of different source types [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The following demonstra- tion of GLASS’s current capabilities uses the LDC2a-v2 data which contain ∼ 30 million galactic UCBs, 37 VGBs, 15 MBHB mergers, and an unspecified instrument noise level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The LDC2a-v2 data span one year of LISA obser- vation time assuming a 100% duty cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' In reality there will be periodic and sporadic interruptions to the data taking which will require further development of GLASS’s noise model and likelihood functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' For each LDC simulation there are “blind” and “train- ing” data sets, where the training data contain the list of signals (injections) used for the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The blind data are simulated using a different realization of the same population that is found in the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' For this demonstration the training data are used, enabling assessment of the pipeline performance through compar- isons of the resulting source catalog to the injected sig- nals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The LDC2a-v2 data TDI channels are generated assuming an equal arm interferometer with stationary instrument noise such that the TDI “A” and “E” chan- nels are noise-orthogonal [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The GLASS noise model correspondingly uses an independent fit to the A and E instrument noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' This simplifying assumption will need to be relaxed for analysis of the observational data but will only effect the computational cost of the analy- sis by introducing correlations between TDI data streams which result in non-zero off diagonal terms in the noise covariance matrix at each frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The resulting like- lihood evaluations require more operations to compute (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 1) but will not effect the overall complexity of the global fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 4: Amplitude spectral density (ASD) of TDI A channel data analyzed by each block of the sampler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The UCB segments are shown in alternating shades of purple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Note the changing bandwidth of the UCB segments, which are larger where the source-density is lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' VGB segments are orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The MBHB (magenta) and noise models (blue) cover the full analysis band, with the noise model extending to slightly higher and lower frequencies for margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' This example uses the first 6 months of the LDC2a-v2 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Fig 4 shows the amplitude spectral density (ASD) of the TDI A channel after the first six months of the LDC2a-v2 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Through the remainder of the paper, the A channel will be used to visualize the data and/or sig- nal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The differences between the A and E channel data are subtle and not informative at this level, though they are crucial for the analysis to decompose the ob- served signal into the the two GW polarization states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The data shown in the figure are colored over the in- tervals being analyzed by different model components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The noise model (light blue) covers the full frequency range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The MBHB model (magenta) spans a similar bandwidth, though there is additional padding for the noise model to ensure that it extends beyond where the MBHB signals are in band during the time they are ob- servable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The UCB segments are shown in alternating light and dark purple bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Though it is difficult to discern from the figure, especially due to the frequency axis being on a log scale, the width of the UCB segments is frequency-dependent, roughly tuned to use narrower segments where the source density is higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Finally the locations of each narrow-band segment for the targeted 10-19 10-20 10-21 10-22 10-23 10-3 10-2 f (Hz)9 VGB analyses are shown in orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Throughout the re- mainder of the paper the same color scheme will be used to identify different model components: Blue for noise, purple for UCBs, orange for VGBs, and magenta for MB- HBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 5: Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 1 but focused on a narrow frequency band near 6 mHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The known binary in this segment of data (orange) is representative of how HM Cnc will appear in the LISA data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' For the headline demonstration of GLASS at work, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 5 shows the reconstructed components of each part of the data model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The figure is showing the same con- tent as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 1 but zoomed in to a narrow interval around 6 mHz containing one of the loudest currently known sources, HM Cnc [29], shown in orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Here we can see all of the model components on display, with a densely- packed collection of UCBs in addition to HM Cnc all overlapping one another (purple), and the MBHB merg- ers sweeping through the band (magenta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The gray curve depicting the residual after all model components have been subtracted from the data is fit by noise model shown in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Note that in this figure the uncertainty in the reconstructions is thinner than the line widths in the figure, as all of the sources in this interval have high signal to noise ratio (S/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Broadening the aperture to the full analysis band of the demonstration, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 6 shows the original data’s ASD which is dominated by the UCBs and is thus shown in purple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Removing the resolved UCBs (and VGBs) leaves the magenta residual containing a bump in the spectrum from the combined signals of the MBHBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The final (light blue) residual is after all of the resolved GW signals in the fit are subtracted from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The remaining bump in the residual spanning ∼3 × 10−4 to ∼5 × 10−3 Hz is due to the foreground of un-resolvable UCBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' As described above the analysis is repeated on increas- ingly long epochs of the full data set, starting with the first 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='5 months of observations going up to the full year of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Analyses are conducted each time the data vol- ume has doubled, resulting in analyses of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='5, 3, 6, and 12 month segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' As the observing time increases the FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 6: ASD of the data including all signals (purple), after removing the fit to the resolvable UCBs leaving behind only the MBHBs (magenta), and then the final residual after all signals in the fit are removed (light blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 7: Number of UCB (top) and MBHB (bottom) detections as a function of observation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The UCB detection number is the number of candidates from the maximum a posteriori model after clustering samples by waveform match and then selecting candidates with z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='5 (lightest shade), z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='9 (medium shade), and those that uniquely correspond to a source in the injected population with a match of m > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' See Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' IV B for a full explanation of the match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' number of detectable signals grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' For UCBs, which are continuous sources, this is due to the source building signal power over time and the improving frequency res- olution of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The MBHBs are transient sources so longer observation times provide more opportunity to catch a black hole merger in the act.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 7 shows the number of candidate detections in the source catalogs for the UCB (purple, left) and MBHB (magenta, right) mod- els as the observing time increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The drop in the total number of UCB detection candidates between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='5 and 3 10-19 10~20 10-21 10~22 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='210 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='216 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='220 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='226 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='230 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='236 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='240 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='246 f (mHz)10-18 10~19 10~20 410-21 10~22 10-23 1024 10-4 10~3 10~2 f (Hz)8000 6000 EDJ 4000 2000 0 15 NE 10 9 12 ntha10 months is unexpected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' However, the number of confident detections that are clearly associated with an injected sig- nal increases, as will be described in section IV B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Time- dependent analyses of data with such short observing times will be particularly sensitive to the initial orienta- tion of the spacecraft constellation relative to the galactic center, and the modeling of the time-varying noise due to the galactic foreground [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' While needing further study, our assessment is that the initial conditions of the LISA orbits and our admittedly incorrect assumption of stationary noise lead to this counter-intuitive result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Having summarized the GLASS performance on a data- wide level, we now take a detailed look at the perfor- mance of the individual model components by studying the properties of the recovered source catalogs and com- paring them to the input populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Noise Model The instrument noise model parameters used in GLASS are not physically meaningful and so the primary diag- nostics for the performance of the sampler are functional tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' In each cycle of the GLASS blocked MH sampler, the PSD model is fitting the residual after the current state of the UCB, VGB, and MBHB models have been sub- tracted from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' That residual includes the instru- ment noise as well as the unresolved galactic foreground, referred to in the LISA literature as “confusion noise,” which is expected to be the dominant source of residual power between ∼10−1 and ∼3 mHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Exactly where the galactic foreground drops below the instrument noise de- pends on details of the galactic population of compact binaries [31], the performance of the LISA instrument, and the observation time [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 8 shows the power spectrum of the 12 month data A channel (dark gray), and the residual after removal of a fair draw from the joint UCB+VGB+MBHB model (light gray).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The colored lines are the PSD fits from the noise model for the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='5, 6, and 12 month runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The 3 month result is omitted for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The black dashed line is the PSD used for simulating the instrument noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' In each of the PSD fits the spline model used ∼15 to ∼30 control points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The results clearly show how the prominent bump in the spectrum where the astrophysical foreground dominates initially grows with time across the band as the joint S/N of the galaxy increases, and then is slowly reduced as the UCB model is able to resolve more binaries, particularly at higher frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Outside of the interval dominated by the astrophysical foreground, the modeled PSD matches the simulated levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='5 month PSD fit is truncated at low frequency because the bandwidth of the noise fit is dynamically set based on the signal content of GLASS which does not include any MBHB merger signals in the first month of the LDC2a-v2 data, alleviating the need to model the PSD below ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='3 mHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' A more quantitative assessment of the PSD model FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 8: Median inferred noise PSD for three (purple), six (green), and 12e (orange) month observing times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The black dashed line is the true PSD used when simulating the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' For reference, the dark gray is the power spectrum of the 12 month data, and the darker gray is the residual after removal of the UCB, VGB, and MBHB models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The difference between the inferred and true PSD between ∼ 2 × 10−4 and ∼ 6 × 10−3 Hz is due to the unresolved galactic foreground, or “confusion noise.” is possible by testing the whitened data ˜w(f) ≡ ˜d(f)/ � Sn(f) where the tilde denotes a Fourier trans- form, d is the data, and Sn(f) is the PSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The PSD is proportional to the frequency-dependant variance of the noise and therefore the whitened data should be consis- tent with a zero mean unit variance normal distribution N[0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 9 shows histograms of the combined real and imaginary components of the Fourier transformed whitened residuals for the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='5 (left, purple), 6 (middle, orange) and 12 (right, pink) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Displayed above each panel is the mean and standard deviation of the whitened residuals, in agreement with the expected results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' While the performance of the noise model passes the tests pre- sented here we know that the model is incomplete and demands further development to meet the challenges of the real observing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The primary limitation of the current model is the implicit assumption that the noise is stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' In practice the LISA noise will have a time- varying PSD due to secular and random fluctuations of the instrument performance, as well as the cyclostation- ary modulations of the galactic foreground imparted by LISA’s orbit [30, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Generalizing the noise model us- ing time-frequency methods [35] is an immediate priority for future development of GLASS, and has ripple effects through the rest of the model components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' UCB Catalog The UCB catalog contains ∼2000 candidate detections after the first 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='5 months of observing, climbing to ∼8500 by the end of the 1 year LDC2a-v2 data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 10 shows 10-36 10-38 TDI A Channel Sn(f) 10-42 44 10-46 10-48 10-4 10-3 10-2 f (Hz)11 (a) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='5 mo (b) 6 mo (c) 12 mo FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 9: Distribution of the whitened data ˜w(f) = ˜d(f)/ � Sn(f) for the 1 month [left], 6 month [middle] and 12 month [right] whitened residual data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' If the PSD model is functioning correctly the whitened data should be distributed as a zero mean unit variance Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The mean and standard deviation computed from the whitened data are printed above each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 10: Scatter plot of frequency f and GW amplitude A of UCBs in the 12 month source catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The black line is an example LISA sensitivity curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' a scatter plot of the point-estimate frequency and GW amplitude parameters (f, A) of the recovered sources in the catalog after analysis of the full 12 month LDC2a-v2 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The black line is a representative LISA sensitivity curve, so that the S/N of each source is proportional to its height above the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The “cavity” of sources above the curve at low frequency is due to the foreground of unresolved galactic binaries becoming the dominant noise source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Distilling the output of the GBMCMC sampler to a dis- crete list of catalog sources is a nuanced and lossy pro- cess, a detailed description of which is found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' To summarize: In each frequency segment the maxi- mum a posteriori (MAP) number of source templates used to fit the data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=', the number of templates used most frequently in the RJMCMC sampler) is selected FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 11: LDC0100498745 parameter estimation over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Green, orange, and pink contours correspond to the measurement after 3, 6, and 12 months of observing with LISA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The contours mark the 1 and 2σ credible intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The sampling parameters from GBMCMC are re-parameterized into orbital period and derivative (P, ˙P) [top left];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' amplitude and inclination (A, ι) [top right];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' galactic latitude and longitude (l, b) [bottom left];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' chirp mass and luminosity distance (M, DL) [bottom right].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The black dashed lines show the injected parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' as the reference model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The posterior samples from that model are clustered into discrete catalog entries using the match m ≡ (hi|hj)/ � (hi|hi)(hj|hj) between the wave- forms computed from the chain samples at step i and j where (·|·) is the standard noise-weighted inner prod- uct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The threshold for considering a chain sample as a member of a cluster is m > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The fraction of the total number of steps in the chain that have a sample in a particular entry is interpreted as a detection confi- dence z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The threshold for inclusion in the final catalog is z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='5 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=', that a catalog entry has a sample in more than half of the total number of chain steps in the MAP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' This is not a strict criteria, roughly equating to sources with a Bayesian odds ratio > 1 being included in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' We therefore use an additional threshold of z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='9 for catalog sources to be considered confident detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Neither the match requirement for inclusion as a sample belonging to a catalog entry, or the frac- tion of samples from the chain that an entry contains, are extensively tested or optimized through large scale injection studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Such critical work must be thoroughly undertaken in advance of using GLASS or anything like it for production analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' As an example of the content contained for a single UCB, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 11 shows a set of marginalized 2-dimensional posterior distributions for a high S/N binary found near 10 mHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' UCBs in the GLASS catalog are identified by their median frequency, so in the 12 month catalog this binary was labeled LDC0100498745.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' See [16] for a dis- w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='98 ww = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='99 w1020, 10~21, 1023 , 10-4 10~3 10~2 f []P [s/s] [×10-10] P [s] [×10-4 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='99007×102A「×10-22 t [deg[deg] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='5 [deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='03 months 06 months 12 months DL [kpc] M [Mo]w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='9912 cussion and demonstration of how UCB candidates are traced through versions of the source catalogs from earlier analysis epochs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' tracking how this particular source was labeled in the 6 month catalog, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Shaded con- tours are the 1 and 2σ credible intervals, and the colors correspond to observing time with blue green, orange, and pink representing 3, 6, and 12 months respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' As with the color-coded source types, throughout the pa- per these colors (along with light-purple for 1 month) will consistently represent the observing times in subsequent figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The posteriors are represented in a different pa- rameterization than is used in the sampling, to better match the observables customarily used by the EM ob- serving community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The top left panel shows the orbital period P and first derivative ˙P of the binary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Note how the measurement precision of ˙P increases more rapidly than other parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' This is because the or- bital evolution of the binary enters the phase as a T 2- dependent term, so the information accumulates more rapidly than the typical √ T scaling due to the increasing S/N of a continuous source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The top right panel shows the gravitational wave amplitude and the binary’s or- bital inclination in degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The bottom left panel is the sky location in galactic coordinates, with l as the galac- tic latitude and b the galactic longitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The bottom right panel are the chirp mass M and luminosity dis- tance DL parameters derived from the GW observables assuming the orbital evolution is purely driven by emis- sion of gravitational waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The horizontal and vertical lines mark the parameter values for the injected signal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' the “right answer” when we have the luxury of know- ing the true source parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' One intriguing opportunity afforded by the LISA UCB catalog is to map the Milky Way’s stellar remnant popu- lation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 12 shows the map of the UCB sky in galactic coordinates after 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='5, 3, 6, and 12 months from top left to bottom right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The maps are constructed by combining the posterior samples from all of the sources in the UCB catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' After only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='5 months of observing large scale galactic features like the bulge and disk are evident in the maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The resolution of the image continues to improve as the observing time increases, revealing a remarkably clear view of the galactic disk and bulge with hundreds of well-localized sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The quality of the image will steadily improve over the LISA mission life time, and will include distance information from the chirping binaries, enabling three-dimensional inferences on the spatial dis- tribution of binaries throughout the galaxy [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' With the benefit of knowing the input source popula- tion, the observed UCB catalog is compared to the in- jected binaries to study the detection efficiency of the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The primary metric for assessing the the qual- ity of the inferred source catalog is the maximum match m between the waveform computed from the point es- timate source parameters in the catalog and the wave- forms from the injected population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' For computational efficiency the match is only computed between the cata- log waveform and an injected waveform if their frequency parameters are within 10 frequency bins of one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 13 shows the distribution of the match values be- tween the sources in the catalog and those in the input population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' For inclusion in the sample we select catalog candidates with detection confidence z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='9 instead of the z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='5 criteria for inclusion in the catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The top panel shows the cumulative distribution function of the matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The vertical axis is thus interpreted as the frac- tion of sources in the catalog with match below m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The bottom panel is the un-normalized survival function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' the total number of sources in the catalog with match greater than m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' If we consider m > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='9 as a criteria for an unambiguous mapping between a source in the catalog and an injection then ∼75% (∼80%) of the confidently identified (z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='9) binaries in the 12 (6) month catalog exceed the criteria equating to ∼5000 (∼4500) sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The most striking feature in the results is a population of sources with matches below ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='9 that emerges in the 12 month catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Previous analyses of simulated galax- ies in LISA data have not shown a similarly-populated low match tail [25, 26], instead finding ≳90% of sources with m > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The comparatively high rate of low-match sources in the GLASS UCB catalog demands further in- vestigation and discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' To begin, compare the matches for each search over different projections of the full parameter space shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 14 displaying the frequency-amplitude plane on the left and the sky location on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Here the sky loca- tion is displayed using the sampling parameters (ecliptic coordinates) rather than galactic coordinates as in Fig 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The source model in GBMCMC is parameterized using eclip- tic coordinates to minimize covariances between parame- ters, and thereby make it easier to sample the posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' In this figure each point in the scatter plot is colored by the waveform mismatch defined as 1 − m, and the color map uses a log scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Thus cool colors are good matches, and warm colors are poor matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The majority of low match sources are found in the frequency interval between 1 and 6 mHz where the un- resolved UCBs are the dominant source of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' One significant difference between the GLASS analysis and pre- vious UCB searches is the global spline noise model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' In references [25, 26] each narrow-band frequency segment independently modeled the noise level effectively using a O(103) parameter piece-wise (and discontinuous) fit to the instrument noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The GLASS noise model is con- strained to be a continuous function, and is using model selection to determine the most parsimonious number of parameters, naturally making the noise model less flexi- ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' It holds together that the differences between a fixed (and high dimensional) piece-wise fit and the parsimo- nious spline model have a larger effect in the foreground- dominated part of the spectrum where there is stronger coupling between the UCB and noise models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Another, and perhaps more impactful difference, is a difference in the prior used for the GW source parameters between the LDC2a-v2 analysis and previous demonstra- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Previous incarnations of the GBMCMC sampler, and 13 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 12: Maps of the source sky locations in galactic coordinates from the UCB catalogs after 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='5 [top left], 3 [top right], 6 [bottom left] and 12 [bottom right] months of observing, showing the increasingly clear reconstruction of the Milky Way disk and bulge structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' it’s ancestors, have used a prior on the sky location pa- rameters derived by assuming the sources followed the spatial distribution of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' While that prior is still an option in GLASS, it was intentionally not used for the LDC2a-v2 analysis in favor of a uniform prior on the sky location parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The choice to not use a galaxy prior was motivated by the idea of eventually performing a hierarchichal analysis where the posterior samples of the binaries in the catalog are used to constrain models for the spatial distribution of sources in the galaxy [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Hierarchichal analyses are complicated by non-trivial pri- ors on the posterior samples and so the choice was made to produce samples for the UCBs in the most accessi- ble form possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The expectation was that this choice would produce larger uncertainties in the position recon- struction of individual sources, but an unintended con- sequence is the effect it had on the detection efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The right hand side of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 14 clearly shows sources with high mismatch are preferentially located outside of the galactic plane, whereas high match sources follow the ex- pected “U” shape of the galaxy in ecliptic coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Mismatching sources in the catalog generally arise from two circumstances: Either a source template is fitting to blended contributions from multiple injections or multi- ple source templates are being used to fit a single injec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The former (one-to-many) can be the most parsimo- nious solution (having the highest Bayesian evidence), or it could be due to modeling issues either from the noise or sky location prior while the latter (many-to-one) is clearly a problem with sampler convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' In the regime where a few templates are fitting a larger number of signals, clear attribution for why this was the preferred configuration of the model is difficult to assign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' However, it is true that from a strict model selection point of view if fewer templates can adequately fit fea- tures in the data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' a larger combination of sources, the parsimonious solution is favored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Additional infor- mation, such as a prior that favors sources in the galac- tic plane, is needed to help further disentangle the over- lapping sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' If the sky location prior and/or noise model were the predominant cause of the poor matches, why would it only appear in the 12 month analysis?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' One possible explanation is that the recovered source catalogs from shorter integration times are going to be dominated by the loudest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' most isolated,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' binaries in the population and as the observing time increases the search is able fit 60° 30° a0 1800 135° :06- 45° o0 45° :06 135° 180° 30° 30° 60°60° 30° 30° a0 18 135° :06- 45° 45° :06 1359 1800 30° 30° 60°60° 30° 30° a0 18 135° 90° 45° 45° 006 135° 1800 30° 30° 60°60° 30° 30° a0 18° 135° 45° 450 900 135° 180° 30° 30° 60°14 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 13: Top: Cumulative distribution function of matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Bottom: Un-normalized survival function of the match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Purple, green, orange, and pink curves are for 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='5, 3, 6, and 12 month observing times respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Results are from confident (z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='9) detections from the source catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' features in the data with lower intrinsic GW amplitude where the injected source density increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Addition- ally, the uniform prior on the sky location of binaries is worse for longer observation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' With short-duration observations the LISA UCB catalog will be dominated by near-by sources, especially at lower frequency, where a uniform distribution on the sky, while still not accurate, is closer to the observed population than at later times when the UCB catalog will sample the full galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The issues of source confusion and the dependence on priors for what constitutes a “detection” are many, nuanced, and require dedicated study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Another possibility is that the jump from 6 months to 12 months was too big a step for the Gaussian mixture model proposal which uses the posteriors from the shorter time span analysis as a pro- posal for the longer time span analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 15 shows a pair of examples suspected of exhibit- ing the parsimony “failure” mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Both show results from a subset of a single UCB analysis segment from the 12 month data roughly bracketing the frequency range where the low match sources are most prevalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The top panels show the ASD of the simulated data after the MBHB signals have been removed (gray), the injected UCB model (black), and the posterior distribution for the recovered UCB model (purple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The middle panels show the injected source parameters in the frequency- amplitude plane (open circles), the point estimate param- eters from the UCB catalog entries (filled purple circles), unfiltered chain samples from GBMCMC (light purple), and the posterior samples colored differently for each indi- vidual source in the catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The bottom panel shows the match for each posterior sample in the same colors as the panel above, and open circles for the point esti- mate match used to construct the curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' In these examples the catalog entries with poor matches are typically found in parts of the frequency segment where there are numerous unresolved injections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The posterior samples do not obviously favor one injected value over the other, yet the overall fit follows the injected signal (top panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' An example of the overfitting problem, where multiple templates are fitting one source, is shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 16 using the same format as 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' In this segment there are six densely-packed injections, two of which are well-fit in the catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The remaining four were fit by 13 templates in the 12 month analysis–a clear convergence error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Exam- ples like figures 15 and 16 will drive future development of the GBMCMC pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' To test our conjecture about the root cause of the high rate of low match sources in the catalog, we reanalyzed five UCB segments from the LDC2a-v2 data with the MBHB injections removed, and used the GBMCMC sam- pler alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The frequency segments were chosen to be evenly spaced between 1 and 6 mHz to cover the regime where the low match catalog entries were most common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Three different configurations were used to measure the effect of the different suspects for the low match popu- lation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The first uses proposals built from the 6 month GLASS run, and the flat sky prior, just like the produc- tion LDC2a-v2 results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The only difference between the first example and the global analysis (ignoring covari- ances with the MBHB model) is the noise model, which in GBMCMC is parameterized as a constant over the frequency interval of the segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Assuming a constant PSD over the each analysis segment effectively amounts to a sig- nificantly more flexible noise model compared to what is used in GLASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The second configuration is the same as the first, but with proposals built from a 9 month GBMCMC analysis of the segments to test whether the low match sources were due to convergence problems stemming from the time steps between analyses being too large, reducing the efficiency of the Gaussian mixture model proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The final configuration reverts to the 6 month proposals but includes the galaxy prior as described in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 17 shows the same type of results as fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 13 but only for the six test segments, and comparing the differ- ent test configurations with the global analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Of the different configurations tested, using the constant PSD model slightly improves the purity of the catalog, but similarly reduces the total number of detections in the catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Including an intermediate 9 month analysis re- sults in a higher number of detections and an improved match distribution, meaning that the overall convergence of the UCB model is heavily dependent on the efficiency of the proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Including the galaxy prior has the largest influence on the results, both improving the cat- alog purity as well as the total number of detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' While a full-scale study is needed to conclusively assess the performance of the different configurations, these re- sults are suggestive that using a model for the galaxy in the analysis and shorter time steps between global fit processing will improve the quality of the UCB source catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The sensitivity to the galaxy prior also rein- forces the fact that modeling choices have a significant match 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content="0 6'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='8 V 20 count 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content="8 6'0 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='0 match15 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 14: Scatter plot of point estimates for candidate sources in the UCB catalog after 12 months of observing colored by the minimum mismatch between the catalog waveform and the injected waveforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Left: GW frequency-amplitude plane showing that most of the high mismatch sources are in the confusion noise regime below ∼6 mHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Right: Sky location parameters in ecliptic coordinates, revealing that the high mismatch sources are preferentially located out of the galactic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 15: Investigation of possible causes for the low match population in the UCB catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Left and right panels focus on different frequency intervals towards the low (∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='8 mHz) and high (∼4 mHz) end of the region where the false alarms were most frequent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The top panel shows ASD of the LDC2a-v2 data with MBHBs removed (gray), input UCB signal (black) and the joint posterior for reconstruction from the GBMCMC sampler (purple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The middle panels are a scatter plot of the UCB frequency-amplitude parameters for the injections (open circles), point estimate catalog entries (filled circles), chain samples (light purple dots), and posterior samples for the individual sources (multi-color dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The bottom panel shows the maximum match between each posterior sample and an injected waveform (same colors as middle panel) and the match from the catalog point estimate used in the summary plots like fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The low-match sources are consistent with fitting combinations of injections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' impact on the resulting inferences, especially near the detection threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' VGB Catalog Analysis and interpretation of the VGB catalog is more straightforward due to the model using informative priors derived from EM observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Because the sampler is as- suming a single source at known orbital period and sky location, the most relevant parameters to be measured by LISA are the GW amplitude and binary inclination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' In the event that the LISA observation time is not long enough for the source to be “detectable,” the posterior samples are useful for setting inclination-dependent up- per limits on the amplitude, and therefore the combined 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='00 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='75 1021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='60 10-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='26 mismatch 1022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='26 10-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='60 10~23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='76 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='00 103 10-3 10~2 0 2 3 f (Hz)10~20, 10~21 10~22 10-22, 10-23, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='0 match 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='802 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='803 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='804 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='806 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='806 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='807 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='808 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='809 f (mHz)10-20 10-21 10~22 ： 10-23 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='0 match 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='6 O 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=" 6f6'E 3." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='960 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='961 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='962 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='963 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='954 f (mHz)16 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 16: Same as fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 15 for an example where multiple templates were fitting a smaller number of injections, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' a clear example of a convergence failure for GBMCMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 17: Same as fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 13 but for different test configurations of GBMCMC on six frequency segments between 1 and 6 mHz to explore possible causes for the high fraction of low match sources in the GLASS UCB catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The dark blue curves are the GLASS results for the test segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Light blue use the flat sky prior but a constant PSD model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Dark orange are the same configuration as light blue but include proposals generated from a 9 month run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Light orange uses the constant PSD model and a prior that prefers sources to be located in the galactic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Results are from confident (z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='9) detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' In this limited test, using the galaxy prior improves both the catalog purity and the number of high-match sources in the catalog and shows that smaller time steps between global fit runs are advantageous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' chirp mass of, and distance to, the binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 18 shows four representative examples from the full set of known binaries comparing inferences between 3, 6, and 12 month observations (green, orange, and ma- genta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The two dimensional posteriors present the 1 and 2σ contours, and the dashed black lines mark the in- jected parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The top two panels show results (a) AM CVn (b) HM Cnc (c) UCXB 4U1820-30 (d) CX0GBS J1751 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 18: Variety of results from targeted analysis of known binaries in the LDC2a-v2 data showing measurement of the GW amplitude and binary inclination as a function of observing time, comparing 3 (green), 6 (orange), and 12 (magenta) month observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' (a) AM CVn is a straightforward example of a strong LISA source properly identified early in the observing campaign with inferences steadily improving over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' (b) HM Cnc shows similar behavior as AM CVn but at higher S/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' (c) UCXB 4U182030 shows how a binary will transition from being undetected, with the analysis providing upper limits on the amplitude (green) to a point where the binary parameters will be constrained (orange, magenta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' (d) CX0GBS J1751 is an example of improving upper limits with observation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' This binary will require longer observing times to be constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' for AM CVn and HM Cnc–two of the canonical known binaries that are identifiable early in the LISA observa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The bottom left panel is for the ultra compact X-ray binary UCXB 4U1820-30 which transitions from a regime where upper limits are set after 3 months of ob- serving, to a constraint in the 6 and 12 month catalogs, indicated by the open contours in green to the closed con- tours in orange and magenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Finally, source CX0GBS J1751 remains undetectable by GLASS after 12 months of observing but note that the upper limit inferred for the amplitude decreases over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' MBHB Catalog The final parts of the GLASS analysis to investigate are the MBHB results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The most interesting single exam- ple is the first MBHB to appear in the LDC2a-v2 data, which merges during the second month of the simulated observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The simulated source also happens to be one of highest S/N binaries in the population and is ob- 10~20 10~22, 10~23 10-24 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' match 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='230 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='231 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='232 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='233 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='234 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='236 f (mHz)match 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content="0 6'0 GLASS 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='8 GBMCMC, flat sky prior 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='6 GBMCMC, flat sky prior, 09mo step 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='6 GBMCMC, galaxy prior 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='4 fraction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='0 match 100 76 A counfs 60 fotal 6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content="8 6'0 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='0 match03 months 06 months 12 months A「×10-22 t (deg)03 months 06 months 12 months 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='90 A「×10-22] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='75 t (deg)03 montHs 06 months 12 monthis A「×10-23 (deg)03 months 06 months 12 months A[×10-22 (deg)17 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 19: Marginalized posterior distributions for the mass m and dimensionless spin χ parameters of the first MBHB to merge in the LDC2a-v2 data, during the second month of the simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The dashed lines mark the parameter values for the simulated signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Note the parameter estimation improves after the signal has left the LISA band because the galactic foreground decreases as the UCB model resolves more binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' servable in three of the different analysis epochs used for this demonstration of GLASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 19 shows the posterior distribution function for the intrinsic source parameters: m1 and m2 are the masses of the black holes in the bi- nary while χ1 and χ2 are their respective dimensionless spin parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Recall that the data and MBHB model both currently assume the binaries have BH spin aligned with the orbital angular momentum vector–an assump- tion that is not valid in nature but made out of conve- nience at this stage of development for simulations and pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' As with other examples, the color indicates observing time and the dashed lines mark the injection values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' What is remarkable about fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 19 are the changes in the posteriors over the observing time even though the binary merged in month 2 of the simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The subtle reduction in the width of the posteriors is due to the improved foreground subtraction from the UCB model, which effectively lowers the noise level, and thereby in- creases the S/N, of the MBHB mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Because of the global nature of LISA analysis, inferences from transient sources will continue to improve long after they have left the measurement band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Moving on to the full MBHB population, fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 20 shows the posterior distribution function for the mass parame- ters from each of the 15 MBHBs injected into, and recov- ered from, the LDC2a-v2 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The MBHBs are labeled in the GLASS catalog by the merger time (in seconds) relative to the start of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The input popula- tion covers a wide range off masses and the posteriors are generally well-constrained due to the large number of in-spiral cycles combined with the strength of the merger signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' To compare against the true values from the sim- ulations, the right-hand panel shows the same posteriors but shifted by the injected mass values, so the point (0,0) marks the truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' For visibility, only the 1σ contours are shown on the right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' All but one of the binaries contain the truth value inside of the 2σ contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The one source whose injected value is outside of the bulk of the GLASS posterior is the lowest mass binary in the in- put population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The same bias is seen in other prototype analyses and is suspected of being the results of artifacts in the LDC2a-v2 data from the waveform simulation pro- cess [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' For the final demonstration of the MBHB catalog, fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 21 displays the sky map for each MBHB source in the catalog, in order of merger time from top left to bottom right, after the full 12 month analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The variety found in the MBHB sky maps is the result of the relative im- portance for each event of the three different “channels” of localization information for these signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The first channel is through the GW phase which is frequency- modulated by the orbital motion of the spacecraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The second channel of localization information comes from the relative arrival time of the GW signal’s wave-fronts at the different spacecraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The third, and least infor- mative, is from the non-uniform detector response over the sky which is encoded in the relative amplitudes of the GW signal in the different TDI channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Sky maps that contain many modes of high probability are typi- cally from higher mass binaries that are shorter lived in the LISA data, missing out on the Doppler modulations induced by the orbital motion of the detector and not reaching high enough frequency to differentiate the sig- nal arrival times at the different spacecraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Lower mass MBHB signals get the best of both worlds, as they are in the measurement band for long enough to have clearly detectable frequency modulation and they reach short enough wavelengths to benefit from the time of arrival measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Note that even for the high mass and short-lived binaries additional information from spin pre- cession and, more importantly, higher harmonics of the waveform will help break degeneracies [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' See Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' [39] for a more comprehensive discussion and demonstration of MBHB parameter estimation with LISA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' DISCUSSION AND FUTURE WORK The global analysis demonstrated here is one impor- tant step towards a fully functioning pipeline ready for LISA observational data but there is still a long way to go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Obviously missing are the other anticipated source types, though the GLASS architecture is designed to seam- lessly accommodate additional modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Extending to other source types is generally expected to be a small perturbation relative to the overall scale of the analysis which is set by the UCBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Another obvious direction of development is to reduce the overall computational cost of the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The cur- rent version of GLASS used O(103) CPUs for O(5) days 03 months 06 months 12 months m2 (Mo)[×105] ×106 m2 (M ×105 X218 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 20: Mass posteriors for the entire observed MBHB population in the 12 month LDC2a-v2 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Left: 1 and 2σ contours for the inferred masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Right: 1σ contours for the binaries shifted by the true values for each simulated source, such that (0,0) marks the injected parameter location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' to process the 12 month data, and those processing times will increase roughly linearly as the observation time grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The current algorithm will become uncomfort- ably expensive for multi-year data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Reducing the computational cost is of crucial importance for further development because the main source of stress on the analysis methods provided by LISA data is the scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Optimal development of the global analysis requires fre- quent processing of full-scale data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The two prongs for reducing the computational cost of the analysis are by lowering the cost of each likelihood evaluation with accelerated compuatational techniques (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Ref [40]), and by reducing the total number of likelihood evalua- tions by developing a more efficient sampling algorithm through further development of data- and domain-driven proposal distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Beyond implementation improvements, there are a number of model assumptions, reflected in the simplic- ity of the likelihood function, that will need to be re- laxed to properly handle observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Generally speaking, it is the assumption of stationary noise that is most problematic, though there are different sources of non-stationarity that each deserve their own strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' As mentioned earlier, there will be periodic (due to the galactic foreground) and secular (due to the instrument) changes in the instrument noise level which introduce non-zero off-diagonal elements of the frequency-domain noise covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' We will mitigate these effects by moving away from conducting the analysis in the Fourier domain, in favor of a discrete wavelet basis, which still yields a diagonal noise covariance matrix if the stationary timescales are longer than the duration of the wavelet ba- sis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Descriptions of waveform and noise models in the wavelet domain are found in Refs [30, 35, 41] and are scheduled to be integrated into GLASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' In the wavelet domain the noise model will be a function of both time and frequency to track the slow drift in the instrument and foreground noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The time-frequency approach renders the heterodyn- ing currently used for both the UCB and MBHB like- lihood calculations redundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Wavelet decompositions incorporate a natural compression of GW signals since the likelihood only integral only changes along the signal track f(t), which has length ∼ √ N for a data set with N data points [35, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Wavelet domain likelihoods are typically faster than their heterodyned analogs without requiring a reference waveform or any pre-computation step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Faster likelihood functions allow for more rapid convergence and better mixing between blocks of the MH sampler for the same computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The wavelet domain is also better suited to handling data gaps than frequency domain analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' In the wavelet domain the basis functions are finite duration with built-in window functions that naturally suppress spectral leakage caused by gaps in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Fourier methods require additional data conditioning to deal with gaps, either through win- dowing or data augmentation [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Another modeling limitation of the current analysis is that it treats the unresolved galactic signals as noise, when in reality they are better described as a cyclo- stationary stochastic signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Going forward we will ex- plore using a physically parameterized instrument noise model [43], while treating the unresolved galactic bina- ries as a separate, time-varying, stochastic signal [30, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Such a treatment requires that we include the (approx- imately, at low frequency) noise-only TDI “T” channel, and not just the A and E channels as is done in the cur- rent version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Modeling of the galactic confusion will be further improved by coupling the resolved UCB popula- tion in the global fit to a physical model of the foreground via parameterized priors for the spatial distribution of bi- naries [36] and the overall number density of sources as a function of frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' A source of non-stationary noise not well suited for modeling with the power spectral density (or time- frequency equivalent) are short duration noise transients MBH004799206 MBH008746626 MBH011167688 0 MBH011258537 MBH011527103 MBH011971300 MBH013616801 MBH016532115 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='2 MBH017244430 (Mo)[×106] MBH018605271 MBH020426287 MBH022228030 MBH023440117 MBH024408799 MBH029515074 5 mi(Mo)[x106]MBH004799206 MBH008746626 MBH011167688 MBH011258537 MBH011527103 MBH011971300 MBH013616801 MBH016532115 MBH017244430 MBH018605271 MBH020426287 MBH022228030 7B023440117 MBH024408799 MBH02951507419 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 21: Marginalized distributions of MBHB sky location in ecliptic coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The sky maps are ordered by merger time from top left to bottom right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The different morphologies of the sky maps are due to the different maximum frequency and duration of the signals, with lower mass binaries reaching high frequency and spending more time in band leading to precise sky localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Degeneracies in the sky location improve when higher harmonics are included in the waveforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' or “glitches.” The path to incorporating a glitch model (and, by corollary, a model for generic GW transients) into GLASS has already been paved by similar work done for LIGO-Virgo data [8] and theoretical demonstrations using simulated LISA data [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' There is already LDC data that contain a simulated glitch population informed by the LISA Pathfinder observations [45] and the in- corporation of a transient noise module into the GLASS framework is a near-term priority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' One final currently planned development direction for GLASS is in the instrument model itself, enabling the global analysis to start with lower level data products than the TDI channels currently used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' A data-driven ap- proach to perform self-calibration coupled with the global fit naturally propagates uncertainties at each stage of the signal processing chain to the astrophysical inferences made with the GW signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Such capabilities may prove ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='of systematic errors are vital,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' the most obvious example of which would be testing the nature of gravity with high S/N MBHBs or EMRIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Already-demonstrated examples of self-calibration methods for LISA include employing UCBs as phase standards [46] and using the phasemeter data to infer the light travel time between spacecraft for cancellation of laser frequency noise and construction of the TDI interferometer combinations [47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Another possible capability to explore is the removal of noise in the inter- and intra-spacecraft interferometer measure- ments caused by angular jitter of the test masses mas- querading as distance fluctuations–the so-called “tilt to length coupling” inherent in the LISA measurement [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' An instrument module in the GLASS is valuable for quan- titatively understanding and, if necessary, mitigating the affect of calibration uncertainties an astrophysical infer- ences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' As is clear from the long list of future work, the GLASS architecture described in this paper does not represent a finished design but instead is the scaffolding upon which further development will be built.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Nevertheless, our demonstrated results are an important way-point on the path towards a fully functional pipeline ready for LISA observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' ACKNOWLEDGEMENTS Software: Results presented here used v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='0 of ldasoft, a public C library which includes the noise, UCB, VGB, and global fit samplers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The MBHB sampler is managed independently at LISA-Massive-Black-Hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Postprocessing and visualization tools for the source cat- alogs are available the python package lisacattools which in turn depends on numpy [50], pandas [51, 52], matplotlib [53], astropy [54], seaborn [55], and ChainConsumer [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' The authors thank K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Gresbach for multithreading the GBMCMC pipeline;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Baker, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Lackeos, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Rob- son, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Slutsky, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Thorpe for their useful dis- cussions and suggestions during the development of the global fit pipeline and the assessment of the results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Malapert and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Thorpe for their role as co-developers of lisacattools;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Thompson for indispensable help deploying the pipeline on the AWS resources;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' and the LISA Data Challenge group for providing and supporting the simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Littenberg is supported by the NASA LISA Study Office.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Cornish appreciates the support of the NASA LISA Preparatory Science Grant 80NSSC19K0320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Amaro-Seoane, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Audley, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Babak, J.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Zhang, 23 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Zonca, and Astropy Project Contributors, The As- tropy Project: Sustaining and Growing a Community- oriented Open-source Project and the Latest Major Re- lease (v5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='0) of the Core Package, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' 935, 167 (2022), arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='14220 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content='IM].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' [55] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Waskom, seaborn: statistical data visualization, Journal of Open Source Software 6, 3021 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' [56] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} +page_content=' Hinton, ChainConsumer, The Journal of Open Source Software 1, 00045 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE2T4oBgHgl3EQfHwb9/content/2301.03673v1.pdf'} diff --git a/jdE5T4oBgHgl3EQfGQ5V/content/tmp_files/2301.05429v1.pdf.txt b/jdE5T4oBgHgl3EQfGQ5V/content/tmp_files/2301.05429v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..cf74a9063a7f19a2d504a90c51bcc03ebf668354 --- /dev/null +++ b/jdE5T4oBgHgl3EQfGQ5V/content/tmp_files/2301.05429v1.pdf.txt @@ -0,0 +1,971 @@ +Motility-induced shear thickening in dense colloidal suspensions +A. G¨ulce Bayram,1, ∗ Fabian Jan Schwarzendahl,2 Hartmut L¨owen,2 and Luca Biancofiore1 +1Department of Mechanical Engineering, Bilkent University, Cankaya, Ankara 06800, Turkey +2Institut f¨ur Theoretische Physik II: Weiche Materie, +Heinrich-Heine-Universit¨at D¨usseldorf, 40225 D¨usseldorf, Germany +(Dated: January 16, 2023) +Phase transitions and collective dynamics of active colloidal suspensions are fascinating topics in +soft matter physics, particularly for out-of-equilibrium systems, which can lead to rich rheological +behaviours in the presence of steady shear flow. In this article, the role of self-propulsion in the rheo- +logical response of a dense colloidal suspension is investigated by using particle-resolved simulations. +First, the interplay between activity and shear in the solid to the liquid transition of the suspension +is analysed. +While both self-propulsion and shear destroy order and melt the system by them- +selves above their critical values, self-propulsion lowers the stress barrier that needs to be overcome +during the transition. Once the suspension reaches a non-equilibrium steady state the rheological +response is analysed. While passive suspensions show a solid-like behaviour, turning on particle +motility fluidises the system and, at low self-propulsion, the suspension behaves as a shear-thinning +fluid. Increasing the self-propulsion of the colloids induces a transition from a shear-thinning to a +shear-thickening behaviour, which we attribute to clustering in the suspensions induced by motility. +This interesting phenomenon of motility-induced shear thickening (MIST) can be used to tailor the +rheological response of colloidal suspensions. +I. +INTRODUCTION +During the last decade, active matter has become a +topic of intense research [1, 2]. In particular, active col- +loids have been investigated [3] since they provide a well- +controlled testing ground for out-of-equilibrium systems. +Experimentally, one among many realizations of active +particles is active Janus colloids [3], which can show fas- +cinating phenomena such as motility-induced phase sep- +aration [4, 5], vortex formation [6], clustering induced +by hydrodynamic fluxes [7] or wall accumulation [8–10]. +Dense suspensions of active particles have been realized +and studied experimentally for Janus colloids[11], for vi- +brated active disks[12, 13] and theoretically [14–16]. Ac- +tive particles that are in a glassy state [17] have given +insights into random close packing [18] and it has been +shown that shearing an active glass former leads to or- +dering [19]. However, the rheological properties of dense +active colloids are largely unknown. +On the side of biological microswimmers, such as bac- +teria or microalgae, it has been shown that the pres- +ence of a small fraction of active swimmers in a fluid +medium can fundamentally change the fluid’s rheological +properties. It was found experimentally that pusher-type +swimmers such as the bacterium Escherichia coli reduce +the effective viscosity [20–24] while puller-type swimmers +such as the microalgae Chlamydomonas reinhardtii can +increase the viscosity [25]. This response to the swim- +ming behavior has also been explained in theoretical and +numerical studies [26–33]. Further, the rheological prop- +erties of active fluids, that can model driven microtubules +or active acting filaments, have been computed using field +theoretical approaches [33–36]. +∗ gulce.bayram@bilkent.edu.tr +(a) +(b) +shear stress +shear rate +shear thinning fuid +shear thickening fuid +Newtonianfuid +Shear thickening +Newtonian +Shear thinning +(c) +FIG. 1. +Schematic representations of (a) the sheared ac- +tive colloidal suspensions with Lees-Edwards boundary condi- +tions (the colour bar represents the distribution of shear force +throughout the computational box), (b) the expected parti- +cle arrangements in different fluids, (c) shear stress-shear rate +curve for different fluids. +Here, we investigate the rheological properties of dense +active colloids in two spatial dimensions using Brown- +ian dynamics computer simulations. +Investigating the +strain-stress curves shows that activity reduces and even +destroys the stress barrier that a shear flow has to over- +come to fluidise the system. +In the steady state, the +system’s shear stress reveals that particle motility fun- +damentally changes the rheological properties of the sys- +tem: at none or low activity the system is shear thinning, +for intermediate activities it becomes Newtonian and at +arXiv:2301.05429v1 [cond-mat.soft] 13 Jan 2023 + +2 +very high activity it is shear thickening (Fig. 1). +The +shear thickening behaviour is induced by particle clus- +ters that stem from the active motion of the particles. +Therefore we refer to this new phenomenon as motility- +induced shear thickening (MIST). In fact, MIST is some- +how a consequence of motility-induced phase separation +(MIPS) [4, 5] in the bulk which shows a pre-clustering +in the one-fluid phase even before full phase separation +is reached. These clusters are responsible for the shear- +thickening under shear. The full rheological response is +further well characterized using a power-law model for +the stress as a function of the shear rate, which shows +the continuous transition from shear thinning to shear +thickening as the activity is increased. Therefore the rhe- +ological behaviour can be tuned by activity. +II. +SIMULATION METHOD +We study a suspension of N self-propelled particles +moving in two spatial dimensions under shear flow, +Fig.1(a). In section II A, we introduce the model to be +used for simulating this system, including the units and +parameters used in it. We then describe in section II B +the observables that we measure in our simulation in or- +der to analyse (i) the phase transition behaviour and (ii) +the rheological response of the system across the different +self-propulsion and shear forces. +A. +Model +In two spatial dimensions, the over-damped dynamics +of the colloids are modeled by active Brownian particles +in the presence of a steady shear rate ˙γ, +dri +dt = − 1 +Γ +� +ij +∂U +∂rij +rx +ijry +ij +rij +� +, +(4) +which is averaging out the stress contribution of every +particle in the steady state time frame. rx +ij and ry +ij are the +distance between the centers of two particles i and j in di- +rection of x and y, respectively. The stress is nondimen- +sionalised by considering the thermal energy kBT and +length scale a as σxya3/kBT, where kB is the Boltzmann +constant and temperature T with kBT = D0Γ. The dy- +namics of the system is controlled by two dimensionless +numbers. +(i) First, the imposed shear rate is given by the shear +Peclet number, Pes = ˙γa2/D0, which measures the +ratio of the shear force to thermal fluctuations. +(ii) The second dimensionless number is the active +Peclet number, Pea = v0a/D0, which measures +the ratio between the self-propulsion force (or the +motility strength) and thermal fluctuations. +The typical magnitude of the dimensionless numbers con- +sidered here are Pea ∈ [0, 150] for the active Peclet num- +ber and Pes ∈ [0, 40] for the shear Peclet number. + +3 +y +x +1 +2 +3 +FIG. 2. (left) Snapshots of particle configurations for (1) solid-like, (2) cubic-like and (3) liquid-like phases of the passive +system, in which the system conditions correspond to the numbered black dots in the plots: (a) The instantaneous shear stress +(σxy) with respect to strain (γ = ˙γt) for passive (Pea = 0) and active systems (Pea > 0). The corresponding structural changes +throughout the bond orientational order parameters, ψ6 (b) and ψ4 (c). The black dotted-horizontal line defines the position +of structural transition for the hexagonal configuration, ψ6 ≈ 0.45[14]. +The structural changes in the system are monitored by +the 2D bond orientational order parameters[39], +ψν = +���� 1 +N +N +� +i=1 +1 +ν +� +j∈N (i) +eiνθij +��� +2 +� +(5) +where N(i) is the set of ν nearest neighbour particles of +the ith particle and θij is the angle between the bond +vector pointing from particle i to j and horizontal fixed +axis. For ν = 6 the order parameter is the hexagonal or- +der parameter, which gives zero in the disordered phase +whereas it equals 1 in a perfect hexagonal crystal. Simi- +larly to the hexatic order parameter ψ6, we also analyse +the cubic orientational order by using ν = 4. +III. +RESULTS +A. +Shear and activity induced melting +Our particular interest is the characterization of self- +propelled particles in the dense regime, ρNa2 = 1.15. We +first analyze the shear stress (σxy) - strain (γ = ˙γt) re- +lation of the system at a constant shear rate Pes = 20 +for different self-propulsions. Meanwhile, we also moni- +tor the bond-orientational order parameters of the sys- +tem. We note that the results represented here show the +system behaviour starting from the time we impose the +shearing force (t > 0.4τ0). +In the absence of particle motility (Pea = 0), the sys- +tem starting out of the perfect hexagonal crystal stays in +this solid-like phase during the relaxation, as in Fig.2 +snapshot(1). +When shearing is turned on thereafter, +shear stress starts building up to a peak value, i.e. point +(2) in Fig.2(a). This brings the system to the onset of +melting, see point (2) in Fig.2(b). +At this point, the +system is not disordered yet and we discuss this struc- +tural transition in detail later on. Here, this nearly lin- +ear increase in shear stress represents that the system +response is nearly elastic at low strains. Further accu- +mulation of the strain melts the crystal and, accordingly, +causes a release of stress due to structural relaxation. +This brings the system to a strain-independent plateau +at higher strains. We can associate this best with the +shear-induced disordering transition where the equilib- +rium melting transition is displaced by the imposed shear +rate[40–42]. +Moreover, including the self-propulsion helps shear by +causing a preliminary melting, which appears as a de- +crease in ψ6 during the relaxation of the system. This +fastens the hexagonal-liquid transition and consequently +decreases the stress barrier, which is defined by the shear +stress peak. +Much higher activities (Pea ≥ 30) melt +the crystal entirely even before imposing the shear rate, +thereby annihilating the stress barrier. After that, the +colloidal suspension is disordered and its behaviour is +liquid-like [in Fig.2 snapshot(3)]. +In the latter case, +the equilibrium melting transition is displaced by self- +propulsion and melting is expected for the parameters of +the system [14]. +Points (2) in Fig.2(a) and (b) together reveal that +the stress peak appears just before the drop of ψ6 be- +low the structural transition, ψ6 = 0.45[14]. +In fact, +the hexatic-to-liquid transition is not direct throughout +this barrier but rather occurs via a structural rearrange- +ment [Fig.2 snapshot(2)]. The zigzag motion of the par- +ticles, which emerges by their alternating motion in one +layer between filling the grooves of the next layer and +hopping up from there[43], temporarily gives a rise in +ψ4 [see Fig.2 (c)]. We might expect this increase of ψ4 +at first sight when we look at the arrangement of parti- +cles in the snapshot (2). Additionally, even in the liquid +regime below ψ6 < 0.45, where the shear melts the crys- + +4 +y +x +1 +2 +3 +4 +FIG. 3. +(top) Snapshots of particle configurations: (1) solid-like, (2) shear-thinning, (3) Newtonian and (4) shear-thickening, +where the particles are coloured with respect to their number of surrounding particles (Nsp). (a) Nondimensional shear stress +� +σxy +� +- shear rate ˙γ curves for different self-propulsion forces. (b) Liquid-like regime shown in log-log scale where the power law +fitting has been applied is indicated by the fitting lines for different rheological behaviours with respect to the corresponding +power indexes(n). The numbered black dots represent the conditions of the simulations where the snapshots are taken. +tal as a whole, some ordered regions show up at times +due to this zig-zag motion of the particles in subsequent +layers. +This is similar to what was observed by Wu +et. +al.[44] in melting passive colloidal suspensions un- +der shear. Although these ordered structures give rise to +local increases in the hexagonal order parameter in the +passive system below the structural transition line [see +Fig.2(b)], the self-propulsion destroys these local ordered +structures, resulting in zero ψ6 [see the Supplementary +Material, Movie Passive Pes20.mp4]. +Furthermore, especially in active cases, we see that +the subsequent melting first starts occurring at the least +sheared domain of the system and spreads toward the +boundaries through the accumulation of interstitial de- +fects as the activity increases. +The self-propulsion re- +veals these interstitial defects at weakly sheared domains +and in this way helps the shear to melt the suspen- +sion. +However, the highly sheared regions of the do- +main, i.e. regions in proximity to the top and bottom +boundaries, are able to temporarily restore their ordered +structures and flow as sliding layers for a longer time +by resisting the self-propulsion force. +Increasing self- +propulsion force amplifies the defects to grow and spread +toward the boundaries, resulting in the total melting of +the system at the end [see the Supplementary Material, +Movie Pea20 interstitialdefects.mp4]. +B. +Motility-induced shear thickening (MIST) and +shear thinning +We now consider the liquid regime and discuss the shear +stress +� +σxy +� +with respect to the imposed shear rate ˙γ [see +in Fig.3(a)], which yields the rheology of the active col- +loidal suspensions. This time, we average the shear stress +data over the time frame corresponding to the strain- +independent plateau of the stress. +The most common rheological model, the power law +model[45] σxy = K ˙γn, is used henceforth to quantify the +possible non-Newtonian behaviour of the system. We fit +our stress-shear rate data from the simulations with dif- +ferent activities to this model [Fig.3(b)]. +Thereby, we +explore the effect of the self-propulsion on the rheologi- +cal response of the system. In this case, we follow up the +structural changes by the number of surrounding parti- +cles Nsp, which is the number of particles counted inside +the surrounding circle with a radius rsurr/a = 3 for every +particle individually [see the top snapshots in Fig.3]. +The passive system, initially in the crystalline phase, +does not melt at very low shear rates (Pes < 2.5) and +shows solid-like behaviour [Fig.3(1)] with non-zero yield +stress, see in Fig.3(a). This is in line with what Chen +et. al. demonstrated for passive ordered suspensions[43]. +Here, thermal fluctuations are responsible for the mo- + +5 +tion of colloids. At this solid regime, we see that turning +on the activity diminishes the yield stress. In addition to +weak activities, the hexagonal order in the system can be +destroyed by more effective shearing, above the so-called +critical shear rate[42, 46]. +On the other hand, activi- +ties Pea > 30 eliminate the solid-like behaviour, since +the hexatic-to-liquid transition is already reached by ac- +tivity [see again in Fig.2(b)]. Thereafter, the resulting +melt starts behaving as a shear-thinning liquid which is +a non-Newtonian behaviour characterized by the slopes +lower than 1 in the logarithmic shear stress-shear rate +curve, then giving power indexes n < 1, see point (2) +in Fig.3(b). Here, the shear force has control over the +motion of the particles and yields sliding layers of par- +ticles aligned in the direction of shear flow [snapshot 2 +in Fig.3]. +Indeed, this layered flow, which is also dis- +cussed in Section III A, has been previously found as be- +ing intrinsic to the shear thinning behaviour of passive +Brownian colloidal suspensions[43, 47, 48]. Moreover, the +local ordered regions temporarily observed as a result +of the zig-zag motion of colloids is known as a shear- +induced ordering phenomenon for 2D shear-thinning col- +loidal suspensions[49]. Although the shear thinning be- +haviour is relatively well understood in Brownian col- +loidal systems, the effect of self-propulsion has not been +addressed. We observe that activities up to Pea ≈ 60 as- +sist only the shear in melting the system, while the result- +ing melt maintains the same shear-thinning behaviour +observed for passive suspensions [see the Supplementary +Material, Movie shearthinning.mp4]. +Interestingly, +we +find +that +increasing +the +self- +propulsion force further introduces a transition in the +rheological response of the suspension. +At moderate +motilities, Pea ≈ 70, the fluid behaviour is not shear- +thinning, but Newtonian, corresponding to the curves (3) +with a linear dependence in Fig.3(a,b). In this regime, +self-propulsion degrades the stability of layered flow and +the suspension becomes completely disordered, see snap- +shot (3) in Fig.3, [also see the Supplementary Material, +Movie Newtonian.mp4]. Similar to the mechanism dis- +cussed in Section III A, some of the interstitial defects +appeared in shear-thinning liquid start spreading with +increasing self-propulsion, being responsible for this tran- +sition. +Above +Pea +≈ +80, +we +observe +that +the +self- +propulsion starts dominating over shear and reveals +structural heterogeneities in the system, +leading to +transient cluster formation as shown by the snapshot +(4) in Fig.3, [also see the Supplementary Material, +Movie shearthickening.mp4]. +In this case, the slope +of shear stress- shear rate curves exceeds 1 [see (4) in +Fig.3(b)], meaning that the suspension reveals a second +transition from Newtonian to shear thickening behaviour. +Here, we also observe that the colloidal suspension starts +showing clusters even before imposing a shear to the sys- +tem, which is expected for active particles in this pa- +rameter regime [50] [see the Supplementary Material, at +the beginning of Movie shearthickening.mp4 in the ab- +sence of shear]. However, we found that these clusters +enable the system to behave as a shear-thickening fluid +when the system is forced to flow. +Therefore, we re- +fer to this behaviour as motility-induced shear thickening +(MIST) in some analogy to motility-induced phase sepa- +ration (MIPS). In fact, the cluster formation generating +MIST is a precursor of MIPS. Additionally, especially for +this behaviour, we observe visually that the number of +surrounding particles Nsp remarkably differs from region +to region in the system [see snapshot (4) in Fig.3]. +The shear thinning-to-thickening transition has been +attained for years in passive colloidal suspensions with +very high shear rates (Pes > 100), by underlining simi- +lar cluster formations in the shear-thickening fluid due to +the hydrodynamic interactions and lubrication forces[51– +54]. In this study, instead, we could reach this transition +by keeping the range of shear rate constant at relatively +low values, while increasing the activity of colloids, with +neglecting the hydrodynamics. +In this work, the clus- +ter formation is not triggered by shearing extremely the +system, rather is triggered by motility. +C. +The effect of particle density +20 +40 +60 +80 +100 +120 +140 +Pea +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +n +Shear Thinning +Newtonian +Shear Thickening +Na2 =1.15 +Na2 =0.83 +Na2 =0.68 +FIG. 4. The power index n vs the active Peclet number Pea +for different particle densities. +The horizontal dashed line +corresponds to the Newtonian case with a slope of 1 on the +log-log scale. +Next, we analysed the stress-shear rate curves for +the suspensions with two lower particle densities, semi- +dilute suspension ρNa2 = 0.83, and dilute suspension +ρNa2 = 0.68, respectively, keeping all other parameters +the same in the system. Decreasing the number of par- +ticles in the computational box gives more space to par- +ticles for disordering, compared to the dense suspension +system. This, consequently, increases the tendency of the +system to be melted. Thus, the solid behaviour is found +as limited to very small parameter ranges. Additionally, +the dilute suspensions show lower yield stresses compared +to the dense case at the same activities. Accordingly, the + +6 +system starts showing a shear thinning behaviour even +with very low self-propulsion forces. At this point, Fig.4 +interprets the trend of the power index with increasing +self-propulsion for all cases. Here, again, the power in- +dexes are obtained by fitting the shear stress/shear rate +data to the power law model for these two other densi- +ties, as done for the dense system in Section III B. The +red line in this figure, which corresponds to the dense sus- +pension case, starts at Pea ≈ 30, since the system retains +mostly solid-like behaviour below this value, followed by +shear-induced disordering with increasing activity. +Regarding the liquid-like regime, it is evident that the +critical self-propulsion for shear thinning to thickening +transition shifts to lower activities with decreasing parti- +cle density. In other words, dilute suspensions can thick- +ens easier than dense ones. The underlying reason for +this effect can be intuitively related to clustering. Keep- +ing the particles very close to each other, as in a dense +system, obviously pronounces their collisions. Therefore, +possible cluster formation becomes difficult in the sys- +tem. Although some clusters are formed with these ac- +tivities, they are not stable enough, such that they can be +damaged by any strike from other particles around them. +However, increasing the self-propulsion further promotes +clustering and finally brings the system into a thickening +regime. As we discussed previously, activity triggers the +colloidal suspension to behave as a shear-thickening fluid +in this study, not strong shearing. On the other hand, ac- +20 +40 +60 +80 +100 +120 +140 +Pea +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +Nsp +Na2 =1.15 +Na2 =0.83 +Na2 =0.68 +FIG. 5. Standard deviations in the number of surrounding +particles, σNsp with increasing self-propulsion Pea for dif- +ferent particle densities, ρNa2 = 1.15 (circle), ρNa2 = 0.83 +(triangle) and ρNa2 = 0.68 (square) at Pes = 20. +tive colloids in a dilute environment can constitute more +stable clusters even with low self-propulsion due to fewer +repulsive collisions coming from particles around the clus- +ters, accordingly undergoing this thinning-to-thickening +transition earlier [see Fig.4]. +This scenario is also confirmed by Fig.5, where we re- +port the standard deviations σNsp = +� +⟨N 2sp⟩ − ⟨Nsp⟩2 of +the number of surrounding particles from the mean ⟨Nsp⟩ +for the three different densities. +Typically there are more fluctuations in the surround- +ing particle density for dilute systems, also for weak self- +propulsion forces. However, here the activity enhances +dramatically (i) the local density inhomogeneities, as re- +vealed by the strong increase of σNsp with Pea, and +(ii) the gap in σNsp between dilute and dense systems. +The motility-induced local density inhomogeneities cor- +respond to pre-clustering which provides evidence that +this is the underlying reason for the shear-thickening be- +haviour consistent with snapshot (4) in Fig.3. Since this +pre-clustering is more pronounced for dilute systems, the +MIST transition occurs earlier in these systems confirm- +ing our previous intuitive explanation. +IV. +CONCLUSIONS +In this work, we have explored the effect of self- +propulsion on the rheological response of the dense col- +loidal suspensions under steady shear by using Brownian +dynamics simulations. +First, the solid-to-liquid transi- +tion of the suspension was characterized, showing that +activity helps to melt the systems, as shown before in +literature[14, 55]. When melting, the self-propulsion of +the colloids reduces the stress barrier that the system +has to overcome in order to transition from the solid +to the liquid state. Self-propulsion is not only respon- +sible for assisting the shear in melting the colloidal sus- +pension but also introduces a transition in the rheology +of the melted suspension. Depending on the activity of +colloids in the suspension, different fluid behaviours are +found: shear thinning, Newtonian and shear-thickening. +The underlying reason behind this response is a well- +known dynamical mechanism of active colloidal particles: +activity-induced clustering. When the self-propulsion is +sufficiently strong to prevail against the shear, or even +before shearing, a cluster formation is observed. The ex- +istence of these clusters creates additional resistance to +flow when the suspension is exposed to shear, causing it +to behave as a shear-thickening fluid. We referred to this +as motility-induced shear-thickening in this article. +The shear-thinning to thickening transition reported +here is different from the transition observed in pas- +sive colloidal systems. Passive colloids tend to stick to- +gether by hydrodynamic and lubrication forces at high +shear rates[53, 56], while active colloids cluster due to +their motility (akin to motility-induced phase separa- +tion). +The simulations presented here were performed +in the absence of hydrodynamic interactions. +Includ- +ing hydrodynamics and lubrication forces would be an +interesting future avenue since hydrodynamics suppress +the tendency for active particles to form clusters [57–59], +while on the other hand, hydrodynamics lead to clusters +of sheared passive colloids. Further, it would be interest- +ing to test the response in a three-dimensional set-up or +to include inertia to the active particles motion [60, 61]. +Experimentally, the shear thinning to thickening transi- + +7 +tion could be tested using active colloids in a ”washing- +machine” set up [62]. +Finally, it would be interesting +to extend our simulation to particles of more complex +shape than spheres such as active polymers or active fil- +aments [63, 64], where new rheological behaviour due to +active entanglements can be expected in dense solutions. +CONFLICTS OF INTEREST +There are no conflicts to declare. +ACKNOWLEDGEMENTS +We thank J¨urgen Horbach for helpful discussions. The +work of A.G.B. was supported within the EU MSCA-ITN +ActiveMatter (Proposal No. 812780). +SUPPLEMENTARY MATERIAL +Movie Passive Pes20.mp4: The movie of the simula- +tion with passive colloids at Pes = 20. The snapshots in +Fig. 3 of the main paper are taken from this movie. The +system starts from the hexagonal crystal phase and the +subsequent shearing promotes the sliding layers through- +out the melting. The local ordered regions emergent due +to the zig-zag motion of particles in these sliding layers +are distinguishable at times. +Movie Pea20 interstitialdefects.mp4: +The movie for +the simulation with active colloids (Pea = 20) under +shearing (Pes = 20), shows that the self-propulsion as- +sists the shear in disordering the system by promoting +the interstitial defects. +These defects start appearing +at the least sheared region of the system, and gradu- +ally spread toward the system boundaries with increas- +ing self-propulsion. However, we show only their emer- +gencies here for the suspension system at a constant self- +propulsion. +Movie shearthinning.mp4: The movie corresponds to +the shear-thinning regime, i.e. Pea = 30. The colloidal +suspension here is melted entirely by self-propulsion. Af- +ter imposing the shear, the particles start to reorganise +and cause a layered flow, which is a peculiar behaviuor +of shear-thinning fluids. Some locally ordered domains +appear at times due to the zig-zag motion of particles +under shear flow. +Movie Newtonain.mp4: The movie corresponds to the +Newtonian regime, i.e. Pea = 70. Colloids are homoge- +neously disordered by self-propulsion. +Movie shearthickening.mp4: +The movie corresponds +to the shear-thickening regime, i.e. +Pea = 150, the +so-called the motility-induced shear thickening (MIST). +Self-propulsion induces structural heterogeneity and clus- +tering in the system. +[1] M. C. Marchetti, J.-F. Joanny, S. Ramaswamy, T. B. +Liverpool, J. Prost, M. Rao, and R. A. Simha, Hydrody- +namics of soft active matter, Reviews of Modern Physics +85, 1143 (2013). +[2] S. 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Gompper, The physics of active +polymers and filaments, The Journal of Chemical Physics +153, 040901 (2020). + diff --git a/jdE5T4oBgHgl3EQfGQ5V/content/tmp_files/load_file.txt b/jdE5T4oBgHgl3EQfGQ5V/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..34af99f6c564db257b9d22e0df94474e1b87776d --- /dev/null +++ b/jdE5T4oBgHgl3EQfGQ5V/content/tmp_files/load_file.txt @@ -0,0 +1,715 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf,len=714 +page_content='Motility-induced shear thickening in dense colloidal suspensions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' G¨ulce Bayram,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' ∗ Fabian Jan Schwarzendahl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content='2 Hartmut L¨owen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content='2 and Luca Biancofiore1 1Department of Mechanical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' Bilkent University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' Cankaya,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' Ankara 06800,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' Turkey 2Institut f¨ur Theoretische Physik II: Weiche Materie,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' Heinrich-Heine-Universit¨at D¨usseldorf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' 40225 D¨usseldorf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' Germany (Dated: January 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' 2023) Phase transitions and collective dynamics of active colloidal suspensions are fascinating topics in soft matter physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' particularly for out-of-equilibrium systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' which can lead to rich rheological behaviours in the presence of steady shear flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' In this article, the role of self-propulsion in the rheo- logical response of a dense colloidal suspension is investigated by using particle-resolved simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' First, the interplay between activity and shear in the solid to the liquid transition of the suspension is analysed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' While both self-propulsion and shear destroy order and melt the system by them- selves above their critical values, self-propulsion lowers the stress barrier that needs to be overcome during the transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' Once the suspension reaches a non-equilibrium steady state the rheological response is analysed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' While passive suspensions show a solid-like behaviour, turning on particle motility fluidises the system and, at low self-propulsion, the suspension behaves as a shear-thinning fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' Increasing the self-propulsion of the colloids induces a transition from a shear-thinning to a shear-thickening behaviour, which we attribute to clustering in the suspensions induced by motility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' This interesting phenomenon of motility-induced shear thickening (MIST) can be used to tailor the rheological response of colloidal suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' INTRODUCTION During the last decade, active matter has become a topic of intense research [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' In particular, active col- loids have been investigated [3] since they provide a well- controlled testing ground for out-of-equilibrium systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' Experimentally, one among many realizations of active particles is active Janus colloids [3], which can show fas- cinating phenomena such as motility-induced phase sep- aration [4, 5], vortex formation [6], clustering induced by hydrodynamic fluxes [7] or wall accumulation [8–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' Dense suspensions of active particles have been realized and studied experimentally for Janus colloids[11], for vi- brated active disks[12, 13] and theoretically [14–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' Ac- tive particles that are in a glassy state [17] have given insights into random close packing [18] and it has been shown that shearing an active glass former leads to or- dering [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' However, the rheological properties of dense active colloids are largely unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' On the side of biological microswimmers, such as bac- teria or microalgae, it has been shown that the pres- ence of a small fraction of active swimmers in a fluid medium can fundamentally change the fluid’s rheological properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' It was found experimentally that pusher-type swimmers such as the bacterium Escherichia coli reduce the effective viscosity [20–24] while puller-type swimmers such as the microalgae Chlamydomonas reinhardtii can increase the viscosity [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' This response to the swim- ming behavior has also been explained in theoretical and numerical studies [26–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' Further, the rheological prop- erties of active fluids, that can model driven microtubules or active acting filaments, have been computed using field theoretical approaches [33–36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' ∗ gulce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content='bayram@bilkent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content='tr (a) (b) shear stress shear rate shear thinning fuid shear thickening fuid Newtonianfuid Shear thickening Newtonian Shear thinning (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' Schematic representations of (a) the sheared ac- tive colloidal suspensions with Lees-Edwards boundary condi- tions (the colour bar represents the distribution of shear force throughout the computational box), (b) the expected parti- cle arrangements in different fluids, (c) shear stress-shear rate curve for different fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' Here, we investigate the rheological properties of dense active colloids in two spatial dimensions using Brown- ian dynamics computer simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' Investigating the strain-stress curves shows that activity reduces and even destroys the stress barrier that a shear flow has to over- come to fluidise the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' In the steady state, the system’s shear stress reveals that particle motility fun- damentally changes the rheological properties of the sys- tem: at none or low activity the system is shear thinning, for intermediate activities it becomes Newtonian and at arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content='05429v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content='soft] 13 Jan 2023 2 very high activity it is shear thickening (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' The shear thickening behaviour is induced by particle clus- ters that stem from the active motion of the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' Therefore we refer to this new phenomenon as motility- induced shear thickening (MIST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' In fact, MIST is some- how a consequence of motility-induced phase separation (MIPS) [4, 5] in the bulk which shows a pre-clustering in the one-fluid phase even before full phase separation is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' These clusters are responsible for the shear- thickening under shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' The full rheological response is further well characterized using a power-law model for the stress as a function of the shear rate, which shows the continuous transition from shear thinning to shear thickening as the activity is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' Therefore the rhe- ological behaviour can be tuned by activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' SIMULATION METHOD We study a suspension of N self-propelled particles moving in two spatial dimensions under shear flow, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content='1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' In section II A, we introduce the model to be used for simulating this system, including the units and parameters used in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' We then describe in section II B the observables that we measure in our simulation in or- der to analyse (i) the phase transition behaviour and (ii) the rheological response of the system across the different self-propulsion and shear forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE5T4oBgHgl3EQfGQ5V/content/2301.05429v1.pdf'} +page_content=' Model In two spatial dimensions, the over-damped dynamics of the colloids are modeled by active Brownian particles in the presence of a steady shear rate ˙γ, dri dt = − 1 Γ � i 0 it indicates that the adjustment direction of the activation value is consistent +with the direction of the demand function. + +The amount of activation energy Ea is assumed to be 100 kJ/mol. Figure 5 shows the result of comparing +the attention utilization between the game model and the cobweb model. The attention amount (attention +is the behavioral and cognitive process of selectively concentrating on a discrete aspect of information, +while ignoring other perceivable information. Attention amount refers to as the allocation size of limited +processing resources), is less than 100 kJ/mol. If the attention amount is insufficient, then attention +resources can only meet part of the node demand, and the resource utilization rate of the SNM will +become higher than that of the cobweb model. When attention supply exceeds the demand of a node, the +cobweb model achieves balance to meet the needs of several nodes after a repetitive cycle. The SNM +meets the needs of all nodes, and the utilization rate of attention resources is higher than that of the +cobweb model. + +Comparisonofloadbalance +9 +8 +nodes +hhhl +6 +Numberof +L +4 +3 +1 +0 +No.1 +No.2 +No.3 +No.4 +No.5 +No.6No.7 +No.8 +No.9 +No.10 +Numberofactivationenergy amount +Iattentiongamemodel +cobwebtheoremmodeFigure 5. Comparison of activation energy values. After the change in the initial value of the activation +energy, the number of iterations increases depending on the difference between the initial activation +energy value in the cobweb model and the balanced energy value. The iteration of the attention game +model can be adjusted according to the difference in the activation energy between supply and demand. A +sizeable adjustment is required to reach the Nash equilibrium state if a large difference exists between the +supply and demand + +Figure 6 shows the cycle times of the SNM and the cobweb model that needs to achieve the Nash +equilibrium. As shown in the figure, the equilibrium activation energy value of the nodes is 20 kJ/mol in +the SNM, and the activation energy is 120 kJ/mol in total. If the initial value of the activation energy is +changed, then the initial activation energy value of the cobweb model is higher than the energy +equilibrium value and requires abundant cycle time. The SNM in each cycle can adjust the activation +energy according to the variance of the activation energy. The variance and adjustment range are large, +and the SNM eventually reaches the Nash equilibrium point. +Figure 6. Cycle times of the SNM and the cobweb model that are needed to achieve the Nash equilibrium. +When the supply falls short of demand, attention resources can only meet the demands of several nodes, +and the resource utilization rate of the SNM becomes higher than that of the cobweb model. If the supply +exceeds demand, then the cobweb model can reach equilibrium after repeated iterations and can meet +only part of the demands of nodes. However, the SNM can meet the demands of all nodes, and its +resource utilization rate is higher than that of the cobweb model +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +40 +60 +80 +100 +120 +Attention Resource Utilization(percentage) +Amount of activation energy(KJ/mol) +Comparison of activation energy values +attention game model +cobweb theorem model + + +CONCLUSION +The authors presented a new model for SL comprehension based on spatial information. This process uses +game theory to simulate the human attention suppression and enhancement process. This process also +joins the forgetting function of human memory traces to compute the initial state of the node. Memory is +encoded with specific (semantic) meaning, or refers to information that is encoded along a spatial and +temporal plane. Although the semantic network provides a functional view of how knowledge may be +organized in the brain, it does not provide a clear model of how semantic memory might be presented in +the brain (see Cacha et al., 2017). Spreading activation reveals that information can be stored in SNs for a +long time, in which a network node is a linguistic concept and the nodes are connected through the +correlation. An algorithmic method is proposed according to selective functions, and its effectiveness was +verified using an example. The results show that the proposed method improves the performance of SL +comprehension. +ACKNOWLEDGMENT +The authors would like to thank Chunda Liu from the National Center for Sign Language and Braille for +helping in stimulus preparation and data collection. This paper forms an expanded and revised version of +a conference paper at the 14th IEEE International Conference on Cognitive Informatics & Cognitive +Computing (ICCI* CC) at Tsinghua University, Beijing, July 6-8, 2015. The authors are grateful to Dr. +Raymond Chiong, and two anonymous referees for their helpful comments. +Conflict of Interest +The authors of this publication declare there is no conflict of interest. +Funding Agency +This research was supported by the Beijing Municipal Natural Science Foundation [4202028]; National +Social Science Foundation of China [21BYY106]; National Natural Science Foundation of China +[62036001, 61866035, 61966033]; Premium Funding Project for Academic Human Resources + +Comparison of cycle times +40 +Number of Iterations +35 +30 +25 +20 +15 +10 +5 +0 +1 +5 +10 +15 +20 +25 +30 +Activationvalue +attentiongamemodel +cobwebtheoremmodelDevelopment in Beijing Union University [BPHR2019CZ05]; Jiangsu Province Key R&D Program +(Industry Prospects and Key Core Technologies) [BE2020047]; and the characteristic-disciplines oriented +research project in Beijing Union University [KYDE40201702]. +REFERENCES +Anderson, J. R., Bothell, D., Byrne, M. 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Gallaudet +University Press. + + + diff --git a/kb_43/content/tmp_files/load_file.txt b/kb_43/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1227a0c0d7b76186284139636e3d746384a817fc --- /dev/null +++ b/kb_43/content/tmp_files/load_file.txt @@ -0,0 +1,836 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf,len=835 +page_content='Semantic Network Model for Sign Language Comprehension Xinchen Kang (22f3d431-dcc0-42a5-8e2b-c26464e0654d) Beijing Union University, China Dengfeng Yao (ae88317c-d091-4f41-97f1-b1e6be00ca68) Beijing Union University, China Minghu Jiang (ea1cc43b-eee9-4185-8d97-edeac9186268) Tsinghua University, China Yunlong Huang (cc1ddbf2-64b9-4c51-b553-0ebe52a8d645) Tsinghua University, China Fanshu Li (64642396-6e95-456f-a4f2-47ff81a23d6e) Beijing Union University, China ABSTRACT In this study, the authors propose a computational cognitive model for sign language (SL) perception and comprehension with detailed algorithmic descriptions based on cognitive functionalities in human language processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The semantic network model (SNM) that represents semantic relations between concepts is used as a form of knowledge representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The proposed model is applied in the comprehension of sign language for classifier predicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The spreading activation search method is initiated by labeling a set of source nodes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' concepts in the semantic network) with weights or "activation,"' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' and then iteratively propagating or "spreading" that activation out to other nodes linked to the source nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The results demonstrate that the proposed search method improves the performance of sign language comprehension in the SNM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Keywords: Attention, Cognitive Processing, Comprehension, Decision-Tree, Game Theory, Linguistics, Perception, Semantic Network, Sign Language INTRODUCTION Sign language (SL) comprehension is a fundamental task for computational linguists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Two types of algorithms have been proposed: (1) rule-based methods (Supalla, 1982), and (2) statistical methods (Bauer & Heinz, 2000; Huenerfauth, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Rule-based methods lack the capability of planning the elements in the entire scene (Liddell, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The method of modeling infinite natural language input through finite rules, especially minor rules, barely meets all requirements of SL processing (Yao et al., 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Therefore, statistical methods are the preferred type of algorithm for SL comprehension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Statistical models can be applied to spoken languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Given the abundant data resources of spoken languages in the digitalized Internet age, statistical models can be applied readily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' However, the raw and annotated corpora of SLs are insufficient because collecting and annotating SL videos are tedious and difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Data sparsity consequently remains as the most serious problem when applying statistical models onto SLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' For example, the real-time factor (RTF) of the SL video corpus is 100; that is, an hour corpus requires at least 100 hours of annotation (Dreuw et al., 2008b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Simulating SL comprehension using traditional statistical models and machine-learning methods isdifficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Thus, reliable methods for establishing a signer’s 3-D model (which is the process of developing a mathematical representation of any three-dimensional surface of moving trajectories of signers in the space for SL via specialized software) for SL corpus building and technologies for annotating a large-scale SL video corpus automatically must be developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Unlike the spoken language that is “a set of values that change with the passage of time” (Huenerfauth, 2005), SL does not have a writing system and thus cannot be saved in any form of written texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The natural language-processing system relies on texts to process spoken languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' This system records only the written text that corresponds to speech flows and relies only on the literacy of the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' On the other hand, the SL system comprises information from multiple modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Examples of such information are the hand shape, hand location, hand movement, hand orientation, head tilting, shoulder tilting, eye gazing, body gestures, and facial expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The considerable information from multiple channels in SL conveys linguistic meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' This multi-modality nature of SL poses difficulties for the coding of SLs into a linear single-channeled character string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' In addition, SLs have writing systems, such as the Sign Writing system (Sutton, 2010), ASL-phabet (Supalla et al., 2008), and HamNoSys (Prillwitz et al., 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' However, these systems have a limited number of users (Johnston, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Many linguistic details are lost because of the multi-modality nature of SL during the translation of SL into its corresponding writing system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' SLs may be understood by directly matching the visual–spatial characteristics of SL with the semantic units in the brain rather than applying written texts as an interpreting medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Here, semantic units are generally used for processing natural languages; these units or nodes contain some information, which are used as knowledge representations form semantic units (Geva et al., 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Such direct matching also represents the most natural way of comprehending SLs in the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' From this perspective, the authors present a computational cognitive model for SL comprehension that is based on the cognitive functionalities of the human brain combined with a knowledge representation theory of artificial intelligence (Shuklin, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Visual–spatial mechanisms are exploited to express the grammatical structures and functions in SL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Visual–spatial perception, memory, and mental transformations are prerequisites to grammatical processing in SL (Emmorey & Corina, 1990) and are central to visual mental imagery (Farah, 1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' A series of experiments have been conducted to investigate visual attention (Neville et al., 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Movement recognition in peripheral vision is important in sign perception because the signers mainly look at the face instead of tracking the hands when they communicate through SL (Siple, 1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Therefore, identification of lexical-level information depends on the peripheral vision system when signs are produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The recognition of movement directions is the selective function of peripheral vision (Bonnet, 1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Whether deaf people only have a strong peripheral vision or efficiently allocate attention to peripheral vision remains unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Stivalet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' (1998) showed that visual attention processing can be changed by auditory deprivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' They determined that deaf people do not shift their attention when processing the information (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='e., alphabet set) presented in the central vision field, whereas hearing subjects must shift their attention to search for the alphabet set continuously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' (1998) also found that lack of auditory input causes weak and selective (or highly distributed) visual attention among deaf children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Stivalet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' (1998) proposed that effective visual processing is caused by intermodal sensory compensation; that is, the strong allocation of visual attention can be attributed to neuron reorganization caused by auditory deprivation from birth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Recent magnetic resonance imaging evidence supports this hypothesis (Bavelier et al., 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' These findings are selective attention cases, in which attention selectively processes certain stimuli but ignores other stimuli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The cases refer to the selective orientation and concentration of the senses (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='e., visual, auditory, taste, and tactile senses) and consciousness (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='e., awareness) of people on certain targets (towards other factors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Studies on attention have failed to describe human attention at the biological level in detail, as a person cannot focus continuously because the brain automatically suppresses activity when attention reaches its limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Emmorey and Reilly (2013) determined that when locations in a signing space (SL expressions streaks the space) function topographically, spatial changes tend to be noticed easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Thus, location information indicating the spatial position of associated referents can be encoded and stored semantically in memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' However, spatial locations with a primary distinguishing function of referents are encoded in a different way and tend to be discarded from memory once the referential function is no longer required by context (Emmorey & Reilly, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Bavelier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' (2001) claimed that only the posterior middle temporal gyrus and the medial superior temporal cortex of deaf signers are highly active while perceiving movements in peripheral vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' This phenomenon is unobservable in hearing signers who have skillfully grasped signs, indicating that auditory deprivation results in a shift to stronger movement attention in the visual periphery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Deaf people can easily reply to the attention and visual monitoring of their peri-personal space (Bavelier et al., 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Neville et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' (1998) determined that the classic language area in the left hemisphere, particularly the left perisylvian, of both deaf and hearing subjects is activated when reading English sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The right hemisphere, including the right perisylvian, of deaf people is also activated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' They argued that spatial processing is of great importance to sign grammar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Thus, the SL comprehension process of deaf people employs neurons at both high and low levels in the neural network, which are connected with each other by edges, and generates high-level features via feature combination processes that are realized by combining the weight on the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' For example, low-level visual edge features are assembled, processed, and sent to the high level to form the angle, shape, and other higher features (Bertasius et al., 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' High- level neurons form features that gradually approximate the semantics in turn, such as simple shapes, simple targets, and real objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The activation of high-level neurons during the reconstruction process also reacts with the low-level neurons and adjusts and corrects deviations and losses (Bertasius et al., 2015); a temporal pattern appears in the horizontal structure connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The neurons can make predictions of the state at the next point of every time point through a horizontal connection based on the information of their current status (Hawkins et al., 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' SEMANTIC NETWORK MODEL (SNM) Model of semantic networks (SNs) are generally used for processing natural languages (Shuklin, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' SNs, as knowledge representations, are extensible and have been used to model mental disturbances (Geva et al., 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The semantic network (which is a network that represents semantic relations between concepts, is used as a form of knowledge representation, here it is based on SL information processing of human brain cortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The edges connect different nodes in the network and represent the strength or weakness of the correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' After being set up, the semantic network is stored in long-term memory for future retrieval and extraction to be encoded as semantic memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Outside stimuli at a certain time can be the demand of a person on specific knowledge and information to activate the demand on the extraction of useful information of long-term memory (Sedikides & Skowronski, 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The activation process of the stored network works in a form of spreading in the memory (Collins & Quillian, 1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The semantic model, which is based on SL information processing of human brain cortex, is developed accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Different areas of the brain cortex are involved in the processing and are connected in a hierarchical manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Low-level information from sense organs is first processed in the primary information-processing regions of the brain cortex and is then transferred to high-level regions for further processing, such as abstracting, integrating, and interpreting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The detailed description and illustration of this hierarchical structure are summarized in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Hierarchical structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Low-level areas in the hierarchy generate specific information that increases speed and contain further details, whereas high-level areas form stable spatial invariance, change slowly, and show high-level semantic object expression (Adapted from Yao et al., 2015) In SL communication, both substantial and semantic information (substantial information includes hand shape, hand location, hand movement, hand orientation, head tilting, shoulder tilting, eye gazing, body gestures, and facial expressions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' and semantic information is represented into semantic concepts by these substantial SL information) almost exclusively relies on signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' However, accurate SL information analysis and prediction remain as challenging tasks in the field of natural SL processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Three main tasks are, namely, capturing, decoding, and extracting the physical characteristics and relationship of signs (perception stage), matching the decoded cognitive representations with the stored semantic information (memory stage), and completing the machine translation process of SL information (judgment stage).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' This process of cognitive processing and understanding during SL communication is based on the PMJ principle of “from the definition and extraction/annotation of cognitive representation (Stage P) to the feature storage in line with the cognitive economy principles (Stage M), and then to the output of the classification and judgment (Stage J).”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The P→M→J (PMJ) principle exhibits a complete fine processing frame, the detailed illustration, and description of SL comprehension frame based on the PMJ principle is summarized in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' SL comprehension frame based on the PMJ principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Perception refers to acquiring sign information through selective attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The information is limitedly processed by the brain if prominence is given to useful and important information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Other information may be filtered out or suppressed when sources for information processing are limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Memory refers to the spreading activation process, in which input information is coded, and one intends to store the information for a short period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Judgment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='targeted ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='wer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='targeted sen antics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='sign semantic concept ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='semantics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='rea ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='representation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='Area H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='hign deve semantid ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='high-level semartic feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='feature detectors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='Area A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='low-evel senantic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='low-level semantic feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='low-level s emanticf eature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='feature detectors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='cogritive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='representation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='Higher leve visual ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='handshape ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='location ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='oriention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='moveme rt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='No-manual feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='AreaV4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='abstractions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='Edge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='outine ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='Simple shape ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='Area V1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='detecton ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='edge feat ure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='ed ge feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='Retina ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='oves ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='pixel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='pixel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='pixel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='pixel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='physical ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='characteristics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='一refers to the process in which the perceived information or the information stored in memory is ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='compared, matched, or classified, and a decision or prediction is made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' After the spreading activation, the network records the attention features of users and activates their future preferences (Yao et al., 2015) Concepts are in the form of network storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The different concepts are stored in different functional areas in both hemispheres of the human brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The same or similar concepts are stored in same or adjacent regions of brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Specific information of entities in the outside world, such as humans, animals, or tools, is represented by the concept network in the human brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' This concept network (A concept is an abstract idea representing the fundamental characteristics of what it represents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Concept network consists of these abstract concepts) is, in turn, connected with the lexical network from mental lexicon in the left temporal lobe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Such specific information from mental lexicon will be employed to facilitate SL production during which the SL users generate classifier hand shapes under the guidance of the knowledge and rules of SL classifier predicates (Valli & Lucas, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Here, classifier predicates are made by combining small meaningful unites to create bigger units, the main units being the hand shape and the movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' This condition implies that findings from brain research can provide knowledge and guidance for the cognitive computational modeling of classifier predicate comprehension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' In order to obtain a deep understanding of sign lexical semantics, a cognitive processing model, which is based on the cognitive mechanism of human brain, is established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The cognitive processing model would activate the concept network of the associated classifier hand shapes in the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Here, classifier predicates differ from traditional linguistic units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Traditional methods, such as the syntactic tree, cannot satisfy the generation of the classifier predicates (Huenerfauth et al., 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' DECISION-TREE BASED ALGORITHMIC METHODS The authors use SNs as the knowledge representation and organization mode of SL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The relationship in semantic networks represents a type of information among nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Nodes with a complicated relationship with other nodes contain additional information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Such nodes require further effort to be understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Consequently, the authors simulate selective attention (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='e., the processing of visual or auditory input based on whether it is relevant or important).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' They selected particular representations to enter perceptual awareness and therefore guide behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Through this process, less relevant information is suppressed by humans using the proposed algorithmic methods to accentuate the nodes selectively and suppress the unessential nodes (Chelazzi et al., 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The emergence of 3-D-based sensors, such as Kinect by Microsoft and Leap Motion (Yao et al., 2014), has improved studies on sign recognition from video-based to 3-D-based sign recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' However, this transformation makes traditional video-based SL recognition methods inapplicable to 3-D-based SL Perception Memon Judgment Attentional enha ncement and suppress ion 0 Spread activation Interactive activationrecognition technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Large training data are required for valid recognition in 3-D-based SL recognition technologies because of the low operation efficiency of the rotatable joint-based sorter and the matching techniques for sign signal recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' (2014) proposed a decision tree-based algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The algorithm aims to achieve a high-precision and real-time performance of SL automatic perception according to the features of Leap Motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The authors adopted this method as the first step of SL comprehension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Attention Function The authors propose the following attention function: ������������������������ = ∑ ������������������������������������ ������������������������������������ ������������ ∑ ������������������������������������ ������������������������ (1) where ∑ ������������������������������������ ������������������������������������ ������������ denotes the sum of the semantic relation weights around the semantic node x, ∑ ������������������������������������ denotes the sum of all semantic relation weights,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' and ������������������������ represents the activation value on the semantic node x after the spreading activation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Semantic Matching Cognitive units in the memory network compete with one another based on certain rules to obtain more of the limited attention resources and more energy for a more active state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' SL comprehension supports interactive activation models (Gutierrez et al., 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Therefore, judgment is the outcome of the attention competition game in the spreading activation, which is a search algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The search algorithm is initiated by labeling a set of source nodes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' concepts in a semantic network) with weights or ”activation,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' and then iteratively propagating or “spreading” that activation out to other nodes linked to the source nodes processes of the human brain (Crestani, 1997; Preece, 1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' A semantic matching algorithm based on activation spreading modes is proposed to determine the most appropriate semantic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Activation starts to spread from the corresponding nodes of the signs presented by the signer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The activation value of the stimulus node (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='e., signs to be perceived before the start of spreading) must be calculated first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' In particular, the increment in the interest value of object concept must be calculated, and this concept node must be used as an initial node for the spread study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Activation spreads to the neighboring nodes, which usually have a lower activation value than the source value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Therefore, introducing an activation attenuation factor for decreasing activation over the path length in the closed interval [0…1] is mandatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' That is, for every propagation through an edge a loss of activation is considered (Neumann et al., 1993; Rocha et al., 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The activation spreading process can be expressed as follows: ������������������������(������������ + 1) = ������������������������(������������)������������������������������������(1 − ������������) (2) where ������������������������(������������ + 1) represents that the value is spread from node x to y at time, t+1, ������������������������(������������) represents the activation value that was spread at node x at time, t, ������������������������������������ signifies the link between nodes x and y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' and δ is an attenuation factor used to describe the energy loss caused in the activation spreading process (Jiang & Tan, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Spreading activation theory states that the activation of human memory “chunks” (the content of any buffer is limited to a single declarative unit of knowledge, called a chunk) is determined by two factors (Anderson et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' al., 2004; Anderson, 2013), namely, the use history of the memory chunk and the correlation between the memory chunk and the current retrieval information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' These two factors calculate the weights and determine whether the chunk is activated and selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' This assumption has been verified by experimental cognitive psychology, and the calculation model has been established (Roelofs, 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The authors must use moments to express the distance in each activation time with the current time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Time units may be per hour as a unit and may also be the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' With the day as a unit, we can count the historical value in the previous day as the activation value of the first day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The algorithm based on the theory of memory activation can improve SL understanding, which is sometimes highly sensitive to time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' At node y, the largest number of neighbor nodes is (n-1); thus, the maximum of ������������������������(������������ + 1) can be expressed as: ������������������������(������������) = [������������1, ������������2, … , ������������������������]������������ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='e., the initial value of the semantic network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Where I1, I2, …, In are these activation value of neighbor nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' If activation spreads from a node in many directions, then its adjacent nodes obtain a low activation value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The adjacent nodes give a feedback value of their resonance energy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='e., contributing structure with the lowest potential energy) to the co-adjacent nodes after they absorb the activation value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The following equation is therefore used: Iz(t + 1) = Oz(t) + ∑ Ox(t)Λxz(1-δ) all actived x (3) where Oz(t) denotes the activation value of node z at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Given that the quantity of activated information is limited, the nodes that obtain less resonance information are equivalently inhibited and are less likely to be activated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The activation value distribution in the resonance process conforms to the human attention model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Attention Game Process Cognitive units in memory network compete with one another by certain rules to increase the possibility of obtaining more human attention resources and more energy that will improve activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' This phenomenon is called a game process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The authors use game theory (Myerson, 1997), which is the study of mathematical models of conflict and cooperation between intelligent rational decision-makers and attempts to achieve the largest cognitive gains with the least energy possible, as a reference to simulate the attention enhancement and suppression processes that are selective attention processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' In other words, when visually searching for a non-spatial feature or a perceptual feature, selectively enhancing the sensitivity to that specific feature plays a role in directing attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' When people are told to look for motion, then motion will capture their attention, but attention is not captured by motion if they are told to look for color (Reynolds & Chelazzi, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Activated results consistent with cognitive features can then be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The authors assume that the game contains n nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' ������������������������ ′ and ������������������������ " are the two selectable strategies for node i, and they represent the acceptance and non-acceptance of the change in the attention function (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='e., ������������������������ ′, ������������������������ " ∈ ������������������������ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The corresponding gain can be represented by ������������������������ ′ and ������������������������ " , and ������������������������ ′, ������������������������ " ∈ ������������������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' N nodes are assumed to reach an agreement before participating in the game to introduce the Nash equilibrium (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='e., each node only selects a specific strategy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The authors let ������������∗ = (������������1 ∗, … , ������������������������∗) represent the agreement, where ������������������������ ∗ is the strategy of the node i specified in the agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Nodes comply with this agreement only when the benefit from complying with the agreement is larger than that from not complying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' This agreement constitutes Nash equilibrium if any node abides by this agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Thus, the Nash equilibrium is written as follows: ������������������������(������������������������ ∗, ������������−������������ ∗ ) ≥ ������������������������(������������������������, ������������−������������ ∗ ), ∀si ∈ Si (4) where the combination of strategy ������������∗ = (������������1 ∗, … , ������������������������∗) is a Nash equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Given that other nodes select ������������−������������ ∗ = (������������1 ∗, … , ������������������������−1 ∗ , ������������������������+1 ∗ , … , ������������������������∗), ������������������������ ∗ is the optimal strategy of each node i (Myerson, 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The attention game process determines whether the nodes need adjustment or need to be changed on the basis of the attention function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The activation energy distribution will reach a state consistent with the human attentive distribution after adjusting the activated value distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Nodes of the spread SNs have their own activation energy threshold values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The source node in the attention game process that represents a presented sign has the maximum activation value O in the present SNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' All equidistant nodes will participate in the game based on the attention function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The nodes with low activation energy (defined as the minimum energy required to start a chemical reaction) of a reaction is denoted by Ea and given in units of kilojoules per mole (kJ/mol) or kilocalories per mole (kcal/mol)), threshold must be removed through a screening process to prevent them from participating in the enhancement and suppression processes of activating the most likely node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' In the proposed screening, the authors ignore the nodes with a significantly low activation value to be activated in the enhancement process instead of lowering the possibility for other nodes to be activated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The difference between the attentive readjustment in the present attention game process and the previous attentive allocation causes the instability in the overall cognitive structure of users to decrease knowledge credibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Thus, a new cognitive structure must be determined at a cost as follows: Cost(t, i, si, ui, SN) = �n−1 ∑ �Ii(t + 1) − Oi(t)� 2 n i=1 (5) where ������������������������(������������ + 1) denotes the activation value that is conveyed from one node at time t+1 to node i, and 0i (t) denotes the activation value of node i at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Therefore, the total cost is attributed to the change in the activation energy of all nodes in the SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The goal of judgment is to achieve the overall optimal gain with a minimal computing cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The gain function in the attention game process must then be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' As the optimal strategy for node i, ������������������������ ∗ must minimize the distribution change that refers to the distribution change in the activation values of the overall network changed by the decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The amount of spreading activation energy is fixed in the total process of activation spread in the SN; thus, the semantic node energy enhancement must be accompanied by reduced node energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The attention parameters are affected by the overall distribution change in activation energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The activation energy enhancement increases the impossibility of activating this node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Such activation is the ultimate purpose of each node (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='e., the node obtains the gain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Accordingly, the gain function is presented as follows: Gain(t, i, si, ui, SN) = �∑ Ix∈{neighbor node}(t+1) num(all x) j=1 −∑ Ox∈{neighbor node}(t) num(all x) j=1 �(1−δ) num(all x) (6) where SN represents the current semantic network, num(all x) represents the number of neighbor nodes x of node i, ∑ Ox∈{neighbor node}(t) num(all x) j=1 denotes the sum of the activation value that was spread of all node i neighboring nodes at time t, the gain function is expressed as the attention gain of neighbor nodes x of node i after the enhancement and suppression processes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' it represents the benefit a node gets by unilaterally changing their strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The utility function of the attention game process can be determined as follows: Max�������������������������(������������, ������������������������ ∗, ������������−������������ ∗ )� = Gain(������������, ������������, ������������������������, ������������������������, ������������������������������������) − Cost(������������, ������������, ������������������������, ������������������������, ������������������������������������) (7) where ������������������������ ∗ is the optimal strategy of each node i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' ������������−������������ ∗ is the strategies set of other nodes except node i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' only when ������������������������(������������, ������������������������ ∗, ������������−������������ ∗ ) reaches the maximum, ������������������������ ∗ is a Nash equilibrium of node i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The utility of the other nodes will be affected by the decision of all other nodes because of the fact that the total quantity of activation energy is fixed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='e., attention is limited) in the attention game process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' When each node selects a decision for itself, it also considers the possible decision of other nodes and selects the “Nash equilibrium point” with maximum utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' This scenario is consistent with classical game theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The authors select a Nash equilibrium decision for each node through the utility function of the attention game that is defined by Equation (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' METHODS Data Sets and Experimental Settings All data from the authors’ experiments are obtained from the Tsinghua University–Chinese SL Corpus (TH–SLC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The data mainly comprise SL expressions of idiom stories and life fragments of deaf students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' No automatic annotation software based on videos is currently available because the annotation process for SL videos is time consuming and requires expert knowledge in dual language (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='e., Chinese language and Chinese SL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Video annotation is also time consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Specifically, it takes about 30 hours for the annotation RTF (real-time factor) of a parliamentary speech (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='e., One hour of speech requires 30 hours of annotation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' However, the annotation RTF (real-time factor) for a full annotation of all manual and non- manual components of an SL video can reach up to 100 hours (Dreuw & Ney, 2008a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Therefore, such a corpus is significantly small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' For example, the Aachen Boston database contains American SL and has annotated 201 English sentences (Dreuw & Ney, 2008a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The authors spent a year collecting more than 2000 sentences, but only 416 sentences containing 2496 signs were marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The authors asked 20 deaf students to select 300 sign pairs from 2469 annotated signs in TH–SLC and to judge the relevance of the sign pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The correlation values range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' For convenience, a five-point scale is used to assess the correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The sign pairs were obtained using a marked correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The authors establish an SN based on the word similarity computing method of HowNet (Liu & Li, 2002) to determine the connection weight of the network to validate the effects of the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The authors introduce the continuous bag-of-words (CBOW that predicts the current word from a window of surrounding context words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The order of context words does not influence the prediction (CBOW assumption) model (Mikolov et al., 2013), and the HowNet (Liu & Li, 2002) method as the baseline methods using the same recommended parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The efficiency of the utility function of the attention game process is evaluated in terms of word correlation computation, and the model complexity is analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Word Relatedness Computation Each model in this task needs to compute the semantic correlation of the given sign pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The correlation between the experimental results of the model and human judgment reflects upon the model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The authors selected 290 signs for the closed set and 10 signs for the open set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Spearman’s correlation between model correlation score and human judgment correlation score was calculated for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Spearman correlation coefficient is defined as the Pearson correlation coefficient among the ranked variables (Myers & Well, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' For a sample of size N, original data ������������������������, ������������������������ are converted into grade data������������������������, ������������������������, the correlation coefficient ρ is defined as follows: ρ = 1 − 6 ∑ di 2 n(n2−1) (8) where the difference between the observations of the two variable levels is set as ������������������������ = ������������������������ − ������������������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' If there is no duplicate value in the data, and two variables are completely monotonic correlation, the Spearman correlation coefficient is +1 or -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' RESULTS For CBOW, the correlation scores of the two words are calculated using the cosine similarity of word embedding (Mikolov et al., 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The evaluative results of the baseline methods and the proposed SNM method in the closed test and in all test sets are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Evaluative results Data Set Closed Test All Test Sets (Including Open Test) Spearman’s Rank Correlation Coefficient Method 290 pairs 300 pairs CBOW (baseline method) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='4843 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='4869 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='4136 Word similarity computing based on HowNet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='6157 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='6174 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='6052 Proposed SNM method 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='6951 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='7063 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='6437 The evaluation results show that the proposed SNM method is better than the baseline method in 290 and 300 word pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' This finding indicates that the cognitive mechanism of sign comprehension is essential to understanding the meaning of signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The internal structure, such as location, orientation, hand shape, and movement, contains rich semantic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' However, deep learning methods, such as CBOW, consider the external context, but ignore the internal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Using the computing method of word similarity based on HowNet results in only a rough semantic computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' For example, adding 10 new sign pairs negligibly changes the performance of these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' In other words, these methods can still handle new signs with improved performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The semantic correlation of these new sign pairs calculated by the proposed method is close to human judgment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Figure 3 shows the quantitative analysis of the attention game process for two signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Each hand shape of the two signs has at least 20 related semantic lexicons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The stimulus information and permutation of each node are shown in the first and second columns from high to low according to the activated value after the activation spreading process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Only 10 semantic lexicons that are maximally activated are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The permutation of each node is shown in columns three to seven from high to low according to the activation value after the end of the first to fifth attention games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The top 10 lexicons are also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The semantic lexicons in the blue background rank high after the games, those in the green background rank low after the games, and those in the white background are unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Examples of attention games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The semantic lexicons in the blue background rank high after the games, those in the green background rank low after the games, and those in the white background are unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' This trend shows that the ranking of other semantic lexicons below slightly changes after the semantic lexicon that ranks highest becomes unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' This condition is due to the source that corresponds to the attention model being determined after several game processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Figure 3 also shows that significant changes occur during the ranking of the semantic lexicons in the first and second instances after the first several games, whereas only a few changes occur in the following stimulus games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' This trend shows that the ranking of lower semantic lexicons slightly change after the semantic lexicon that ranks highest becomes unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' This condition is due to the source that corresponds to the attention model being determined after several game processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Attention is also assigned to other nodes in accordance with the attention game process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Humans reach a steady state after thinking about problems constantly, and the result negligibly changes if they rethink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Nearly no change is observed in the result after several rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Several semantic lexicons related to the signs are contained in the text set; thus, a few possible changes occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The result of the attention game model conforms to human cognitive rules to a certain degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Attention is also assigned to other nodes in accordance with the attention game process (here, efforts have been made in modeling according to the mechanism of human attention).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The result of the SNM conforms to human cognitive rules to a certain degree (Gutierrez et al., 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' For example, the authors assume that deaf people understand the signs shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Deaf people usually search for many familiar and specific nouns or signs in a spreading activation mode to comprehend classifier predicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' After all activated values are calculated; the activated nodes are graded and sorted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' A high-activated value of the node indicates the importance of the interested object or concept represented by the node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' This shows that deaf people are familiar with the concept node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Similar to the attention game process shown in Figure 3, ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='activation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='activation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='activation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='activation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='activation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='activation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='value ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='value ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='sorting after ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='sorting after ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='after ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='the first ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='the second ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='the 3rd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='the 4th ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='the 5th ' metadata={'source': 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+page_content=' This result shows that the most common subjects for deaf people are typical subjects that represent classifier predicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' DISCUSSION Compared with that of existing models, the complexity of the proposed model is reflected mainly on the computational cost of the memory stage and the judgment stage (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='e., the computational cost of spreading activation and the attention game at time (t + 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The cost is a dynamic value and related to two factors, namely, the activation state of the current sign and the current cycle as the first activation of the sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Therefore, the value changes regardless of the choice of the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' This outcome is consistent with the strong dynamics of sign information, which can reflect the influence of information in different periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' In the memory stage, the time complexity of computing ������������������������(������������) is unity; thus, the time complexity is related to the total amount N of activation energy and cycle times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The time complexity of each activation in each cycle is n × 1 = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Space complexity is the storage space of each node and the semantic relation weight according to semantic similarity (semantic similarity can be estimated by defining a topological similarity, by using ontologies to define the distance between terms/concepts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Therefore, unlike the general model such as cobweb theorem model and vector space model, where the SNM increases the overhead in time complexity and space complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The model also increases the matching time of query nodes and weights in the current activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' However, the overhead at this time can provide more effective results than an invalid spreading and can be accepted by users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' In the judgment stage, when the node selects the game strategy to change its activated energy value, the convergence speed of adjusting the cognitive benefits to its own utility maximum “Nash equilibrium” is an important measure of evaluating the SNM (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='e., the cycle times of an attention game process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' For the attention game, the Nash decision of different semantic nodes must minimize the change cost of the activation energy distribution of the entire network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The Nash equilibrium point decision for each node is selected using the utility function defined in the SNM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' This process is repeated until the overall network activation energy distribution change is less than the specified threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The node needs to solve n-order nonlinear equations in every cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Therefore, the performance of the convergence speed of the SNM is indicated by the number of game cycles that the network requires to reach the Nash equilibrium point (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='e., the computing times of calculating the corresponding equation by each node in a game process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The square root of the sum of the variance of activation value ������������������������(������������ + 1) of each adjusted node is directly reflected by the rate of convergence in the game process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' To verify its effectiveness, the attention game model is compared with the traditional model in terms of load balancing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' In the traditional method, the activation value of each node is certain (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='e., the value is not enhanced or inhibited).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The experimental results are shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The results show that the load balance performance of the attention game model is better than that of the traditional model because the attention game model adjusts the activation strategy after the activation of each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' When the change cost of the energy distribution of the entire network activation is larger than the specified threshold, the human brain adjusts the strategy to inhibit the activation energy value in the next cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' In doing so, the free competition and distribution of attention for each node according to the attention game model can be assured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The result is obtained through the overall competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The load of attention of the network is balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The traditional model assumes that the activation energy value of each node is certain because the brain activation energy resource amount is constant in a period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The brain selects the node with a low activation energy value and performs the allocation of attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' This allocation causes the attention load of several nodes to be excessively large or unutilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Comparison of load balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The load balance performance of the SNM is better than that of the traditional model because the SNM adjusts the activation strategy after the activation of each node The proposed SNM model used Nash equilibrium to simulate the energy activation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' In order to quantitatively analyze the effects of Nash equilibrium, the authors compared the SNM with the cobweb theorem model (Pashigian, 2008) in terms of different activation energy amounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The cobweb theorem is expressed as follows: ������������(������������ + 1) = ������������(������������) + ������������ ��������������������������(������������)� − �������������������������′(������������)�� (9) where r is the adjustment parameter of the activation value, �������������������������(������������)� is the activation function of a node, ������������(������������) is the activation value at time t, �������������������������′(������������)� is the attention allocation function, ������������′(������������) is the expectation activation value at time t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' and �������������������������(������������)� − �������������������������′(������������)�is the excessive demand function that represents the actual gaps between the activation value and activated allocated value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' A large gap indicates a high activation value of the Nash Equilibrium of node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The parameter (r) indicates the actual speed and strength of adjusting the activation value according to the attention distribution condition in the last moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' When r > 0 it indicates that the adjustment direction of the activation value is consistent with the direction of the demand function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The amount of activation energy Ea is assumed to be 100 kJ/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Figure 5 shows the result of comparing the attention utilization between the game model and the cobweb model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The attention amount (attention is the behavioral and cognitive process of selectively concentrating on a discrete aspect of information, while ignoring other perceivable information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Attention amount refers to as the allocation size of limited processing resources), is less than 100 kJ/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' If the attention amount is insufficient, then attention resources can only meet part of the node demand, and the resource utilization rate of the SNM will become higher than that of the cobweb model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' When attention supply exceeds the demand of a node, the cobweb model achieves balance to meet the needs of several nodes after a repetitive cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The SNM meets the needs of all nodes, and the utilization rate of attention resources is higher than that of the cobweb model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Comparisonofloadbalance 9 8 nodes hhhl 6 Numberof L 4 3 1 0 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='1 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='2 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='3 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='4 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='5 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='6No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='7 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='8 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='9 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='10 Numberofactivationenergy amount Iattentiongamemodel cobwebtheoremmodeFigure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Comparison of activation energy values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' After the change in the initial value of the activation energy, the number of iterations increases depending on the difference between the initial activation energy value in the cobweb model and the balanced energy value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The iteration of the attention game model can be adjusted according to the difference in the activation energy between supply and demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' A sizeable adjustment is required to reach the Nash equilibrium state if a large difference exists between the supply and demand Figure 6 shows the cycle times of the SNM and the cobweb model that needs to achieve the Nash equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' As shown in the figure, the equilibrium activation energy value of the nodes is 20 kJ/mol in the SNM, and the activation energy is 120 kJ/mol in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' If the initial value of the activation energy is changed, then the initial activation energy value of the cobweb model is higher than the energy equilibrium value and requires abundant cycle time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The SNM in each cycle can adjust the activation energy according to the variance of the activation energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The variance and adjustment range are large, and the SNM eventually reaches the Nash equilibrium point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Cycle times of the SNM and the cobweb model that are needed to achieve the Nash equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' When the supply falls short of demand, attention resources can only meet the demands of several nodes, and the resource utilization rate of the SNM becomes higher than that of the cobweb model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' If the supply exceeds demand, then the cobweb model can reach equilibrium after repeated iterations and can meet only part of the demands of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' However, the SNM can meet the demands of all nodes, and its resource utilization rate is higher than that of the cobweb model 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content='2 40 60 80 100 120 Attention Resource Utilization(percentage) Amount of activation energy(KJ/mol) Comparison of activation energy values attention game model cobweb theorem model CONCLUSION The authors presented a new model for SL comprehension based on spatial information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' This process uses game theory to simulate the human attention suppression and enhancement process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' This process also joins the forgetting function of human memory traces to compute the initial state of the node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Memory is encoded with specific (semantic) meaning, or refers to information that is encoded along a spatial and temporal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Although the semantic network provides a functional view of how knowledge may be organized in the brain, it does not provide a clear model of how semantic memory might be presented in the brain (see Cacha et al., 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Spreading activation reveals that information can be stored in SNs for a long time, in which a network node is a linguistic concept and the nodes are connected through the correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' An algorithmic method is proposed according to selective functions, and its effectiveness was verified using an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The results show that the proposed method improves the performance of SL comprehension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' ACKNOWLEDGMENT The authors would like to thank Chunda Liu from the National Center for Sign Language and Braille for helping in stimulus preparation and data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' This paper forms an expanded and revised version of a conference paper at the 14th IEEE International Conference on Cognitive Informatics & Cognitive Computing (ICCI* CC) at Tsinghua University, Beijing, July 6-8, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' The authors are grateful to Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Raymond Chiong, and two anonymous referees for their helpful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' Conflict of Interest The authors of this publication declare there is no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_43/content/kb_43.pdf'} +page_content=' 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Under the assumption that the structure of this wind +is constant in time and corotates with the Sun, solar wind and thereby space +weather forecasts have been made quite effectively. Such corotation forecasts +are well studied with decades of observations from STEREO and near-Earth +spacecrafts. Forecast accuracy depends upon the latitudinal separation (or offset +∆θ) between source and spacecraft, forecast lead time (∆t) and the solar cycle +via the sunspot number (SSN). The precise dependencies factoring in uncertain- +ties however, are a mixture of influences from each of these factors. And for +high precision forecasts, it is important to understand what drives the forecast +accuracy and its uncertainty. Here we present a causal inference approach based +on information theoretic measures to do this. Our framework can compute not +only the direct (linear and non-linear) dependencies of the forecast mean absolute +error (MAE) on SSN, ∆θ and ∆t, but also how these individual variables combine +to enhance or diminish the MAE. We provide an initial assessment of this with +potential of aiding data assimilation in the future. +Keywords: Solar Wind, Corotation Forecast; Causality +� Chakraborty +ae0221@coventry.ac.uk +Turner +h.turner3@pgr.reading.ac.uk +Owens +m.j.owens@reading.ac.uk +Lang +matthew.lang@reading.ac.uk +1 +School of Computing, Electronics and Mathematics, Coventry University, United +Kingdom +2 +Department of Meteorology, University of Reading, Earley Gate, PO Box 243, +Reading, RG6 6BB, UK +SOLA: sola_example_6.tex; 30 January 2023; 1:30; p. 1 +arXiv:2301.11904v1 [astro-ph.SR] 27 Jan 2023 + +Author-a et al. +1. Introduction +Forecasting terrestrial space weather impacts (e.g. Cannon et al., 2013) neces- +sitates knowledge of the up-stream solar wind conditions which will encounter +the Earth’s magnetosphere in the future. Currently, direct (in situ) solar wind +observations are only routinely available near the Earth-Sun line at the first +Lagrange point, L1, giving less than 40 minutes forecast lead time. Physics-based +simulations of the whole Sun-Earth system can potentially provide forecast lead +times of 2 to 5 days, but there remain many technical and scientific challenges +to this approach (Luhmann et al., 2004; Toth et al., 2005; Merkin et al., 2007). +A simple, yet robust, alternative forecast of near-Earth solar wind conditions +can made using observations anywhere in the ecliptic plane by assuming the +structure of the solar wind is fixed in time and corotates with the Sun. For +example, observations in near-Earth space can be used to predict conditions at +the same location a whole solar (synodic) rotation ahead, approximately 27.27 +days (Bartels, 1934; Owens et al., 2013; Kohutova et al., 2016). Of course, the +structure of the corona and solar wind does evolve over such time scales, partic- +ularly around solar maximum. From the L5 Lagrange point, approximately 60◦ +behind Earth in its orbit, the corotation time is approximately 5 days. This is +sufficiently long that the forecast lead time is useful, but sufficiently short that +the corotation approximation is generally appropriate (Simunac et al., 2009; +Thomas et al., 2018). Partly for these reasons, Vigil, the upcoming operational +space-weather monitor, will make routine observations at L5 (Kraft, Puschmann, +and Luntama, 2017). +Assessing and quantifying the factors which influence the accuracy of coro- +tation forecasts is important directly for improved corotation forecasting, but +also for effective data assimilation of the solar wind observations into solar wind +models (Lang and Owens, 2019; Lang et al., 2021), as it informs the expected +observational errors. Longitudinal separation between the observing spacecraft +and the forecast point – and hence the forecast lead time – is obviously expected +to increase forecast error, as the steady-state assumption becomes increasingly +invalid. We may also expect that this effect would be more pronounced (and +corotation forecasts generally less accurate) around sunspot maximum, when +the corona is known to be more dynamic and the occurrence of time-dependent +coronal mass ejections (CMEs) increases (Yashiro et al., 2004). (However, see +Owens et al., 2022, +for evidence that this effect is reduced near the ecliptic +plane). Similarly, it has been argued using simulation data that corotation fore- +cast error should increase with latitudinal separation of observing spacecraft +from forecast position (Owens et al., 2019), and that this effect is maximised at +sunspot minimum (Owens et al., 2020). +The OMNI dataset of near-Earth solar wind observations (King and Papi- +tashvili, 2005) allows us to assess corotation forecasts over nearly five complete +solar cycles. As near-Earth observations are used to make near-Earth forecasts +one solar rotation ahead, the forecast lead time is fixed at 27.27 days and the +latitudinal separation, caused by Earth’s motion over a solar rotation, reaches +a maximum value of around 3.5◦. The twin spacecraft of the Solar-Terrestrial +Relations Observatory (STEREO) (Kaiser, 2005) provide a means to assess the +SOLA: sola_example_6.tex; 30 January 2023; 1:30; p. 2 + +Example paper +performance of corotation forecasts over a larger parameter range. The spacecraft +launched into Earth-like orbits in late 2006, with STEREO-A moving ahead of +Earth in its orbit, and STEREO-B behind, separating from Earth at a rate of +22.5◦ per year. This allows the corotation forecast to be assessed for a full range +of longitudinal separations – and hence forecast lead times between 0 and 27.27 +days – and, due to the inclination of the ecliptic plane to the solar equator, +latitudinal separations covering the range ±15◦. More than a solar cycle of data +is available (although the STEREO-B spacecraft was lost in 2014), allowing the +effect of increasing solar activity to be estimated. +However, while uniquely valuable, assessing corotation forecasts with the +STEREO dataset does present a number of challenges. Longitudinal and lat- +itudinal separation from Earth are interdependent, as both are due to the same +orbital geometry. Due to timing of launch and the orbital period, solar activity +also varies approximately in step with the orbit; the spacecraft launched just +before sunspot minimum and reached maximum separation just after sunspot +maximum. Thus it is difficult to isolate and quantify the individual sources of +error in corotation forecasting (Turner et al., 2021). This kind of problem is ripe +for causal analysis. +Study of cause and effect is central to all branches of sciences and there are +questions in solar physics – such as factors affecting corotation forecasts – that +can be cast in those terms. In non-interventional (or observational) systems +like the Sun, causal discovery is the process of inferring mechanisms or models +relating cause and effects from data. But even when principal mechanisms are +known from physics, causal frameworks can also be used as a diagnostic tool to +determine how uncertainty in one or more variable influences another. This is +very useful in making forecasts. Typically, establishing a causal relationship be- +tween variables entails determining their conditional dependency (Granger, 1969; +Pearl, 2000). For random variables, both continuous and discrete, this is done +via probabilistic measures. Conditional dependency has traditionally been estab- +lished with Granger causality (Granger, 1969) and these measures are mostly +derived from information theory, i.e., they are ‘Shannon based’ (Schreiber, 2000; +Kraskov, St¨ogbauer, and Grassberger, 2004; Williams and Beer, 2010). In addi- +tion, for time-series data, the time order of events is also critical to establishing +causality. Time-lags between different variables need to be carefully evaluated. +Therefore the temporal resolution of time-series must be sufficient for estab- +lishing the direction of information flow; missing data can lead to spurious +correlations (Runge, 2018). Non-linear correlations between multiple drivers can +be very difficult to disentangle. We here attempt to address and demonstrate +this with a framework (van Leeuwen et al., 2021) which uses a transformed infor- +mation theoretic measure that applies to both discrete and continuous variables. +Typically the current state-of-the-art causal estimates are point estimates: Data +is used to produce a single number to quantify the causal relationships. There +is no robust uncertainty quantification. Addressing this in general, is a work in +progress (for eg., Heckerman, 2020; Runge, 2018). However, we will provide an +elementary estimate of the distribution of the strength of causal relationships - +the causal strength, cs from hereinafter. +SOLA: sola_example_6.tex; 30 January 2023; 1:30; p. 3 + +Author-a et al. +Our goal in this work is to provide an initial assessment of the causal depen- +dencies between the accuracy of a forecast, the target or “effect” variable, with +the driver or “cause” variables. For reasons explained above, the driver variables +are assumed to be solar activity (quantified by sunspot number), forecast lead +time (which is primarily determined by longitudinal spacecraft separation for +the OMNI and STEREO observations) and latitudinal spacecraft separation. +The typical approach would be to cross-correlate these variables, or rather the +time-series associated with them, pairwise. However, as these relationships can +often be nonlinear and multivariate we need more advanced estimators such +as those based on information theoretic measures like mutual information and +higher order terms (Chakraborty and van Leeuwen, 2022). So the approach we +follow here is to start with the analog of pairwise correlation, but with the +non-linear estimator; mutual information. We then introduce a third variable, +via conditional mutual information, to disentangle inter-dependencies amongst +three driver/cause variables, in order that mediated or induced effects can be +isolated. In principle, a full causal network (Runge, 2018; van Leeuwen et al., +2021) can be constructed using time-series observations. But this comes with +computational and, in certain situations, interpretation challenges. Hence we +leave this for future work. +We describe the solar wind observations from OMNI and STEREO A and B +spacecraft in section 2. Next, we introduce the causal inference methods, demon- +strating their application to the OMNI observations in section 3. We compute +the distribution of causal relationships, first pairwise, quantified in terms of the +mutual information using a non-linear information theoretic measure (subsec- +tion 3.2), examine the time averaging effect on sunspot number (subsection 3.3), +followed by the conditional mutual information to separate influence of the +third variable (subsection 3.4). We use 27-day corotation forecasts (also called +‘recurrence’ or ‘27-day persistence’ forecasts) using only OMNI data first, as +it eliminates the lead-time as a variable by design ; this leaves us with testing +2 (instead of 3) drivers:the solar activity encoded in the (smoothed) Sunspot +Number (or SSN27) and the latitudinal offset. By first learning dependencies in +this simpler dataset, we then compare effects of this same subset of drivers in +the STEREO datasets ignoring at first the lead time (subsection 3.6). Following +this, in Section 4, we study induced or mediated dependencies with lead time +included, by using the STEREO datasets. Finally we interpret the results and +conclude whilst looking at future opportunities to improve forecasts in section 5. +2. Observations +Two primary data sets are used in this study. Firstly, the OMNI dataset of +near-Earth solar wind conditions (King and Papitashvili, 2005). Data are avail- +able from https://omniweb.gsfc.nasa.gov/. Prior to 1995, data coverage varies +significantly, so the period of study is limited to 1995 to present. Secondly, the +STEREO dataset, which is available from https://stereo-ssc.nascom.nasa.gov/ +data.shtml. STEREO-A data are used from the whole mission, 2007-present, +while STEREO-B data are only available until 2014. All data are averaged to +SOLA: sola_example_6.tex; 30 January 2023; 1:30; p. 4 + +Example paper +Figure 1. A summary of the corotation forecast of solar wind speed obtained by using OMNI +near-Earth observations to forecast near-Earth conditions 27.27 days ahead. Top: Sunspot +number. Middle: The absolute value of the latitudinal separation between observation and +forecast location (∆θ). Bottom: The mean absolute error in the solar wind speed corotation +forecast. All properties are calculated at 1-day resolution (dotted lighter curves), then averaged +over 27 days (solid darker curves). +1-day resolution to remove the effect of small-scale stochastic structure, such as +waves and turbulence (Verscharen, Klein, and Maruca, 2019). +Solar wind speed corotation forecasts are produced by ballistically mapping +data from the observation radial distance to 1 AU, then applying a corotation +delay consistent with the longitude separation. By far the dominant factor is the +longitudinal separation. Further details can be found in Turner et al. (2021). For +each forecast we compute the mean absolute error (MAE) between the forecast +and observed solar wind speed. +For solar cycle context, we use the daily sunspot number (SN), provided by +SILSO (Clette and Lef`evre, 2016) and available from https://www.sidc.be/silso/. +Figure 1 shows a summary of OMNI data used to make a 27.27-day lead time +forecast of near-Earth conditions. By eye, some correlation can be seen between +the MAE and SN. E.g. There are few intervals of MAE above 250 km/s during the +solar minima of 1996-97, 2009-10 or 2019-20. Conversely, there is no immediately +obvious relation between MAE and the absolute latitudinal separation between +observation and forecast location, ∆θ. However, the ∆θ variation here is very +small, arising from Earth’s latitudinal orbital motion over a 27.27-day interval +and reaching a maximum magnitude of around 3.5◦. +Figure 2 shows the summary of STEREO-B observations used to forecast solar +wind speed at STEREO-A. As the spacecraft separate in longitude, the forecast +SOLA: sola_example_6.tex; 30 January 2023; 1:30; p. 5 + +Time-Series : 27 day average time-series +SSN +200 +0 +0 +MAE [km s-1] +400 +200 +i +0 +1995-02-23 +2000-08-21 +2006-02-15 +2011-08-08 +2017-01-28 +TimeAuthor-a et al. +Figure 2. A summary of the corotation forecast of solar wind speed obtained by using +STEREO-B observations to forecast conditions at the STEREO-A spacecraft. Top: Sunspot +number. Second row: The absolute value of the latitudinal separation between observation and +forecast location (∆θ). Third row: Forecast lead time (directly proportional to longitudinal +separation, and to a much lesser extent, radial separation of spacecraft). Bottom: The mean +absolute error in the solar wind speed corotation forecast. All properties are calculated at +1-day resolution (dotted lighter curves), then averaged over 27 days (solid darker curves). +lead time, ∆t, increases almost linearly. The maximum value of ∆θ grows as the +spacecraft increase their absolute longitudinal separation until mid 2010, then +declines as the spacecraft move closer together (behind the Sun, from Earth’s +point of view). There is a somewhat linear growth in MAE from 2007 to 2012, +though without further analysis it is not possible to say whether this is the result +of sunspot number (post smoothing as we will see), ∆t or the amplitude of ∆θ +increasing through this time. Or some combination of those variables. +3. Methods : Causal Dependencies of Corotation Forecasts +We wish to study the principle drivers of the error in the corotation forecasts. In +order to do that, we perform a causal analysis on the mean absolute error (MAE) +as the target/effect variable and the sunspot number (SN), latitudinal offset +(|∆θ|[◦]), and the forecast lead time (∆t [days]) as the principal driver/cause +variables. With this setup we can use a non-linear measure of dependency to +compute the causal relationships between these variables. There are a number +of choices for such measures: those based on information theory as (conditional) +SOLA: sola_example_6.tex; 30 January 2023; 1:30; p. 6 + +Time-Series : 27 day average time-series +200 +SSN +10 +MAE [km s-1j△ t [days] +500 +250 +27/03/2007 +14/08/2008 +30/12/2009 +18/05/2011 +03/10/2012 +20/02/2014 +TimeExample paper +mutual information (Kraskov, St¨ogbauer, and Grassberger, 2004), transfer en- +tropy (Schreiber, 2000), directed information transfer (Amblard and Michel, +2009), etc. We chose the mutual information (and its conditional variants) as it +is well studied (e.g., van Leeuwen et al., 2021; Runge, 2015) and there are robust +estimators available, along with an analytical result for Gaussian variables. The +mutual information I(x; y1:N) between a target process x and a possible driver +process y, or a whole range of driver processes denoted in our general formalism +(van Leeuwen et al., 2021) by y1:N (or sometimes y, z, w,...etc.) is defined via +the Shannon entropy H(..) as +I(x; y1:N) = H(x) − H(x|y1:N) +(1) += +� +p(x, y1:N) log +� p(x, y1:N) +p(x) p(y1:N) +� +dxdy1:N +(2) +Mathematically, the mutual information I(x; y1:N) is a positive definite quantity. +It can be thought of as the reduction in entropy (or uncertainty) in the target +(here x) in presence of information content from the driver variables (here y1:N). +3.1. Symbolic representation : Venn Diagram Visualisation +This is shown graphically as an Information Venn Diagram in figures 3 (and +5 for higher order terms that we will discuss later). The circles represent the +conditional entropy (H(x|y)) of the individual variables and the intersection +(shaded region with lines) represents the reduction in entropy of variable due to +the presence of the other, which is the mutual information (I(x; y)) defined in +eqn. 1. For our application, one of these variables is the target and the other a +driver ; hence the superscripts n + 1 and n showing different time indices. The +labels also show the specific case on hand with the solar wind variables (MAE, +SSN, ∆θ), but we will elaborate on these in upcoming subsections. The drivers +that causally influence the target would reduce the entropy and the extent of +this reduction is viewed as the extent of causal influence. On the other hand, if +a driver does not have a causal influence, it does not reduce the entropy and the +mutual information of the target with that driver is zero. Graphically this would +mean a separation of the two circles with zero overlap. There are limitations to +a formal interpretation of all situations in terms of Venn diagrams - this will +become clear for higher order terms like interaction information described in +subsection 3.4 (Ghassami and Kiyavash, 2017). Hence, these Venn diagrams serve +as a visualisation to build up our intuition rather than be a formal representation. +3.2. Mutual Information : Pairwise dependency between Latitudinal +Offset, Sunspot Number and MAE +With the goal of disentangling causal influences of drivers in corotation forecasts, +we begin with OMNI data used to make a forecast at Earth. In this case, coro- +tation forecasts have a fixed lead time of 27.27 days and forecast error, MAE, +inherently has two primary drivers, the time variation of the sun – approximated +SOLA: sola_example_6.tex; 30 January 2023; 1:30; p. 7 + +Author-a et al. + xn+1=MAEn+127 +zn=∆θn
 +or +yn=SSNn +27 +I( xn+1 ; zn ) = I(MAEn+127 ; ∆θn) +I( xn+1 ;yn ) = I(MAEn+127 ; SSNn +27) +or +Figure 3. The figure shows a graphical illustration of the mutual information via Venn +diagrams of the conditional entropies H(x|y) or H(x|z). One circle represents the entropy +content in the target - forecast accuracy (at time n + 1) and the second represents that of one +driver - either SSN or ∆θ. The intersection represents the reduction of entropy in the target +by knowledge of the driver. +by the sunspot number (SN) – and latitudinal offset (∆θ) between the observa- +tion and forecast position (i.e. between Earth’s location 27.27 days apart). This +provides a relatively simple causal network to explore with our framework. +We compute the mutual information (MI) between pairs of the target and one +of the drivers, e.g., I(MAE ; |∆θ|). Given the length of the observation time series, +we can empirically estimate the distribution of these quantities as histograms. +The mutual information serves as the measure of causal dependency between +pairs of one of the drivers and the target variable. Once again we refer to figure 3 +for a visualisation graphically via Venn diagram described in section 3.1. In this +figure, the example is given for variables x and y representing target MAE, and +driver either ∆θ or SN. In other words, we determine the reduction in entropy (or +random uncertainty) in MAE, due to ∆θ or SN. It must be noted, that we break +the symmetry between the two variables (target v/s driver), with the driver +(cause) as lagging in time with respect to the target (effect). The quantities +(or rather their distributions) represented by these information diagrams are +estimated in figure 4. +As a positive definite quantity with no upper limit, MI can take very large +values. Thus it is useful to normalise this measure, which is possible in a number +of ways. One option is to normalise it with the total entropy or uncertainty in the +SOLA: sola_example_6.tex; 30 January 2023; 1:30; p. 8 + +Example paper +Figure 4. Histograms of causal strengths, cs, of drivers – sunspot number(SN) and latitudinal +offset (∆θ) – on the driver, corotation MAE obtained from 27.27-day forecasts using OMNI +data. Dashed lines represent mean cs values used in causal diagrams. We take the 27-day +smoothed MAE as the target in both cases. On the Left: SN and ∆θ are used at daily resolution +; cs computed from daily SN. Right: cs computed from 27-day smoothed SN. cs for SN shows +a greater dependence on |∆θ| than on SN, whereas the 27 day smoothed SN clearly shows the +greater association of solar cycle with MAE by suppressing the stochasticity and emphasising +the solar activity. +variable x, giving the causal strength, cs(x; y1:N) = I(x;y1:N) +H(x) +or simply cs(x; z) = +I(x;z) +H(x) for two variables : target x and driver z. There is a challenge here ; the +entropy we use is for continuous variables, also known as the differential entropy, +which can acquire negative values. In practice, we do not encounter this here in +our applications. However, to mitigate this effect – and for general interpretation +– we will ultimately use relative causal strengths to the total over all the drivers +combined; in these relative causal strengths we ignore the contribution of noise +or unmodeled drivers to merely focus on interpreting selected drivers. +3.3. Influence of Sunspot Number - Timescale Matters +The measured or observed quantity for solar activity is the daily sunspot number. +These observations display large variability as seen in figures 1 and +2. As we +will demonstrate here, the stochasticity has an impact on the causal association +with forecast accuracy term, MAE. Figure 4 shows the corresponding distri- +bution of causal strengths of the pairs of MAE with 27-day smoothed right +and daily unsmoothed Left SN and the latitudinal offset (∆θ). The unsmoothed +daily sunspot number (SN) has lower cs (= MI/H) than the latitudinal offset +(∆θ). However, upon performing a rolling mean on the daily SN to yield 27- +day smoothed averaged SN or the SSN27, the hierarchy reverses. As shown in +figure 4, we see that the total causal strength of SN, cs(SSN27 → MAE27), goes +from 0.03 to 0.14 upon averaging, compared to cs(∆θ → MAE27) with a mean +value of 0.05. As expected, the stage of solar cycle and overall time variability +of the Sun is better represented by the smoothed SN, which has a significant +influence on MAE. That the daily SN has a significant stochastic component is +also confirmed by / evident from the entropy estimates. The entropy is reduced +upon smoothing or averaging SN and is lower than that of ∆θ by a factor of a +few (for example ≈ 2 for STB-STA ) ; however, this is not a significant effect. +SOLA: sola_example_6.tex; 30 January 2023; 1:30; p. 9 + +4.0 +Meanl cs(MAE27;SN) )=0.03 +Mean cs(MAE27; △)=0.05 +3.5 +cs(MAE27; Sunspot Number) +CS(MAE27; △) +3.0 +2.5 +2.0 +1.5 +1.0 +0.5 +0.0 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +Causal Strength = Information Term / EntropyMean cs(MAE27; SSN27) )=0.14 +Mean cs(MAE27: e) )=0.05 +cs(MAE2z;SmoothedSunspotNumber2z) +4 +Cs(MAE27; △0) +3 +2 +1 +0 +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +0.16 +0.18 +Causal Strength = Information Term / EntropyAuthor-a et al. + xn+1=MAEn+127 +yn=SSNn +27 +zn=∆θn +I( xn+1 ; zn ) = I(MAEn+127 ; ∆θn) +I( xn+1 ; yn ; zn ) = I( MAEn+127 ; SSNn +27 ; ∆θn ) +H( xn+1 | yn, zn ) = H(MAEn+127) 
 +- I(MAEn+127 ; SSNn +27) - I(MAEn+127 ; ∆θn) 
 ++ I(MAEn+127 ; SSNn +27 ; ∆θn)
 + +Figure 5. The figure shows the different information components for three variables. The +intersections represent information shared between variables. The black striped region is the +mutual information shared between target MAE and driver ∆θ. The yellow dots denote inter- +action information : information shared between MAE27 and both ∆θ and SSN27. The pink +circles show the entropy or uncertainty in MAE27 that is not explained or shared by either +SSN27 or ∆θ or their interaction. +3.4. Conditional and Interaction Information : Higher order terms +In presence of multiple causes or drivers (say y and z), the aforementioned +causal strength term, cs(x; z) will come to represent the fractional reduction +in uncertainty in the target due to the driver, z. And there’s a similar term for y. +To further disentangle and isolate the influence of each driver we also compute +the conditional mutual information (CMI), e.g., I(x; z|y). For two drivers y and z +(and a single target x), conditional mutual information, I(x; z|y) given in eqn 3, +‘conditions out’ the effect of one driver (y), leaving the direct influence of the +other one (z). This can be visualised in terms of Venn diagrams in figure 5. It is +the difference between the intersection of x and z circles (black stripes) and that +of x, z and y circles (yellow spots). In our application to corotation forecasts, +the example used for illustration has x as MAE27 and the y and z as the drivers +∆θ and SSN27, respectively. We will keep the same normalisation with entropy +for all information terms so that they can be combined or compared. +The conditional mutual information can be defined in terms of the conditional +entropies as, +I(x; y|z) = H(x|z) − H(x|y, z) +(3) +The above equation for the conditional mutual information of x with respect +to y and z represents the difference in entropy of x “conditioning out” z alone +(H(x|z)) and entropy of x “conditioning out” y and z together (H(x|y, z)). This +SOLA: sola_example_6.tex; 30 January 2023; 1:30; p. 10 + +Example paper +leaves us with the direct influence of driver y on target x, excluding any indirect +influence mediated by or shared with z. The distributions of such conditional +information terms for the triplet (MAE27,∆θ,SSN27) are estimated in figure 6 ; +these provide the so-called direct causal influence contribution of SSN27 and ∆θ +on MAE27. These are symbolised by the black arrows in the causal summary +diagram in figure 7. The causal summary diagram, as the name suggests, provides +a summary of the information flow from (and therefore the causal influence of) +the driver variables ; in this case the latitudinal offset (∆θ) and the smoothed +sunspot number (SSN27). Now the interaction information can be written in +terms of the mutual and conditional mutual information as, +I(x; y; z) = I(x; y) − I(x; y|z) += I(x; z) − I(x; z|y) +(4) +This equation for the interaction information of x with y and z gives the dif- +ference between the mutual information shared between x and y (I(x; y)) and +information shared between them, upon conditioning out z (I(x; y|z)). This is +the interaction information shared between the 3 variables, x, y and z and is +symmetric in all three variables. If we fix one as the target with the other two as +drivers, as we do for our application, then the expression for interaction infor- +mation is symmetric in the two drivers as demonstrated by the two equivalent +expressions for I(x; y; z) in equation 4. So it doesn’t matter which driver we +condition on. We will exploit this later on as estimates from actual measurements +may not converge to the same value as eqn 4. So we can take the average of the +two symmetric expressions to represent the interaction information between one +target and two drivers. This is seen later in the observational estimates. +This quantity can be interpreted as the information shared between x and y, +less the information shared between them when z is known. If the interaction +information is non-negative, or I(x; y) ≥ I(x; y|z), it implies that the depen- +dency of x on z partially or entirely (equality) constitutes the dependency on +y (Ghassami and Kiyavash, 2017). If the interaction information is negative, +or I(x; y) < I(x; y|z), then each one of the variables induces and increases +correlation between the other two. +In the previous subsection, we ascertained that the smoothed sunspot number +or SSN27 is more appropriate as a proxy for the solar activity in evaluating +its causal influence on the average corotation forecast accuracy, MAE27. Now +we wish to disentangle the direct and indirect effects of both SSN27 and the +latitudinal offset, ∆θ on MAE27. Their joint effect, or one mediating through +the other, is naturally a higher order effect and hence we need the higher +order information terms. We compute the higher order information theoretic +quantities, namely conditional mutual information and interaction information +between drivers SSN27 and ∆θ and the target, MAE27, still using only the +OMNI dataset. For MAE27 (x), ∆θ (y) and ∆t (z), the interaction information +corresponds to the common part with yellow circles in the Venn diagram in +figure 5. This is therefore the information shared across all three variables in +general. +Formally, causality necessitates there be a time lag between the cause and +effect such that the former precedes the latter. And indeed there is a time +SOLA: sola_example_6.tex; 30 January 2023; 1:30; p. 11 + +Author-a et al. +Figure 6. The figure shows the Left: distribution of conditional causal strengths with OMNI +data. cs here is associated with conditional mutual information normalised by the entropy, +H, for the combinations of MAE27 with Sunspot Number (SSN27) (27 day rolling averages +for both) and Latitudinal Offset for the full dataset, namely cs(MAE27; SSN27|∆Θ) and +cs(MAE27; ∆Θ|SSN27). +∆θn +MAEn+127 +SSNn +27 +0.06 +-0.01 +0.14 +OMNI_to_OMNI +Figure 7. The figure summarises the mean MAE27 dependence on latitudinal offset ∆θ and +SSN27 individually (black) and in combination (red) for OMNI-OMNI. This is done in terms +of the causal strength (cs = Information Term +Entropy +) values defined earlier - the numbers attached +to the arrows. The superscripts merely indicate the formal need for the causes (SSNn +27, ∆θn) +to precede the effect (MAEn+1 +27 +) - in practice for this application, a single time-step makes +negligible difference. +SOLA: sola_example_6.tex; 30 January 2023; 1:30; p. 12 + +4.0 +Meancs(MAE27:SSN27|△e))=0.15 +Mean cs(MAE27;|SSN27))=0.03 +3.5 +cs(MAE27:SmoothedSunspotNumber2zl△e) +cs(MAE27:△|SmoothedSunspotNumber27) +3.0 +2.5 +2.0 +1.5 +1.0 +0.5 +0.0 +0.000 +0.025 +0.050 +0.075 +0.100 +0.125 +0.150 +0.175 +0.200 +Causal Strength= Information Term/ EntropyExample paper +lag between the forecast accuracy of a future step, MAEn+1 +27 +, and the drivers +∆θn and SSN n +27. This is innate/intrinsic to the way the time-series observa- +tions are done. However, in this particular application, we are considering daily +variations and the drivers – latitudinal separation, longitudinal separation and +27-day smoothed sunspot number – vary over much longer timescales. Thus a +single time-step between n and n+1 makes negligible difference to the computed +information components. However, the notation involving target at n+1 and +drivers at n is maintained to demonstrate the general principle. +3.5. Symbolic representation : Causal Summary diagrams +The causal information flow between the variables is summarised in figure 7. +The nodes (or ovals) represent the variables and the arrows represent the flow +of information to the target variable, MAE27 one time step in the future (n+1). +Black arrows represent the influence of single driver conditioning out influence of +the other drivers. The red segments ending in an arrowhead on the target repre- +sents the joint causal influence of the drivers. The confluence of the segments out +of the drivers into a point symbolises this join or combined effect ; the arrowhead +as usual points to the information flow into the target. This represents the com- +ponent of influence that is driven by the combination of drivers together, distinct +from their individual, direct influences on the target, shown by black arrows. This +combined or joint effect could be a positive one showing a redundancy in driver +or that one driver partially or entirely captures the influence due to another. +It could be negative suggesting that one driver induces an influence from the +other driver. These can be mathematically quantified in terms of the interaction +information. Here for 27-day corotation forecasts using only OMNI data, the +only drivers are ∆θ and SSN27, now considered simultaneously. (As we will see +in an upcoming section, the lead time ∆t – related to the longitudinal separation +– will have a role to play for STEREO data.) We find that the direct influences of +SSN27 and ∆θ (black arrows) are more important than the joint influence (red +arrow) on MAE27. In general, we can compute joint influence due to multiple +drivers starting from pairs (the red arrows) to the joint influence of all n drivers +simultaneously. However, to use full general mathematical framework in van +Leeuwen et al. (2021) is computationally expensive and complex. It is also not +essential in our work here to get the main dependencies. We compute the causal +strengths (defined earlier) from the mutual and conditional mutual information +terms in accordance with van Leeuwen et al. (2021). The black arrows are given +by: +(∆θn → MAEn+1 +27 )1link = I(MAEn+1 +27 ; ∆θn|SSN n +27) +(5) +(SSN n +27 → MAEn+1 +27 )1link = I(MAEn+1 +27 ; SSN n +27|∆θn) +(6) +The red arrow symbolising the joint influence of ∆θ and ∆t represents and is +related to the interaction information shown in the figure 5. Graphically this +represents the intersection of the information component common to each of the +three variables in our triplet i.e. two drivers (latitudinal offset and lead time) +SOLA: sola_example_6.tex; 30 January 2023; 1:30; p. 13 + +Author-a et al. +and target (MAE27 ). This is therefore symmetric and is, theoretically, indepen- +dent of the variable that it is conditioned on. However, when estimating from +measured quantities, this symmetry, indicated both graphically in figure 5 and +equation 4 is not strictly adhered to. Hence, we can express the joint influence +indicated by the red arrow as the average of the two equivalent ways of estimating +it as: +(∆θn → MAEn+1 +27 +)2link + (SSN n +27 → MAEn+1 +27 +)2link += +1/2 [ I(MAEn+1 +27 +; ∆θn) − I(MAEn+1 +27 +; ∆θn|SSN n +27) ++ +I(MAEn+1 +27 +; SSN n +27) − I(MAEn+1 +27 +; SSN n +27|∆θn)] +(7) +3.6. Consistency across Datasets +We next test this relative influence of SSN27 and ∆θ on MAE27 across the avail- +able datasets, namely STB-STA, STB-OMNI and OMNI-STA pairings. This is +shown in figure 8. In each case, we find the interaction information, I(MAEn+1 +27 +; ∆θn; SSN n +27) +to be positive. This is an indication that SSN27 partially constitutes the depen- +dency of MAE27 on ∆θ and vice versa, but it is not very significant. And across +these three datasets (as well as OMNI-OMNI recurrence forecasts), we found +that the direct causal strengths of latitudinal offset, I(MAEn+1 +27 +; ∆θn|SSN n +27), is +around 60−70% of the direct causal strength of SSN27, I(MAEn+1 +27 +; SSN n +27|∆θn). +Furthermore, estimates of the interaction information, given by I(MAEn+1 +27 +; SSN n +27)− +I(MAEn+1 +27 +; SSN n +27|∆θn), are merely ∼ 7% of the direct causal influence, as +was also shown in OMNI dataset in figure 7. This suggests that to a good +approximation, the causal influence of the solar activity is decoupled from that +of the latitudinal offset. This will aide us in considering the causal influence of +lead time in turns with these two variables, simplifying the causal network. +4. STEREO : Effect of Lead Time +As explained in the previous section, the OMNI (27-day) corotation forecast +dataset allows us to focus on the causal influence of ∆θ and SSN27 as proxy +of the solar activity on MAE27. Having learnt that the interaction information +between these three driver variables is small, we can assume their influence to +be largely independent. We will now proceed to pair ∆θ and SSN27 by turns, +with the lead time ∆t. This will give us the direct and interaction terms for each +case, analogous to the causal network in figure 7. +4.1. Conditional Causal Influence : Lead Time +Analogous to the causal analysis of the OMNI time-series, we estimate causal +linkages using the STA-OMNI corotation time-series. We begin with the time- +series of ∆t and ∆θ as drivers along with the MAE27 as the target. We compute +causal strength terms corresponding to mutual information terms I(MAEn+1 +27 +; ∆θn) +SOLA: sola_example_6.tex; 30 January 2023; 1:30; p. 14 + +Example paper +Figure 8. MAE27 dependence on principle drivers for corotation forecasts; latitudinal off- +set ∆θ conditioned on smoothed sunspot number or SSN27 and vice versa in pairs of Top: +STB-STA, Middle: STB-OMNI and Bottom: OMNI-STA. +SOLA: sola_example_6.tex; 30 January 2023; 1:30; p. 15 + +4.0 +MeancS(MAE27:SSN2zl△0))=0.15 +MeancS(MAE27;|SSN27))=0.06 +3.5 +cs(MAE27;SmoothedSunspotNumber27|△) +cs(MAE27△e|SmoothedSunspotNumber27) +3.0 +2.5 +2.0 +1.5 +1.0 +0.5 +0.0 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +Causal Strength = Information Term / Entropy3.0 +Mean cs(MAE27;SSN2zl△0))=0.13 +Meancs(MAE27;△|SSN27))=0.08 +2.5 +cs(MAE27;SmoothedSunspotNumber27|△) +cs(MAE27;△e|SmoothedSunspotNumber27) +2.0 +1.5 +1.0 +0.5 +0.0 +0.000 +0.025 +0.050 +0.075 +0.100 +0.125 +0.150 +0.175 +Causal Strength=InformationTerm/Entropy4.0 +Meancs(MAE27;SSN2zl△))=0.12 +Meancs(MAE27;△0|SSN27))=0.08 +3.5 +cs(MAE27;Smoothed SunspotNumber27|△) +cs(MAE27△e|SmoothedSunspotNumber27) +3.0 +2.5 +2.0 +1.5 +1.0 +0.5 +0.0 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +Causal Strength = Information Term / EntropyAuthor-a et al. +Figure 9. The figure shows the histogram of causal strengths, cs, of drivers – the lead time, +∆t27 and ∆θ – with the target, MAE27 using STA-OMNI data. On the Left:, are the distri- +butions of pairwise strengths (MI/H) and on the Right: are the conditional causal strengths +(CMI/H). +and I(MAEn+1 +27 +; ∆tn) and the conditional information terms, I(MAEn+1 +27 +; ∆θn|∆tn) +and I(MAEn+1 +27 +; ∆tn|∆θn). +I(MAEn+1 +27 +; ∆θn; ∆tn) is actually the influence of ∆θ on MAE27 that is +shared by (or mediated through / induced by) ∆t, in general. From figure 9, it is +evident from the histograms of cs(MAEn+1 +27 +; ∆θn) and cs(MAEn+1 +27 +; ∆θn|∆tn) +(or equivalently the corresponding information terms) that the interaction infor- +mation, I(MAEn+1 +27 +; ∆θn; ∆tn) is positive. Using the mean values of each term in +the information triplet, direct (i.e. conditional mutual) and interaction informa- +tion for (MAE27,∆θ,∆t) we get an interaction information amounts to ∼ 20%. +The direct influence of ∆t on MAE27 is ∼ 50% and the direct influence of ∆θ is +the remaining ∼ 30%. These numbers are the relative proportions of influence of +the two drivers considered ∆θ and ∆t, without considering SSN27. The ‘missing’ +20% is likely to have a significant contribution from the other driver, SSN27. +However, we also have noise, which includes the statistical fluctuations. Also, +the averaged sunspot number is used as a proxy for the time evolving nature of +the solar wind ; it’s not an exact proxy. +Histograms with 15 bins as before are used to estimate the distribution of +these causal strength terms and shown in figure 9. It is clear that the lead time, +∆t, has an influence on the MAE27. Upon conditioning on ∆θ, we find a sizable +direct influence of ∆t on MAE27, around 1.5 times greater than the direct +influence of ∆θ ; this is shown by the black arrows in the summary diagram in +figure 10. +Next we look at the driver pair of lead time and (smoothed or rolling 27 day +average) sunspot number (∆t, SSN27). We again estimate the pairwise (mutual +information) and direct dependencies (conditional mutual information) of ∆t +and SSN27 on MAE27. Once again, the two drivers SSN27 and ∆t have shared +dependencies. This is summarised in figure 10. +The black arrows are given by : +(∆tn → MAEn+1 +27 )1link = I(MAEn+1 +27 ; ∆tn|SSN n +27) +(8) +SOLA: sola_example_6.tex; 30 January 2023; 1:30; p. 16 + +4.0 +Mean cs(MAE27;△0) )=0.12 +Mean cs(MAE27; △t27) )=0.23 +3.5 +Cs(MAE27; △0) +Cs(MAE27; △t27) +3.0 +2.5 +2.0 +1.5 +1.0 +0.5 +0.0 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +Causal Strength = Information Term / Entropy4.0 +Mean cs(MAE27; △0|△t27)=0.04 +Mean cs(MAE27; △t27l△0)=0.15 +3.5 +cs(MAE27; △|△t27) +cs(MAE27; △t27|△) +3.0 +2.5 +2.0 +1.5 +1.0 +0.5 +0.0 +0.00 +0.05 +0.10 +0.15 +0.20 +Causal Strength = Information Term / EntropyExample paper +∆θn +MAEn+127 +∆tn +0.12 +0.08 +0.19 +SSNn +27 +MAEn+127 +∆tn +0.08 +~0.09 +0.12 +STA_to_OMNI +∆θn +MAEn+127 +SSNn +27 +0.13 +~0.03 +0.08 +Figure 10. The figure summarises the mean causal dependence of forecast accuracy MAE27 +on latitudinal offset ∆θ, lead time ∆t and average/smoothed sunspot number SSN27 +individually (black) and in pairwise combinations (red) for STA-OMNI. +SOLA: sola_example_6.tex; 30 January 2023; 1:30; p. 17 + +Author-a et al. +and +(SSN n +27 → MAEn+1)1link = I(MAEn+1; SSN n +27|∆tn) +(9) +The red arrows symbolising the joint influence can be written symmetrically as +a sum of +(∆tn → MAEn+1 +27 +)2link + (SSN n +27 → MAEn+1 +27 +)2link += +1/2 [A I(MAEn+1 +27 +; ∆tn) − I(MAEn+1 +27 +; ∆tn|SSN n +27) ++ +I(MAEn+1 +27 +; SSN n +27) − I(MAEn+1 +27 +; SSN n +27|∆tn)] +(10) +The quantitative analyses is summarised in the causal summary diagram in +fig 10. The direct causal influence of ∆t and SSN27 are ≈ 41% and ≈ 28%, +respectively, in relative terms. And the joint influence is ≈ 31%. +4.2. Dependencies of the driver triplet +Here we put aside the target variable, MAE27, and apply the causal measures to +explore the dependencies between the drivers themselves. The goal is to directly +probe the statistical (in)dependence of the drivers without any lag i.e. we do not +seek causal information flow in time, from one variable to another. Instead we +look simply for information overlap which tells us the how the drivers relate to +each other. Hence, the three variables are SSN n +27, ∆θn and ∆tn. These dependen- +cies between drivers are symbolised by line segments instead of arrows. So with +no natural target variable, we treat SSN27 effectively as the target. The reason +for this, is that apriori we might expect a stronger dependence between ∆θn +and ∆tn and hence we wish to see how these two affect SSN27. Our approach is +reflected in the causal diagram shown in figure 11. We find that the direct effect +(black line segment) of ∆t on SSN27 is greater than that of ∆θ by over an order +of magnitude. This is because the solar activity does depend upon the phase and +hence the lead time across cycles. And while the joint effect (red segment) of ∆t +and ∆θ on SSN27 is lower than the direct effect of ∆t, it is still non-trivial. As +the corresponding interaction information term is positive, it suggests that ∆t +mediates the dependency on ∆θ. +5. Conclusion +In this paper, we probe what drives the accuracy of corotation forecasts of the +solar wind observations. As we do not have means to make interventional ex- +periments, we apply causal inference methods to the available observations. The +causal drivers of relevance are the latitudinal separation or offset (∆θ) between +the observing and forecast locations, the forecast lead time (∆t) and the solar +activity, as measured by 27 day rolling average of the sunspot number (SSN27). +The target variable is the forecast accuracy measured in terms of the mean +absolute error between observation and forecast. We use information theoretic +measures to estimate the strength of the causal influence of the drivers on the +target. These do not merely estimate correlations between pairs of variables, but +SOLA: sola_example_6.tex; 30 January 2023; 1:30; p. 18 + +Example paper + xn=SSNn +27 +yn=∆tn +zn=∆θn +I( xn ; yn ; zn ) = I(SSNn +27 ; ∆θn ; ∆tn) +∆θn +SSNn +27 +∆tn +0.03 +0.14 +0.15 +STB_to_OMNI +Figure 11. The figure summarises the mean (in)dependence of the smoothed sunspot number +(SSN27) on the latitudinal offset ∆θ and corotation time ∆t individually (black) and in +combination (red) for STB-OMNI. Here we test the interdependence of the drivers and not a +relation between causes preceding the effect symbolically shown by the absence of the arrow +heads ; hence each is at n. +can disentangle the influence of a third variable via conditioning the information +content shared between three of them. Depending on the different information +terms or components, we can estimate the influence of the individual drivers +as well as their joint effect, induced effects, and redundancy due to correlation +amongst drivers. +We draw the following conclusions: +i) The decomposition of information flow between the different drivers (or causes) +and the target is effective in identifying cause and effect relationships driving +dynamical systems in presence of complex, non-linear relationships between +multiple variables. This approach is perfectly suited to trace what drives the +corotation forecast accuracy or rather the uncertainty. +ii) The pairwise causal relationship between drivers – latitudinal offset (∆θ), +forecast lead time (∆t), and the average sunspot number (SSN27) – and +SOLA: sola_example_6.tex; 30 January 2023; 1:30; p. 19 + +Author-a et al. +target MAE27 given by mutual information normalised to the target entropy +confirms our understanding from previous results (for eg. Turner et al., 2021). +The solar activity levels measured via SSN27 and the lead time ∆t have a big +influence on the average forecast error, MAE27, followed by the latitudinal +offset, ∆θ. Statistical noise levels impacts the absolute values of the depen- +dencies. The relative values of the causal strengths, however, teach us of the +hierarchy of influence of the different driver combinations. +iii) Exploiting higher order measures like interaction information, we can probe +deeper into causal influence of multiple drivers in conjunction with one an- +other. A non-zero interaction information has two possible cases with cor- +responding interpretations. Negative values show an induced effect (i.e. one +driver showing a coupling to the target induced only due to the presence and +influence of another driver) whereas positive values show a redundant / shared +influence. We can be quantitative about the relative importance of joint causal +(positive or negative) influence and therefore potentially impact forecasting. +Also qualitatively, knowledge of whether drivers act independently of one +another, can help design future data experiments. For instance, from fig. 8, it +is clear from the consistency of MAE27 dependence on SSN27 and Latitudinal +Offset – from OMNI alone and OMNI with STEREO – that solar activity has +a stronger, direct influence on MAE27 than latitudinal offset, independent +of lead time. Hence, the appropriate weight can be given to each of these +drivers in an assimilation forecast. On the other hand, from fig. (11) while +solar activity and latitudinal offset have a weaker direct association (black line +between them) relative to its stronger, direct coupling with lead time,they do +have an indirect coupling. The association of sunspot number with latitudinal +offset is owed predominantly to the lead time. This implies, that the phase of +the solar cycle is important. +iv) The interaction information terms, such as I(MAEn+1 +27 +; ∆θn; ∆tn) and +I(MAEn+1 +27 +; SSN n +27; ∆θn) or equivalently the corresponding causal strength +terms (I/H) quantify the information content shared between the 3 variables. +From these terms, we can learn that the SSN27 and ∆θ share very little +information. i.e. their influence on MAE27 is predominantly independent +of one another. On the other hand, for the STEREO dataset, ∆θ and ∆t +share a non-trivial fraction (∼ 20%) of the their total information content +(or influence on MAE27). And in this case, the +ve sign of the interaction +information indicates that ∆θ partially contributes to the influence of ∆t +on MAE27, and vice versa. These effects could not be revealed by standard +correlation analyses. +Using a causal inference approach, disentangles the drivers of the forecast +accuracy in ways that standard statistical analyses or data assimilation can- +not. Rather one can improve upon these latter two, using causal diagnostics. +It allows us to not only disentangle individual sources of uncertainty in the +forecast, but also calculate partial and complete redundancies in drivers of this +uncertainty. These learning can potentially be applied to improve the solar wind +data assimilation forecasts. +SOLA: sola_example_6.tex; 30 January 2023; 1:30; p. 20 + +Example paper +Acknowledgments +This work was part-funded by the Science and Technology Facilities +Council (STFC) grant numbers ST/R000921/1 and ST/V000497/1. +Appendix +References +Amblard, P., Michel, O.J.J.: 2009, Measuring information flow in networks of stochastic +processes. 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DOI. +Turner, +H., +Owens, +M.J., +Lang, +M.S., +Gonzi, +S.: +2021, +The +influence +of +spacecraft +latitudinal +offset +on +the +accuracy +of +corotation +forecasts. +Space +Weather +19(8), +e2021SW002802. +e2021SW002802 +2021SW002802. +DOI. +https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2021SW002802. +van Leeuwen, P.J., DeCaria, M., Chakaborty, N., Pulido, M.: 2021, A framework for causal +discovery in non-intervenable systems. +Verscharen, D., Klein, K.G., Maruca, B.A.: 2019, The multi-scale nature of the solar wind. +Living Rev. Sol. Phys. 16(1), 5. DOI. +Williams, P.L., Beer, R.D.: 2010, Nonnegative decomposition of multivariate information. +arXiv:1004.2515. +Yashiro, S., Gopalswamy, N., Michalek, G., St Cyr, O.C., Plunkett, S.P., Rich, N.B., Howard, +R.A.: 2004, A catalog of white light coronal mass ejections observed by the SOHO spacecraft. +J. Geophys. Res. 109. DOI. +SOLA: sola_example_6.tex; 30 January 2023; 1:30; p. 22 + diff --git a/ktFKT4oBgHgl3EQfxy4K/content/tmp_files/load_file.txt b/ktFKT4oBgHgl3EQfxy4K/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4df03e0870b4dcb97f76ef512328828b9211fc0f --- /dev/null +++ b/ktFKT4oBgHgl3EQfxy4K/content/tmp_files/load_file.txt @@ -0,0 +1,1190 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf,len=1189 +page_content='Solar Physics DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='1007/•••••-•••-•••-••••-• Causal Analysis of Influence of the Solar Cycle and Latitudinal Solar-Wind Structure on Corotation Forecasts Nachiketa Chakraborty1,2 · Harriet Turner2 · Mathew Owens2 · Mathew Lang2 © Springer •••• Abstract Studying solar wind conditions is central to forecasting impact of space weather on Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Under the assumption that the structure of this wind is constant in time and corotates with the Sun, solar wind and thereby space weather forecasts have been made quite effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Such corotation forecasts are well studied with decades of observations from STEREO and near-Earth spacecrafts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Forecast accuracy depends upon the latitudinal separation (or offset ∆θ) between source and spacecraft, forecast lead time (∆t) and the solar cycle via the sunspot number (SSN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The precise dependencies factoring in uncertain- ties however, are a mixture of influences from each of these factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' And for high precision forecasts, it is important to understand what drives the forecast accuracy and its uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Here we present a causal inference approach based on information theoretic measures to do this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Our framework can compute not only the direct (linear and non-linear) dependencies of the forecast mean absolute error (MAE) on SSN, ∆θ and ∆t, but also how these individual variables combine to enhance or diminish the MAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' We provide an initial assessment of this with potential of aiding data assimilation in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Keywords: Solar Wind, Corotation Forecast;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Causality � Chakraborty ae0221@coventry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='uk Turner h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='turner3@pgr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='reading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='uk Owens m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='owens@reading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='uk Lang matthew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='lang@reading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='uk 1 School of Computing, Electronics and Mathematics, Coventry University, United Kingdom 2 Department of Meteorology, University of Reading, Earley Gate, PO Box 243, Reading, RG6 6BB, UK SOLA: sola_example_6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 30 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 1:30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='11904v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='SR] 27 Jan 2023 Author-a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Introduction Forecasting terrestrial space weather impacts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Cannon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=', 2013) neces- sitates knowledge of the up-stream solar wind conditions which will encounter the Earth’s magnetosphere in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Currently, direct (in situ) solar wind observations are only routinely available near the Earth-Sun line at the first Lagrange point, L1, giving less than 40 minutes forecast lead time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Physics-based simulations of the whole Sun-Earth system can potentially provide forecast lead times of 2 to 5 days, but there remain many technical and scientific challenges to this approach (Luhmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=', 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Toth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Merkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=', 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' A simple, yet robust, alternative forecast of near-Earth solar wind conditions can made using observations anywhere in the ecliptic plane by assuming the structure of the solar wind is fixed in time and corotates with the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' For example, observations in near-Earth space can be used to predict conditions at the same location a whole solar (synodic) rotation ahead, approximately 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='27 days (Bartels, 1934;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Owens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Kohutova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Of course, the structure of the corona and solar wind does evolve over such time scales, partic- ularly around solar maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' From the L5 Lagrange point, approximately 60◦ behind Earth in its orbit, the corotation time is approximately 5 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' This is sufficiently long that the forecast lead time is useful, but sufficiently short that the corotation approximation is generally appropriate (Simunac et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Thomas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Partly for these reasons, Vigil, the upcoming operational space-weather monitor, will make routine observations at L5 (Kraft, Puschmann, and Luntama, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Assessing and quantifying the factors which influence the accuracy of coro- tation forecasts is important directly for improved corotation forecasting, but also for effective data assimilation of the solar wind observations into solar wind models (Lang and Owens, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Lang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=', 2021), as it informs the expected observational errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Longitudinal separation between the observing spacecraft and the forecast point – and hence the forecast lead time – is obviously expected to increase forecast error, as the steady-state assumption becomes increasingly invalid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' We may also expect that this effect would be more pronounced (and corotation forecasts generally less accurate) around sunspot maximum, when the corona is known to be more dynamic and the occurrence of time-dependent coronal mass ejections (CMEs) increases (Yashiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=', 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' (However, see Owens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=', 2022, for evidence that this effect is reduced near the ecliptic plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Similarly, it has been argued using simulation data that corotation fore- cast error should increase with latitudinal separation of observing spacecraft from forecast position (Owens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=', 2019), and that this effect is maximised at sunspot minimum (Owens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The OMNI dataset of near-Earth solar wind observations (King and Papi- tashvili, 2005) allows us to assess corotation forecasts over nearly five complete solar cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' As near-Earth observations are used to make near-Earth forecasts one solar rotation ahead, the forecast lead time is fixed at 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='27 days and the latitudinal separation, caused by Earth’s motion over a solar rotation, reaches a maximum value of around 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='5◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The twin spacecraft of the Solar-Terrestrial Relations Observatory (STEREO) (Kaiser, 2005) provide a means to assess the SOLA: sola_example_6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 30 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 1:30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 2 Example paper performance of corotation forecasts over a larger parameter range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The spacecraft launched into Earth-like orbits in late 2006, with STEREO-A moving ahead of Earth in its orbit, and STEREO-B behind, separating from Earth at a rate of 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='5◦ per year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' This allows the corotation forecast to be assessed for a full range of longitudinal separations – and hence forecast lead times between 0 and 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='27 days – and, due to the inclination of the ecliptic plane to the solar equator, latitudinal separations covering the range ±15◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' More than a solar cycle of data is available (although the STEREO-B spacecraft was lost in 2014), allowing the effect of increasing solar activity to be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' However, while uniquely valuable, assessing corotation forecasts with the STEREO dataset does present a number of challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Longitudinal and lat- itudinal separation from Earth are interdependent, as both are due to the same orbital geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Due to timing of launch and the orbital period, solar activity also varies approximately in step with the orbit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' the spacecraft launched just before sunspot minimum and reached maximum separation just after sunspot maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Thus it is difficult to isolate and quantify the individual sources of error in corotation forecasting (Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' This kind of problem is ripe for causal analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Study of cause and effect is central to all branches of sciences and there are questions in solar physics – such as factors affecting corotation forecasts – that can be cast in those terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' In non-interventional (or observational) systems like the Sun, causal discovery is the process of inferring mechanisms or models relating cause and effects from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' But even when principal mechanisms are known from physics, causal frameworks can also be used as a diagnostic tool to determine how uncertainty in one or more variable influences another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' This is very useful in making forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Typically, establishing a causal relationship be- tween variables entails determining their conditional dependency (Granger, 1969;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Pearl, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' For random variables, both continuous and discrete, this is done via probabilistic measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Conditional dependency has traditionally been estab- lished with Granger causality (Granger, 1969) and these measures are mostly derived from information theory, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=', they are ‘Shannon based’ (Schreiber, 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Kraskov, St¨ogbauer, and Grassberger, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Williams and Beer, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' In addi- tion, for time-series data, the time order of events is also critical to establishing causality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Time-lags between different variables need to be carefully evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Therefore the temporal resolution of time-series must be sufficient for estab- lishing the direction of information flow;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' missing data can lead to spurious correlations (Runge, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Non-linear correlations between multiple drivers can be very difficult to disentangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' We here attempt to address and demonstrate this with a framework (van Leeuwen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=', 2021) which uses a transformed infor- mation theoretic measure that applies to both discrete and continuous variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Typically the current state-of-the-art causal estimates are point estimates: Data is used to produce a single number to quantify the causal relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' There is no robust uncertainty quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Addressing this in general, is a work in progress (for eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=', Heckerman, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Runge, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' However, we will provide an elementary estimate of the distribution of the strength of causal relationships - the causal strength, cs from hereinafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' SOLA: sola_example_6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 30 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 1:30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 3 Author-a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Our goal in this work is to provide an initial assessment of the causal depen- dencies between the accuracy of a forecast, the target or “effect” variable, with the driver or “cause” variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' For reasons explained above, the driver variables are assumed to be solar activity (quantified by sunspot number), forecast lead time (which is primarily determined by longitudinal spacecraft separation for the OMNI and STEREO observations) and latitudinal spacecraft separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The typical approach would be to cross-correlate these variables, or rather the time-series associated with them, pairwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' However, as these relationships can often be nonlinear and multivariate we need more advanced estimators such as those based on information theoretic measures like mutual information and higher order terms (Chakraborty and van Leeuwen, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' So the approach we follow here is to start with the analog of pairwise correlation, but with the non-linear estimator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' mutual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' We then introduce a third variable, via conditional mutual information, to disentangle inter-dependencies amongst three driver/cause variables, in order that mediated or induced effects can be isolated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' In principle, a full causal network (Runge, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' van Leeuwen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=', 2021) can be constructed using time-series observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' But this comes with computational and, in certain situations, interpretation challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Hence we leave this for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' We describe the solar wind observations from OMNI and STEREO A and B spacecraft in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Next, we introduce the causal inference methods, demon- strating their application to the OMNI observations in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' We compute the distribution of causal relationships, first pairwise, quantified in terms of the mutual information using a non-linear information theoretic measure (subsec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='2), examine the time averaging effect on sunspot number (subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='3), followed by the conditional mutual information to separate influence of the third variable (subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' We use 27-day corotation forecasts (also called ‘recurrence’ or ‘27-day persistence’ forecasts) using only OMNI data first, as it eliminates the lead-time as a variable by design ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' this leaves us with testing 2 (instead of 3) drivers:the solar activity encoded in the (smoothed) Sunspot Number (or SSN27) and the latitudinal offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' By first learning dependencies in this simpler dataset, we then compare effects of this same subset of drivers in the STEREO datasets ignoring at first the lead time (subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Following this, in Section 4, we study induced or mediated dependencies with lead time included, by using the STEREO datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Finally we interpret the results and conclude whilst looking at future opportunities to improve forecasts in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Observations Two primary data sets are used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Firstly, the OMNI dataset of near-Earth solar wind conditions (King and Papitashvili, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Data are avail- able from https://omniweb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='gov/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Prior to 1995, data coverage varies significantly, so the period of study is limited to 1995 to present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Secondly, the STEREO dataset, which is available from https://stereo-ssc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='nascom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='gov/ data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='shtml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' STEREO-A data are used from the whole mission, 2007-present, while STEREO-B data are only available until 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' All data are averaged to SOLA: sola_example_6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 30 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 1:30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 4 Example paper Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' A summary of the corotation forecast of solar wind speed obtained by using OMNI near-Earth observations to forecast near-Earth conditions 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='27 days ahead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Top: Sunspot number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Middle: The absolute value of the latitudinal separation between observation and forecast location (∆θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Bottom: The mean absolute error in the solar wind speed corotation forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' All properties are calculated at 1-day resolution (dotted lighter curves), then averaged over 27 days (solid darker curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 1-day resolution to remove the effect of small-scale stochastic structure, such as waves and turbulence (Verscharen, Klein, and Maruca, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Solar wind speed corotation forecasts are produced by ballistically mapping data from the observation radial distance to 1 AU, then applying a corotation delay consistent with the longitude separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' By far the dominant factor is the longitudinal separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Further details can be found in Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' For each forecast we compute the mean absolute error (MAE) between the forecast and observed solar wind speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' For solar cycle context, we use the daily sunspot number (SN), provided by SILSO (Clette and Lef`evre, 2016) and available from https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='sidc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='be/silso/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Figure 1 shows a summary of OMNI data used to make a 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='27-day lead time forecast of near-Earth conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' By eye, some correlation can be seen between the MAE and SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' There are few intervals of MAE above 250 km/s during the solar minima of 1996-97, 2009-10 or 2019-20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Conversely, there is no immediately obvious relation between MAE and the absolute latitudinal separation between observation and forecast location, ∆θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' However, the ∆θ variation here is very small, arising from Earth’s latitudinal orbital motion over a 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='27-day interval and reaching a maximum magnitude of around 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='5◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Figure 2 shows the summary of STEREO-B observations used to forecast solar wind speed at STEREO-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' As the spacecraft separate in longitude, the forecast SOLA: sola_example_6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 30 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 1:30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 5 Time-Series : 27 day average time-series SSN 200 0 0 MAE [km s-1] 400 200 i 0 1995-02-23 2000-08-21 2006-02-15 2011-08-08 2017-01-28 TimeAuthor-a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' A summary of the corotation forecast of solar wind speed obtained by using STEREO-B observations to forecast conditions at the STEREO-A spacecraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Top: Sunspot number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Second row: The absolute value of the latitudinal separation between observation and forecast location (∆θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Third row: Forecast lead time (directly proportional to longitudinal separation, and to a much lesser extent, radial separation of spacecraft).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Bottom: The mean absolute error in the solar wind speed corotation forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' All properties are calculated at 1-day resolution (dotted lighter curves), then averaged over 27 days (solid darker curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' lead time, ∆t, increases almost linearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The maximum value of ∆θ grows as the spacecraft increase their absolute longitudinal separation until mid 2010, then declines as the spacecraft move closer together (behind the Sun, from Earth’s point of view).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' There is a somewhat linear growth in MAE from 2007 to 2012, though without further analysis it is not possible to say whether this is the result of sunspot number (post smoothing as we will see), ∆t or the amplitude of ∆θ increasing through this time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Or some combination of those variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Methods : Causal Dependencies of Corotation Forecasts We wish to study the principle drivers of the error in the corotation forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' In order to do that, we perform a causal analysis on the mean absolute error (MAE) as the target/effect variable and the sunspot number (SN), latitudinal offset (|∆θ|[◦]), and the forecast lead time (∆t [days]) as the principal driver/cause variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' With this setup we can use a non-linear measure of dependency to compute the causal relationships between these variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' There are a number of choices for such measures: those based on information theory as (conditional) SOLA: sola_example_6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 30 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 1:30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 6 Time-Series : 27 day average time-series 200 SSN 10 MAE [km s-1j△ t [days] 500 250 27/03/2007 14/08/2008 30/12/2009 18/05/2011 03/10/2012 20/02/2014 TimeExample paper mutual information (Kraskov, St¨ogbauer, and Grassberger, 2004), transfer en- tropy (Schreiber, 2000), directed information transfer (Amblard and Michel, 2009), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' We chose the mutual information (and its conditional variants) as it is well studied (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=', van Leeuwen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Runge, 2015) and there are robust estimators available, along with an analytical result for Gaussian variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The mutual information I(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' y1:N) between a target process x and a possible driver process y, or a whole range of driver processes denoted in our general formalism (van Leeuwen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=', 2021) by y1:N (or sometimes y, z, w,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=') is defined via the Shannon entropy H(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='.) as I(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' y1:N) = H(x) − H(x|y1:N) (1) = � p(x, y1:N) log � p(x, y1:N) p(x) p(y1:N) � dxdy1:N (2) Mathematically, the mutual information I(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' y1:N) is a positive definite quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' It can be thought of as the reduction in entropy (or uncertainty) in the target (here x) in presence of information content from the driver variables (here y1:N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Symbolic representation : Venn Diagram Visualisation This is shown graphically as an Information Venn Diagram in figures 3 (and 5 for higher order terms that we will discuss later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The circles represent the conditional entropy (H(x|y)) of the individual variables and the intersection (shaded region with lines) represents the reduction in entropy of variable due to the presence of the other, which is the mutual information (I(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' y)) defined in eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' For our application, one of these variables is the target and the other a driver ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' hence the superscripts n + 1 and n showing different time indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The labels also show the specific case on hand with the solar wind variables (MAE, SSN, ∆θ), but we will elaborate on these in upcoming subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The drivers that causally influence the target would reduce the entropy and the extent of this reduction is viewed as the extent of causal influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' On the other hand, if a driver does not have a causal influence, it does not reduce the entropy and the mutual information of the target with that driver is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Graphically this would mean a separation of the two circles with zero overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' There are limitations to a formal interpretation of all situations in terms of Venn diagrams - this will become clear for higher order terms like interaction information described in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='4 (Ghassami and Kiyavash, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Hence, these Venn diagrams serve as a visualisation to build up our intuition rather than be a formal representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Mutual Information : Pairwise dependency between Latitudinal Offset, Sunspot Number and MAE With the goal of disentangling causal influences of drivers in corotation forecasts, we begin with OMNI data used to make a forecast at Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' In this case, coro- tation forecasts have a fixed lead time of 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='27 days and forecast error, MAE, inherently has two primary drivers, the time variation of the sun – approximated SOLA: sola_example_6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 30 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 1:30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 7 Author-a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' xn+1=MAEn+127 zn=∆θn or yn=SSNn 27 I( xn+1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' zn ) = I(MAEn+127 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' ∆θn) I( xn+1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='yn ) = I(MAEn+127 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' SSNn 27) or Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The figure shows a graphical illustration of the mutual information via Venn diagrams of the conditional entropies H(x|y) or H(x|z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' One circle represents the entropy content in the target - forecast accuracy (at time n + 1) and the second represents that of one driver - either SSN or ∆θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The intersection represents the reduction of entropy in the target by knowledge of the driver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' by the sunspot number (SN) – and latitudinal offset (∆θ) between the observa- tion and forecast position (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' between Earth’s location 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='27 days apart).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' This provides a relatively simple causal network to explore with our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' We compute the mutual information (MI) between pairs of the target and one of the drivers, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=', I(MAE ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' |∆θ|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Given the length of the observation time series, we can empirically estimate the distribution of these quantities as histograms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The mutual information serves as the measure of causal dependency between pairs of one of the drivers and the target variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Once again we refer to figure 3 for a visualisation graphically via Venn diagram described in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' In this figure, the example is given for variables x and y representing target MAE, and driver either ∆θ or SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' In other words, we determine the reduction in entropy (or random uncertainty) in MAE, due to ∆θ or SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' It must be noted, that we break the symmetry between the two variables (target v/s driver), with the driver (cause) as lagging in time with respect to the target (effect).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The quantities (or rather their distributions) represented by these information diagrams are estimated in figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' As a positive definite quantity with no upper limit, MI can take very large values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Thus it is useful to normalise this measure, which is possible in a number of ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' One option is to normalise it with the total entropy or uncertainty in the SOLA: sola_example_6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 30 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 1:30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 8 Example paper Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Histograms of causal strengths, cs, of drivers – sunspot number(SN) and latitudinal offset (∆θ) – on the driver, corotation MAE obtained from 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='27-day forecasts using OMNI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Dashed lines represent mean cs values used in causal diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' We take the 27-day smoothed MAE as the target in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' On the Left: SN and ∆θ are used at daily resolution ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' cs computed from daily SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Right: cs computed from 27-day smoothed SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' cs for SN shows a greater dependence on |∆θ| than on SN, whereas the 27 day smoothed SN clearly shows the greater association of solar cycle with MAE by suppressing the stochasticity and emphasising the solar activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' variable x, giving the causal strength, cs(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' y1:N) = I(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='y1:N) H(x) or simply cs(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' z) = I(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='z) H(x) for two variables : target x and driver z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' There is a challenge here ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' the entropy we use is for continuous variables, also known as the differential entropy, which can acquire negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' In practice, we do not encounter this here in our applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' However, to mitigate this effect – and for general interpretation – we will ultimately use relative causal strengths to the total over all the drivers combined;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' in these relative causal strengths we ignore the contribution of noise or unmodeled drivers to merely focus on interpreting selected drivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Influence of Sunspot Number - Timescale Matters The measured or observed quantity for solar activity is the daily sunspot number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' These observations display large variability as seen in figures 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' As we will demonstrate here, the stochasticity has an impact on the causal association with forecast accuracy term, MAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Figure 4 shows the corresponding distri- bution of causal strengths of the pairs of MAE with 27-day smoothed right and daily unsmoothed Left SN and the latitudinal offset (∆θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The unsmoothed daily sunspot number (SN) has lower cs (= MI/H) than the latitudinal offset (∆θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' However, upon performing a rolling mean on the daily SN to yield 27- day smoothed averaged SN or the SSN27, the hierarchy reverses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' As shown in figure 4, we see that the total causal strength of SN, cs(SSN27 → MAE27), goes from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='03 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='14 upon averaging, compared to cs(∆θ → MAE27) with a mean value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' As expected, the stage of solar cycle and overall time variability of the Sun is better represented by the smoothed SN, which has a significant influence on MAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' That the daily SN has a significant stochastic component is also confirmed by / evident from the entropy estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The entropy is reduced upon smoothing or averaging SN and is lower than that of ∆θ by a factor of a few (for example ≈ 2 for STB-STA ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' however, this is not a significant effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' SOLA: sola_example_6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 30 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 1:30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='0 Meanl cs(MAE27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='SN) )=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='03 Mean cs(MAE27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' △)=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='5 cs(MAE27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Sunspot Number) CS(MAE27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' △) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='08 Causal Strength = Information Term / EntropyMean cs(MAE27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' SSN27) )=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='14 Mean cs(MAE27: e) )=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='05 cs(MAE2z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='SmoothedSunspotNumber2z) 4 Cs(MAE27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' △0) 3 2 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='18 Causal Strength = Information Term / EntropyAuthor-a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' xn+1=MAEn+127 yn=SSNn 27 zn=∆θn I( xn+1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' zn ) = I(MAEn+127 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' ∆θn) I( xn+1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' yn ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' zn ) = I( MAEn+127 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' SSNn 27 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' ∆θn ) H( xn+1 | yn, zn ) = H(MAEn+127) I(MAEn+127 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' SSNn 27) - I(MAEn+127 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' ∆θn) + I(MAEn+127 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' SSNn 27 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' ∆θn) Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The figure shows the different information components for three variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The intersections represent information shared between variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The black striped region is the mutual information shared between target MAE and driver ∆θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The yellow dots denote inter- action information : information shared between MAE27 and both ∆θ and SSN27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The pink circles show the entropy or uncertainty in MAE27 that is not explained or shared by either SSN27 or ∆θ or their interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Conditional and Interaction Information : Higher order terms In presence of multiple causes or drivers (say y and z), the aforementioned causal strength term, cs(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' z) will come to represent the fractional reduction in uncertainty in the target due to the driver, z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' And there’s a similar term for y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' To further disentangle and isolate the influence of each driver we also compute the conditional mutual information (CMI), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=', I(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' z|y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' For two drivers y and z (and a single target x), conditional mutual information, I(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' z|y) given in eqn 3, ‘conditions out’ the effect of one driver (y), leaving the direct influence of the other one (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' This can be visualised in terms of Venn diagrams in figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' It is the difference between the intersection of x and z circles (black stripes) and that of x, z and y circles (yellow spots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' In our application to corotation forecasts, the example used for illustration has x as MAE27 and the y and z as the drivers ∆θ and SSN27, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' We will keep the same normalisation with entropy for all information terms so that they can be combined or compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The conditional mutual information can be defined in terms of the conditional entropies as, I(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' y|z) = H(x|z) − H(x|y, z) (3) The above equation for the conditional mutual information of x with respect to y and z represents the difference in entropy of x “conditioning out” z alone (H(x|z)) and entropy of x “conditioning out” y and z together (H(x|y, z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' This SOLA: sola_example_6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 30 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 1:30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 10 Example paper leaves us with the direct influence of driver y on target x, excluding any indirect influence mediated by or shared with z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The distributions of such conditional information terms for the triplet (MAE27,∆θ,SSN27) are estimated in figure 6 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' these provide the so-called direct causal influence contribution of SSN27 and ∆θ on MAE27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' These are symbolised by the black arrows in the causal summary diagram in figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The causal summary diagram, as the name suggests, provides a summary of the information flow from (and therefore the causal influence of) the driver variables ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' in this case the latitudinal offset (∆θ) and the smoothed sunspot number (SSN27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Now the interaction information can be written in terms of the mutual and conditional mutual information as, I(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' z) = I(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' y) − I(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' y|z) = I(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' z) − I(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' z|y) (4) This equation for the interaction information of x with y and z gives the dif- ference between the mutual information shared between x and y (I(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' y)) and information shared between them, upon conditioning out z (I(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' y|z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' This is the interaction information shared between the 3 variables, x, y and z and is symmetric in all three variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' If we fix one as the target with the other two as drivers, as we do for our application, then the expression for interaction infor- mation is symmetric in the two drivers as demonstrated by the two equivalent expressions for I(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' z) in equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' So it doesn’t matter which driver we condition on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' We will exploit this later on as estimates from actual measurements may not converge to the same value as eqn 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' So we can take the average of the two symmetric expressions to represent the interaction information between one target and two drivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' This is seen later in the observational estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' This quantity can be interpreted as the information shared between x and y, less the information shared between them when z is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' If the interaction information is non-negative, or I(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' y) ≥ I(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' y|z), it implies that the depen- dency of x on z partially or entirely (equality) constitutes the dependency on y (Ghassami and Kiyavash, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' If the interaction information is negative, or I(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' y) < I(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' y|z), then each one of the variables induces and increases correlation between the other two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' In the previous subsection, we ascertained that the smoothed sunspot number or SSN27 is more appropriate as a proxy for the solar activity in evaluating its causal influence on the average corotation forecast accuracy, MAE27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Now we wish to disentangle the direct and indirect effects of both SSN27 and the latitudinal offset, ∆θ on MAE27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Their joint effect, or one mediating through the other, is naturally a higher order effect and hence we need the higher order information terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' We compute the higher order information theoretic quantities, namely conditional mutual information and interaction information between drivers SSN27 and ∆θ and the target, MAE27, still using only the OMNI dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' For MAE27 (x), ∆θ (y) and ∆t (z), the interaction information corresponds to the common part with yellow circles in the Venn diagram in figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' This is therefore the information shared across all three variables in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Formally, causality necessitates there be a time lag between the cause and effect such that the former precedes the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' And indeed there is a time SOLA: sola_example_6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 30 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 1:30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 11 Author-a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The figure shows the Left: distribution of conditional causal strengths with OMNI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' cs here is associated with conditional mutual information normalised by the entropy, H, for the combinations of MAE27 with Sunspot Number (SSN27) (27 day rolling averages for both) and Latitudinal Offset for the full dataset, namely cs(MAE27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' SSN27|∆Θ) and cs(MAE27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' ∆Θ|SSN27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' ∆θn MAEn+127 SSNn 27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='14 OMNI_to_OMNI Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The figure summarises the mean MAE27 dependence on latitudinal offset ∆θ and SSN27 individually (black) and in combination (red) for OMNI-OMNI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' This is done in terms of the causal strength (cs = Information Term Entropy ) values defined earlier - the numbers attached to the arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The superscripts merely indicate the formal need for the causes (SSNn 27, ∆θn) to precede the effect (MAEn+1 27 ) - in practice for this application, a single time-step makes negligible difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' SOLA: sola_example_6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 30 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 1:30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='0 Meancs(MAE27:SSN27|△e))=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='15 Mean cs(MAE27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='|SSN27))=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='03 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='5 cs(MAE27:SmoothedSunspotNumber2zl△e) cs(MAE27:△|SmoothedSunspotNumber27) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='200 Causal Strength= Information Term/ EntropyExample paper lag between the forecast accuracy of a future step, MAEn+1 27 , and the drivers ∆θn and SSN n 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' This is innate/intrinsic to the way the time-series observa- tions are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' However, in this particular application, we are considering daily variations and the drivers – latitudinal separation, longitudinal separation and 27-day smoothed sunspot number – vary over much longer timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Thus a single time-step between n and n+1 makes negligible difference to the computed information components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' However, the notation involving target at n+1 and drivers at n is maintained to demonstrate the general principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Symbolic representation : Causal Summary diagrams The causal information flow between the variables is summarised in figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The nodes (or ovals) represent the variables and the arrows represent the flow of information to the target variable, MAE27 one time step in the future (n+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Black arrows represent the influence of single driver conditioning out influence of the other drivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The red segments ending in an arrowhead on the target repre- sents the joint causal influence of the drivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The confluence of the segments out of the drivers into a point symbolises this join or combined effect ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' the arrowhead as usual points to the information flow into the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' This represents the com- ponent of influence that is driven by the combination of drivers together, distinct from their individual, direct influences on the target, shown by black arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' This combined or joint effect could be a positive one showing a redundancy in driver or that one driver partially or entirely captures the influence due to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' It could be negative suggesting that one driver induces an influence from the other driver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' These can be mathematically quantified in terms of the interaction information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Here for 27-day corotation forecasts using only OMNI data, the only drivers are ∆θ and SSN27, now considered simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' (As we will see in an upcoming section, the lead time ∆t – related to the longitudinal separation – will have a role to play for STEREO data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=') We find that the direct influences of SSN27 and ∆θ (black arrows) are more important than the joint influence (red arrow) on MAE27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' In general, we can compute joint influence due to multiple drivers starting from pairs (the red arrows) to the joint influence of all n drivers simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' However, to use full general mathematical framework in van Leeuwen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' (2021) is computationally expensive and complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' It is also not essential in our work here to get the main dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' We compute the causal strengths (defined earlier) from the mutual and conditional mutual information terms in accordance with van Leeuwen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The black arrows are given by: (∆θn → MAEn+1 27 )1link = I(MAEn+1 27 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' ∆θn|SSN n 27) (5) (SSN n 27 → MAEn+1 27 )1link = I(MAEn+1 27 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' SSN n 27|∆θn) (6) The red arrow symbolising the joint influence of ∆θ and ∆t represents and is related to the interaction information shown in the figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Graphically this represents the intersection of the information component common to each of the three variables in our triplet i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' two drivers (latitudinal offset and lead time) SOLA: sola_example_6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 30 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 1:30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 13 Author-a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' and target (MAE27 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' This is therefore symmetric and is, theoretically, indepen- dent of the variable that it is conditioned on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' However, when estimating from measured quantities, this symmetry, indicated both graphically in figure 5 and equation 4 is not strictly adhered to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Hence, we can express the joint influence indicated by the red arrow as the average of the two equivalent ways of estimating it as: (∆θn → MAEn+1 27 )2link + (SSN n 27 → MAEn+1 27 )2link = 1/2 [ I(MAEn+1 27 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' ∆θn) − I(MAEn+1 27 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' ∆θn|SSN n 27) + I(MAEn+1 27 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' SSN n 27) − I(MAEn+1 27 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' SSN n 27|∆θn)] (7) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Consistency across Datasets We next test this relative influence of SSN27 and ∆θ on MAE27 across the avail- able datasets, namely STB-STA, STB-OMNI and OMNI-STA pairings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' This is shown in figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' In each case, we find the interaction information, I(MAEn+1 27 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' ∆θn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' SSN n 27) to be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' This is an indication that SSN27 partially constitutes the depen- dency of MAE27 on ∆θ and vice versa, but it is not very significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' And across these three datasets (as well as OMNI-OMNI recurrence forecasts), we found that the direct causal strengths of latitudinal offset, I(MAEn+1 27 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' ∆θn|SSN n 27), is around 60−70% of the direct causal strength of SSN27, I(MAEn+1 27 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' SSN n 27|∆θn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Furthermore, estimates of the interaction information, given by I(MAEn+1 27 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' SSN n 27)− I(MAEn+1 27 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' SSN n 27|∆θn), are merely ∼ 7% of the direct causal influence, as was also shown in OMNI dataset in figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' This suggests that to a good approximation, the causal influence of the solar activity is decoupled from that of the latitudinal offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' This will aide us in considering the causal influence of lead time in turns with these two variables, simplifying the causal network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' STEREO : Effect of Lead Time As explained in the previous section, the OMNI (27-day) corotation forecast dataset allows us to focus on the causal influence of ∆θ and SSN27 as proxy of the solar activity on MAE27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Having learnt that the interaction information between these three driver variables is small, we can assume their influence to be largely independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' We will now proceed to pair ∆θ and SSN27 by turns, with the lead time ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' This will give us the direct and interaction terms for each case, analogous to the causal network in figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Conditional Causal Influence : Lead Time Analogous to the causal analysis of the OMNI time-series, we estimate causal linkages using the STA-OMNI corotation time-series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' We begin with the time- series of ∆t and ∆θ as drivers along with the MAE27 as the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' We compute causal strength terms corresponding to mutual information terms I(MAEn+1 27 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' ∆θn) SOLA: sola_example_6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 30 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 1:30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 14 Example paper Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' MAE27 dependence on principle drivers for corotation forecasts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' latitudinal off- set ∆θ conditioned on smoothed sunspot number or SSN27 and vice versa in pairs of Top: STB-STA, Middle: STB-OMNI and Bottom: OMNI-STA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' SOLA: sola_example_6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 30 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 1:30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='0 MeancS(MAE27:SSN2zl△0))=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='15 MeancS(MAE27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='|SSN27))=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='06 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='5 cs(MAE27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='SmoothedSunspotNumber27|△) cs(MAE27△e|SmoothedSunspotNumber27) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='30 Causal Strength = Information Term / Entropy3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='0 Mean cs(MAE27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='SSN2zl△0))=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='13 Meancs(MAE27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='△|SSN27))=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='08 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='5 cs(MAE27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='SmoothedSunspotNumber27|△) cs(MAE27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='△e|SmoothedSunspotNumber27) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='175 Causal Strength=InformationTerm/Entropy4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='0 Meancs(MAE27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='SSN2zl△))=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='12 Meancs(MAE27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='△0|SSN27))=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='08 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='5 cs(MAE27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='Smoothed SunspotNumber27|△) cs(MAE27△e|SmoothedSunspotNumber27) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='25 Causal Strength = Information Term / EntropyAuthor-a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The figure shows the histogram of causal strengths, cs, of drivers – the lead time, ∆t27 and ∆θ – with the target, MAE27 using STA-OMNI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' On the Left:, are the distri- butions of pairwise strengths (MI/H) and on the Right: are the conditional causal strengths (CMI/H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' and I(MAEn+1 27 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' ∆tn) and the conditional information terms, I(MAEn+1 27 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' ∆θn|∆tn) and I(MAEn+1 27 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' ∆tn|∆θn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' I(MAEn+1 27 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' ∆θn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' ∆tn) is actually the influence of ∆θ on MAE27 that is shared by (or mediated through / induced by) ∆t, in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' From figure 9, it is evident from the histograms of cs(MAEn+1 27 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' ∆θn) and cs(MAEn+1 27 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' ∆θn|∆tn) (or equivalently the corresponding information terms) that the interaction infor- mation, I(MAEn+1 27 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' ∆θn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' ∆tn) is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Using the mean values of each term in the information triplet, direct (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' conditional mutual) and interaction informa- tion for (MAE27,∆θ,∆t) we get an interaction information amounts to ∼ 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The direct influence of ∆t on MAE27 is ∼ 50% and the direct influence of ∆θ is the remaining ∼ 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' These numbers are the relative proportions of influence of the two drivers considered ∆θ and ∆t, without considering SSN27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The ‘missing’ 20% is likely to have a significant contribution from the other driver, SSN27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' However, we also have noise, which includes the statistical fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Also, the averaged sunspot number is used as a proxy for the time evolving nature of the solar wind ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' it’s not an exact proxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Histograms with 15 bins as before are used to estimate the distribution of these causal strength terms and shown in figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' It is clear that the lead time, ∆t, has an influence on the MAE27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Upon conditioning on ∆θ, we find a sizable direct influence of ∆t on MAE27, around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='5 times greater than the direct influence of ∆θ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' this is shown by the black arrows in the summary diagram in figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Next we look at the driver pair of lead time and (smoothed or rolling 27 day average) sunspot number (∆t, SSN27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' We again estimate the pairwise (mutual information) and direct dependencies (conditional mutual information) of ∆t and SSN27 on MAE27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Once again, the two drivers SSN27 and ∆t have shared dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' This is summarised in figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The black arrows are given by : (∆tn → MAEn+1 27 )1link = I(MAEn+1 27 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' ∆tn|SSN n 27) (8) SOLA: sola_example_6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 30 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 1:30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='0 Mean cs(MAE27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='△0) )=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='12 Mean cs(MAE27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' △t27) )=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='23 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='5 Cs(MAE27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' △0) Cs(MAE27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' △t27) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='30 Causal Strength = Information Term / Entropy4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='0 Mean cs(MAE27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' △0|△t27)=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='04 Mean cs(MAE27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' △t27l△0)=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='5 cs(MAE27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' △|△t27) cs(MAE27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' △t27|△) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='20 Causal Strength = Information Term / EntropyExample paper ∆θn MAEn+127 ∆tn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='19 SSNn 27 MAEn+127 ∆tn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='08 ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='12 STA_to_OMNI ∆θn MAEn+127 SSNn 27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='13 ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='08 Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The figure summarises the mean causal dependence of forecast accuracy MAE27 on latitudinal offset ∆θ, lead time ∆t and average/smoothed sunspot number SSN27 individually (black) and in pairwise combinations (red) for STA-OMNI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' SOLA: sola_example_6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 30 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 1:30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 17 Author-a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' and (SSN n 27 → MAEn+1)1link = I(MAEn+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' SSN n 27|∆tn) (9) The red arrows symbolising the joint influence can be written symmetrically as a sum of (∆tn → MAEn+1 27 )2link + (SSN n 27 → MAEn+1 27 )2link = 1/2 [A I(MAEn+1 27 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' ∆tn) − I(MAEn+1 27 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' ∆tn|SSN n 27) + I(MAEn+1 27 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' SSN n 27) − I(MAEn+1 27 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' SSN n 27|∆tn)] (10) The quantitative analyses is summarised in the causal summary diagram in fig 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The direct causal influence of ∆t and SSN27 are ≈ 41% and ≈ 28%, respectively, in relative terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' And the joint influence is ≈ 31%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Dependencies of the driver triplet Here we put aside the target variable, MAE27, and apply the causal measures to explore the dependencies between the drivers themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The goal is to directly probe the statistical (in)dependence of the drivers without any lag i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' we do not seek causal information flow in time, from one variable to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Instead we look simply for information overlap which tells us the how the drivers relate to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Hence, the three variables are SSN n 27, ∆θn and ∆tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' These dependen- cies between drivers are symbolised by line segments instead of arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' So with no natural target variable, we treat SSN27 effectively as the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The reason for this, is that apriori we might expect a stronger dependence between ∆θn and ∆tn and hence we wish to see how these two affect SSN27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Our approach is reflected in the causal diagram shown in figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' We find that the direct effect (black line segment) of ∆t on SSN27 is greater than that of ∆θ by over an order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' This is because the solar activity does depend upon the phase and hence the lead time across cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' And while the joint effect (red segment) of ∆t and ∆θ on SSN27 is lower than the direct effect of ∆t, it is still non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' As the corresponding interaction information term is positive, it suggests that ∆t mediates the dependency on ∆θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Conclusion In this paper, we probe what drives the accuracy of corotation forecasts of the solar wind observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' As we do not have means to make interventional ex- periments, we apply causal inference methods to the available observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The causal drivers of relevance are the latitudinal separation or offset (∆θ) between the observing and forecast locations, the forecast lead time (∆t) and the solar activity, as measured by 27 day rolling average of the sunspot number (SSN27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The target variable is the forecast accuracy measured in terms of the mean absolute error between observation and forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' We use information theoretic measures to estimate the strength of the causal influence of the drivers on the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' These do not merely estimate correlations between pairs of variables, but SOLA: sola_example_6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 30 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 1:30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 18 Example paper xn=SSNn 27 yn=∆tn zn=∆θn I( xn ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' yn ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' zn ) = I(SSNn 27 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' ∆θn ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' ∆tn) ∆θn SSNn 27 ∆tn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='15 STB_to_OMNI Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The figure summarises the mean (in)dependence of the smoothed sunspot number (SSN27) on the latitudinal offset ∆θ and corotation time ∆t individually (black) and in combination (red) for STB-OMNI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Here we test the interdependence of the drivers and not a relation between causes preceding the effect symbolically shown by the absence of the arrow heads ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' hence each is at n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' can disentangle the influence of a third variable via conditioning the information content shared between three of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Depending on the different information terms or components, we can estimate the influence of the individual drivers as well as their joint effect, induced effects, and redundancy due to correlation amongst drivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' We draw the following conclusions: i) The decomposition of information flow between the different drivers (or causes) and the target is effective in identifying cause and effect relationships driving dynamical systems in presence of complex, non-linear relationships between multiple variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' This approach is perfectly suited to trace what drives the corotation forecast accuracy or rather the uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' ii) The pairwise causal relationship between drivers – latitudinal offset (∆θ), forecast lead time (∆t), and the average sunspot number (SSN27) – and SOLA: sola_example_6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 30 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 1:30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 19 Author-a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' target MAE27 given by mutual information normalised to the target entropy confirms our understanding from previous results (for eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The solar activity levels measured via SSN27 and the lead time ∆t have a big influence on the average forecast error, MAE27, followed by the latitudinal offset, ∆θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Statistical noise levels impacts the absolute values of the depen- dencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The relative values of the causal strengths, however, teach us of the hierarchy of influence of the different driver combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' iii) Exploiting higher order measures like interaction information, we can probe deeper into causal influence of multiple drivers in conjunction with one an- other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' A non-zero interaction information has two possible cases with cor- responding interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Negative values show an induced effect (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' one driver showing a coupling to the target induced only due to the presence and influence of another driver) whereas positive values show a redundant / shared influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' We can be quantitative about the relative importance of joint causal (positive or negative) influence and therefore potentially impact forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Also qualitatively, knowledge of whether drivers act independently of one another, can help design future data experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' For instance, from fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 8, it is clear from the consistency of MAE27 dependence on SSN27 and Latitudinal Offset – from OMNI alone and OMNI with STEREO – that solar activity has a stronger, direct influence on MAE27 than latitudinal offset, independent of lead time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Hence, the appropriate weight can be given to each of these drivers in an assimilation forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' On the other hand, from fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' (11) while solar activity and latitudinal offset have a weaker direct association (black line between them) relative to its stronger, direct coupling with lead time,they do have an indirect coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' The association of sunspot number with latitudinal offset is owed predominantly to the lead time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' This implies, that the phase of the solar cycle is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' iv) The interaction information terms, such as I(MAEn+1 27 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' ∆θn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' ∆tn) and I(MAEn+1 27 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' SSN n 27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' ∆θn) or equivalently the corresponding causal strength terms (I/H) quantify the information content shared between the 3 variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' From these terms, we can learn that the SSN27 and ∆θ share very little information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' their influence on MAE27 is predominantly independent of one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' On the other hand, for the STEREO dataset, ∆θ and ∆t share a non-trivial fraction (∼ 20%) of the their total information content (or influence on MAE27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' And in this case, the +ve sign of the interaction information indicates that ∆θ partially contributes to the influence of ∆t on MAE27, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' These effects could not be revealed by standard correlation analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Using a causal inference approach, disentangles the drivers of the forecast accuracy in ways that standard statistical analyses or data assimilation can- not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Rather one can improve upon these latter two, using causal diagnostics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' It allows us to not only disentangle individual sources of uncertainty in the forecast, but also calculate partial and complete redundancies in drivers of this uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' These learning can potentially be applied to improve the solar wind data assimilation forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' SOLA: sola_example_6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='tex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 30 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 1:30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' 20 Example paper Acknowledgments This work was part-funded by the Science and Technology Facilities Council (STFC) grant numbers ST/R000921/1 and ST/V000497/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' Appendix References Amblard, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=', Michel, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' : 2009, Measuring information flow in networks of stochastic processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktFKT4oBgHgl3EQfxy4K/content/2301.11904v1.pdf'} +page_content=' CoRR abs/0911.' metadata={'source': 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b/mNFKT4oBgHgl3EQfEC1Y/content/tmp_files/2301.11714v1.pdf.txt @@ -0,0 +1,555 @@ +1 +Distributed Consensus in Wireless Networks with +Probabilistic Broadcast Scheduling +Daniel Pérez Herrera, Zheng Chen and Erik G. Larsson +Abstract—We consider distributed average consensus in a wire- +less network with partial communication to reduce the number of +transmissions in every iteration/round. Considering the broadcast +nature of wireless channels, we propose a probabilistic approach +that schedules a subset of nodes for broadcasting information +to their neighbors in every round. We compare several heuristic +methods for assigning the node broadcast probabilities under +a fixed number of transmissions per round. Furthermore, we +introduce a pre-compensation method to correct the bias between +the consensus value and the average of the initial values, and +suggest possible extensions for our design. Our results are partic- +ularly relevant for developing communication-efficient consensus +protocols in a wireless environment with limited frequency/time +resources. +Index Terms—Average consensus, broadcast transmission, +scheduling, wireless networks. +I. INTRODUCTION +Distributed consensus deals with the problem of reaching +an agreement among networked agents on a certain quantity +of interest that depends on the initial states of all agents. +The interaction rule that specifies the information exchange +between an agent (node) and its neighbors is known as +consensus protocol or algorithm [1]. Traditional studies of +consensus problems usually assume a fixed topology with +perfect communication between the nodes in the network. For +this ideal case, the convergence conditions and convergence +speed of discrete-time average consensus are provided in [2] +and [3]. As an extension to the main theme, different methods +can be used to accelerate convergence, such as extrapola- +tion [4] and stepsize adjustment [5]. For varying topologies, +necessary and sufficient conditions to reach consensus over +random networks are presented in [6], using ergodicity and +probabilistic arguments. +Conventional works on consensus algorithms tend to rely +on idealized assumptions (e.g., instantaneous, error-free) on +the communication links between connected nodes. Consensus +algorithms may suffer from imperfect communication between +the nodes due to, for example, link failures [7], [8], [9]. To deal +with the missing information, two compensation (biased and +balanced) methods are presented in [10]. The main difference +between these methods is how the weights of the failed links +are re-distributed among the successful links. Both algorithms +converge almost surely and in the mean-square sense to a value +that generally is not equal to the average of the initial values. +The authors are with the Department of Electrical Engineering, Linköping +University, Linköping, SE-58183 Sweden. E-mail: {daniel.perez.herrera, +zheng.chen, erik.g.larsson}@liu.se. This work was supported by Zenith, +ELLIIT, and the KAW foundation. +In addition to the conventional consensus protocols where +every node communicates with all their neighbors in every iter- +ation, gossip algorithms can also be used to achieve consensus +with randomized and asynchronous communication [11], [12]. +A key feature of these algorithms is the asynchronous time +model, where information is exchanged between a subset of +nodes. This information is processed by the receiving nodes to +compute local update via a linear combination of its own value +and the received one. There are different variants of gossip +algorithms, such as geographic gossip [13], randomized gossip +[14] and broadcast gossip [15]. Particularly, broadcast gossip +is more relevant to the implementation of gossip algorithms +in wireless networks, where the transmission of one node can +reach any other nodes within its communication range. +Unlike the case with broadcast gossip, where the nodes +update their states immediately after receiving information +from another node, we propose a probabilistic broadcast-based +method to schedule a subset of nodes for broadcasting in +every communication round. The nodes update their values +after receiving information from all the scheduled broadcasting +nodes. Since only a subset of nodes is selected for transmission +in every round, the number of transmission slots needed in +every round is reduced as compared to the full communication +case. To compensate for the lack of information with our par- +tial communication approach, we use a biased compensation +method similar to the one proposed in [10]. Our objective is to +achieve fast convergence with fewer transmission slots while +maintaining small deviation from the average of the initial +values. We investigate several heuristic methods for assigning +the broadcast probabilities of the nodes, combined with a +pre-compensation mechanism to eliminate the bias caused by +unbalanced information exchange. +II. SYSTEM MODEL +A. Graph Model +We consider a wireless network modeled by an undirected +and connected graph G = (N, E), where N is the set of nodes +and E is the set of edges. The total number of nodes is N. +Each edge {i, j} ∈ E indicates the existence of a link between +the pair of nodes i and j. The set of neighbors of node i is +Ni = {j|{i, j} ∈ E}.1 +The connectivity of the graph can be represented by its +adjacency matrix A, with elements aij = 1 if {i, j} ∈ E, +1In a wireless network, the notion of connectivity is not strictly defined +since every node can hear the transmission of other nodes in the network. We +consider a simplified model where a link between two nodes does not exist +when the distance is larger than some predefined threshold. +arXiv:2301.11714v1 [cs.MA] 27 Jan 2023 + +2 +and aij = 0 otherwise. Another matrix representation of the +graph is the Laplacian matrix, defined as L = D − A, where +D is a diagonal matrix whose i-th entry is di = |Ni|. +B. Consensus Algorithm +We consider the problem of distributed average consensus, +where the goal is for all nodes in the network to reach +consensus on the arithmetic mean of their initial values, by +only communicating with their neighbors. To accomplish this, +we use the following distributed linear iteration +x(t + 1) = Wx(t), +(1) +where x(t) = [x1(t), x2(t), ..., xN(t)]T contains the values of +every node in iteration t ≥ 0. W is the mixing matrix, defined +as W = I − ϵL, where I is the identity matrix and ϵ ∈ R is +the step size. Let ∆(G) be the maximum degree of the graph +G. When 0 < ϵ < 1/∆(G), the following are necessary and +sufficient conditions for x(t) to converge to the average of the +initial values [2]: +1) Wu = u, +2) uT W = uT , +3) ρ(W − uuT /N) < 1, +where ρ(·) denotes the spectral radius of a matrix and u +the all 1’s column vector. Note that since W is symmetric, +conditions 1 and 2 are equivalent. Condition 3 is also related +to the convergence speed, since minimizing the second largest +eigenvalue of W can speed up convergence of distributed +averaging algorithms [2]. +III. PARTIAL COMMUNICATION WITH PROBABILISTIC +BROADCAST SCHEDULING +Distributed consensus algorithms rely on an iterative process +where in each iteration, every node must communicate with all +its neighbors before the updating step in (1). The communi- +cation design of consensus algorithms in wireless networks +is non-trivial due to the “many-to-many” communication +topology. If all nodes transmit information simultaneously +using the same frequency-time resources, then all signals will +be garbled. One simple way to avoid interference between +concurrent transmissions is to assign orthogonal resources +to each transmitting node. For instance, each node occupies +one dedicated time slot for broadcasting its information to +the neighbors. In this way, it takes N times slots in total +for the entire network to complete information fusion in one +consensus iteration with full communication. +We refer to one communication round as one iteration +interval during which all nodes exchange information with the +neighbors and update their state values, and one transmission +slot as the time interval during which one node broadcasts +its value. With full communication over the network, after +J iterations of the consensus algorithm, the total number of +required transmission slots is NJ. +To reduce the communication costs (time slots), we consider +a partial communication design where in every round we only +select a subset of nodes for broadcasting their information. Let +p = [p1, p2, ..., pN]T be a vector that contains the broadcast +probabilities of the nodes with �N +i=1 pi = K and K < N. +This means that in every communication round, K nodes are +selected on average for broadcasting. Therefore, the average +number of transmission slots for completing J iterations is +KJ < NJ. +Since in every iteration, some nodes are not scheduled for +broadcasting their values, we compensate the missing infor- +mation by using the biased compensation method proposed in +[10]. The idea is to use a new mixing matrix W(t) in every +round t, whose elements are given by +wij(t) = +� +� +� +wijvj(t), +if +j ̸= i. +1 − �N +k=1,k̸=i wikvk(t), +if +j = i. +(2) +Here, vj(t) = 1 if node j is selected for broadcasting in the +t-th communication round, and vj(t) = 0 otherwise. With +our probabilistic scheduling approach, we have pi = E[vi(t)]. +Though the biased compensation method ensures convergence +in mean square, in general the consensus value is not equal to +the average of the initial values, since the new mixing matrix +is only row stochastic but not necessarily doubly stochastic. +A. Heuristic Designs for Broadcast Probability Vector +Intuitively, the broadcast probabilities should be determined +in a way that can reflect the “importance” of the nodes +in a network. As a first approach, we consider different +methods of creating the broadcast probability vector based +on commonly used metrics for network connectivity, such as +degree, PageRank and betweenness centrality [16]. +1) Degree-based method: Degree centrality of a node is +the number of neighbors of the node: the degree of node j is +dj = |Nj|. We can choose the broadcast probability of node +i as +pi = min{1, γdi} +(3) +where the constant γ is chosen such that �N +j=1 pj = K, for +a given value of K. +2) PageRank-based method: PageRank was first proposed +to rank webpages using the hyperlink network structure of +the web. It can be used in any network as a measure of +the node importance. The PageRank-based method gives us a +probability vector p in the same way, but using the PageRank +centrality of the nodes instead of their degrees. +3) Betweenness-based method: Betweenness centrality of +a node is defined as the fraction of shortest paths between +pairs of other nodes that pass through it. Let bi represent the +betweenness centrality of node i. Since bi is zero if node i +has degree one, a small positive value is added to this metric, +i.e., the new score ˜bi of node i is: +˜bi = bi + β, +(4) +where β is a small positive number. The probability vector +p is then created in the same way as for the degree-based +method, but using this new score instead of the node degree. + +3 +B. Bias Correction +With our probabilistic scheduling of broadcasting nodes in +every round, the matrix W(t) is row stochastic with positive +diagonals (as long as pi > 0, ∀i). As a consequence of +Theorem 2.1 in [10], the product �J +t=0 W(t) converges almost +surely, when J → ∞, to a random rank-one matrix, uαT , for +some vector α = [α1, α2, ..., αN]T . In general, a bias will +be introduced, meaning that the consensus value will not be +the average of the initial values. There are several existing +bias correction approaches in the literature of distributed +consensus, such as the corrective consensus in [17], and the +re-scaling method in [18]. The method proposed in [18] is not +directly applicable in our system because of the time-varying +communication topology. However, it is possible to correct the +bias by using a pre-compensation method as follows. +For a given topology G = (N, E), a set of initial values +{xi(0)}N +i=1 and a scheduling probability vector p: +1) Run N consensus processes with probabilistic schedul- +ing of broadcast transmissions. In the i-th consensus +process, the initial value of every node is zero, except for +node i, whose initial value is one. In this case all nodes +will converge to αi, since �∞ +t=0 W(t)ei = uαT ei = +αiu, where ei is the i-th column of the N × N identity +matrix. +2) Pre-compensate the initial values: for node i, the new +initial value becomes ˜xi(0) = xi(0) +αiN , i = 1, ..., N. +3) Run another consensus algorithm with initial node val- +ues {˜x1(0), ˜x2(0), ..., ˜xN(0)}. This consensus process +will converge to +1 +N +�N +i=1 xi(0). +Importantly, the same seed must be used for the generation +of (pseudo-)random numbers in the consensus mechanisms in +Steps 1 and 3. This can be achieved, for example, by using a +source of shared randomness. Note that the pre-compensation +weights computation (Step 1) before the actual consensus pro- +cess (Step 3) requires extra communication and computation. +Specifically, it will consume JKN transmission slots, where +J is the number of iterations in Step 1. However, this extra +cost is incurred only once, since the pre-compensation weights +can be reused for any number of future consensus processes, +as long as the network topology and the probability vector +remain the same. +C. Possible Extensions +1) Numerical Optimization of Broadcast Probabilities: As +mentioned in Sec. II-B, for a fixed graph, we can optimize the +second largest eigenvalue of the mixing matrix to achieve fast +convergence as shown in [2]. However, following our approach +with partial communication, W(t) is a time-varying random +matrix whose elements are determined by the scheduling +decision in every communication round. It is generally difficult +to analyze the eigenvalues of �∞ +t=0 W(t). Therefore, we focus +on the expectation of W(t) and optimize the probability vector +p by minimizing the second largest eigenvalue of E[W(t)]. +The elements of E[W(t)] are: +E[wij(t)] = +� +� +� +wijpj, +if +j ̸= i. +1 − �N +k=1,k̸=i wikpk, +if +j = i. +(5) +The optimization problem can be defined as: +min +p +ρ(E[W(t)] − uuT /N) +s.t. +N +� +i=1 +pi = K +0 ≤ pi ≤ 1, +i = 1, ..., N +(6) +for a given value of K. +To the best of our knowledge, there is no tractable expres- +sion that can characterize the relation between the second +largest eigenvalue of E[W(t)] and the broadcast probability +vector p. It is possible to apply some derivative-free opti- +mization method, such as Simultaneous Perturbation Stochas- +tic Approximation (SPSA), for optimizing the broadcasting +probabilities [19], [20]. The convergence analysis of SPSA has +been established in the literature with different assumptions on +the objective function. However, due to the unknown structure +of the objective function in (6), the convergence to the optimal +solution in this case is not guaranteed. Another challenge of +using this method for the eigenvalue optimization problem is +that some broadcast probabilities can be zero depending on +the network topology and the given K, which is undesirable. +In practice, this can be handled by imposing a minimum value +of the probabilities. +2) Alternative Bias Correction Methods: Another possible +bias correction method is to apply the corrective consensus +mechanism proposed in [17]. In that mechanism, a new set of +auxiliary variables φij(t) is introduced to represent the amount +of change that node i has made to its state value xi(t), due +to the information received from its neighbor j at iteration t. +Periodically, one corrective iteration takes place, where every +node i transmits φij(t) to the corresponding neighbor j and +computes ∆ij(t) = φij(t) + φji(t). This new set of variables +∆ij represents the bias accumulated in both directions and +it is used to correct the values of xi and φij. Note that +this bias correction method introduces extra communication +overhead. In every corrective iteration, �N +i=1 di transmission +slots are needed for the exchange of φij. Another issue with +this method is that convergence is not guaranteed, even for +a complete graph. (The convergence criteria given in [17] +assume retransmissions). +IV. SIMULATION RESULTS +For simulations, we generate one instance of an Erdös-Rényi +random graph with N = 100 nodes, that is undirected and +connected. The initial values of the nodes are created by using +a normal distribution with zero mean and unit variance. Every +plot is obtained by averaging 10 realizations with the same +graph but different initial values. The mixing matrix W = +I − ϵL is computed for ϵ = +1 +∆(G)+1. +A. Broadcast Probability Vector Design +First, we show the performance of our partial communica- +tion design with different heuristic choices for the broadcast +probabilities. We choose K = 80, which corresponds to 80% +of the nodes scheduled for broadcasting their values in every +communication round. The results are presented in Fig. 1. We + +4 +observe a clear gain in terms of convergence speed (measured +in transmission slots) with our partial communication design, +especially with betweenness-based method. However, the con- +verged value is not equal to the average of the initial values, +which implies a tradeoff between the convergence speed and +the bias in the consensus value. +0 +1 +2 +3 +4 +5 +10 +4 +10 +2 +100 +Standard Deviation +no_scheduling, K = 100 +pagerank, K = 80 +betweenness, K = 80 +degree, K = 80 +0 +1 +2 +3 +4 +5 +Transmission slots (×104) +10 +2 +100 +RMSE +Fig. 1. +Comparison of the standard deviation of the nodes values and its +distance from true average (RMSE) for different heuristic methods. +B. Partial Communication with Bias Correction +To deal with the bias introduced by our partial communica- +tion design, we implement the pre-compensation mechanism +introduced in Sec. III-B. The results are presented in Fig. 2. +We can see that by introducing a pre-compensation step before +the actual consensus process, we can eliminate the bias in +the consensus value while maintaining the advantage of our +proposed design in reducing the communication costs. +0 +2 +4 +6 +8 +10 +12 +Transmission Slots (×104) +10 +11 +10 +7 +10 +3 +RMSE +0 +2 +4 +6 +8 +10 +12 +10 +11 +10 +7 +10 +3 +Standard Deviation +no_scheduling, K = 100 +betweenness pre-comp., K = 80 +betweenness, K = 80 +Fig. 2. Performance with the proposed bias correction mechanism, using the +betweenness-based probability vector design. +C. Potential Improvement by Using SPSA +Finally, we show the performance of our partial communi- +cation design with the scheduling probabilities obtained with +SPSA. In Fig. 3, we compare the SPSA-based method and the +0 +1 +2 +3 +4 +5 +10 +5 +10 +2 +101 +Standard Deviation +no_scheduling, K = 100 +betweenness, K = 80 +SPSA, K = 80 +0 +1 +2 +3 +4 +5 +Transmission slots (×104) +10 +2 +100 +RMSE +Fig. 3. +Comparison of the standard deviation of the nodes values and the +RMSE for no scheduling, betweenness-based and SPSA-based methods. +heuristic design of the probability vector using betweenness +centrality, as detailed in Sec. IV-A. +We observe that with the SPSA-based method, we can +achieve faster convergence as compared to the heuristic design, +while keeping similar RMSE. This result is well supported +by the fact that reducing the second largest eigenvalue of the +expected mixing matrix improves convergence speed. +V. CONCLUSION +We proposed a partial communication design for distributed +average consensus over wireless networks using probabilistic +broadcast scheduling. 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Sadegh, “Constrained optimization via stochastic approximation +with a simultaneous perturbation gradient approximation,” Automatica, +vol. 33, pp. 889–892, 1997. + diff --git a/mNFKT4oBgHgl3EQfEC1Y/content/tmp_files/load_file.txt b/mNFKT4oBgHgl3EQfEC1Y/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0f096e5a566a1bcde64669831e7a77c14ebd8f93 --- /dev/null +++ b/mNFKT4oBgHgl3EQfEC1Y/content/tmp_files/load_file.txt @@ -0,0 +1,385 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf,len=384 +page_content='1 Distributed Consensus in Wireless Networks with Probabilistic Broadcast Scheduling Daniel Pérez Herrera, Zheng Chen and Erik G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Larsson Abstract—We consider distributed average consensus in a wire- less network with partial communication to reduce the number of transmissions in every iteration/round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Considering the broadcast nature of wireless channels, we propose a probabilistic approach that schedules a subset of nodes for broadcasting information to their neighbors in every round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' We compare several heuristic methods for assigning the node broadcast probabilities under a fixed number of transmissions per round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Furthermore, we introduce a pre-compensation method to correct the bias between the consensus value and the average of the initial values, and suggest possible extensions for our design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Our results are partic- ularly relevant for developing communication-efficient consensus protocols in a wireless environment with limited frequency/time resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Index Terms—Average consensus, broadcast transmission, scheduling, wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' INTRODUCTION Distributed consensus deals with the problem of reaching an agreement among networked agents on a certain quantity of interest that depends on the initial states of all agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' The interaction rule that specifies the information exchange between an agent (node) and its neighbors is known as consensus protocol or algorithm [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Traditional studies of consensus problems usually assume a fixed topology with perfect communication between the nodes in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' For this ideal case, the convergence conditions and convergence speed of discrete-time average consensus are provided in [2] and [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' As an extension to the main theme, different methods can be used to accelerate convergence, such as extrapola- tion [4] and stepsize adjustment [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' For varying topologies, necessary and sufficient conditions to reach consensus over random networks are presented in [6], using ergodicity and probabilistic arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Conventional works on consensus algorithms tend to rely on idealized assumptions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=', instantaneous, error-free) on the communication links between connected nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Consensus algorithms may suffer from imperfect communication between the nodes due to, for example, link failures [7], [8], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' To deal with the missing information, two compensation (biased and balanced) methods are presented in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' The main difference between these methods is how the weights of the failed links are re-distributed among the successful links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Both algorithms converge almost surely and in the mean-square sense to a value that generally is not equal to the average of the initial values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' The authors are with the Department of Electrical Engineering, Linköping University, Linköping, SE-58183 Sweden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' E-mail: {daniel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content='perez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content='herrera, zheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content='chen, erik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content='larsson}@liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content='se.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' This work was supported by Zenith, ELLIIT, and the KAW foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' In addition to the conventional consensus protocols where every node communicates with all their neighbors in every iter- ation, gossip algorithms can also be used to achieve consensus with randomized and asynchronous communication [11], [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' A key feature of these algorithms is the asynchronous time model, where information is exchanged between a subset of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' This information is processed by the receiving nodes to compute local update via a linear combination of its own value and the received one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' There are different variants of gossip algorithms, such as geographic gossip [13], randomized gossip [14] and broadcast gossip [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Particularly, broadcast gossip is more relevant to the implementation of gossip algorithms in wireless networks, where the transmission of one node can reach any other nodes within its communication range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Unlike the case with broadcast gossip, where the nodes update their states immediately after receiving information from another node, we propose a probabilistic broadcast-based method to schedule a subset of nodes for broadcasting in every communication round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' The nodes update their values after receiving information from all the scheduled broadcasting nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Since only a subset of nodes is selected for transmission in every round, the number of transmission slots needed in every round is reduced as compared to the full communication case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' To compensate for the lack of information with our par- tial communication approach, we use a biased compensation method similar to the one proposed in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Our objective is to achieve fast convergence with fewer transmission slots while maintaining small deviation from the average of the initial values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' We investigate several heuristic methods for assigning the broadcast probabilities of the nodes, combined with a pre-compensation mechanism to eliminate the bias caused by unbalanced information exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' SYSTEM MODEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Graph Model We consider a wireless network modeled by an undirected and connected graph G = (N, E), where N is the set of nodes and E is the set of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' The total number of nodes is N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Each edge {i, j} ∈ E indicates the existence of a link between the pair of nodes i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' The set of neighbors of node i is Ni = {j|{i, j} ∈ E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content='1 The connectivity of the graph can be represented by its adjacency matrix A, with elements aij = 1 if {i, j} ∈ E, 1In a wireless network, the notion of connectivity is not strictly defined since every node can hear the transmission of other nodes in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' We consider a simplified model where a link between two nodes does not exist when the distance is larger than some predefined threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content='11714v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content='MA] 27 Jan 2023 2 and aij = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Another matrix representation of the graph is the Laplacian matrix, defined as L = D − A, where D is a diagonal matrix whose i-th entry is di = |Ni|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Consensus Algorithm We consider the problem of distributed average consensus, where the goal is for all nodes in the network to reach consensus on the arithmetic mean of their initial values, by only communicating with their neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' To accomplish this, we use the following distributed linear iteration x(t + 1) = Wx(t), (1) where x(t) = [x1(t), x2(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=', xN(t)]T contains the values of every node in iteration t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' W is the mixing matrix, defined as W = I − ϵL, where I is the identity matrix and ϵ ∈ R is the step size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Let ∆(G) be the maximum degree of the graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' When 0 < ϵ < 1/∆(G), the following are necessary and sufficient conditions for x(t) to converge to the average of the initial values [2]: 1) Wu = u, 2) uT W = uT , 3) ρ(W − uuT /N) < 1, where ρ(·) denotes the spectral radius of a matrix and u the all 1’s column vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Note that since W is symmetric, conditions 1 and 2 are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Condition 3 is also related to the convergence speed, since minimizing the second largest eigenvalue of W can speed up convergence of distributed averaging algorithms [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' PARTIAL COMMUNICATION WITH PROBABILISTIC BROADCAST SCHEDULING Distributed consensus algorithms rely on an iterative process where in each iteration, every node must communicate with all its neighbors before the updating step in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' The communi- cation design of consensus algorithms in wireless networks is non-trivial due to the “many-to-many” communication topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' If all nodes transmit information simultaneously using the same frequency-time resources, then all signals will be garbled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' One simple way to avoid interference between concurrent transmissions is to assign orthogonal resources to each transmitting node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' For instance, each node occupies one dedicated time slot for broadcasting its information to the neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' In this way, it takes N times slots in total for the entire network to complete information fusion in one consensus iteration with full communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' We refer to one communication round as one iteration interval during which all nodes exchange information with the neighbors and update their state values, and one transmission slot as the time interval during which one node broadcasts its value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' With full communication over the network, after J iterations of the consensus algorithm, the total number of required transmission slots is NJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' To reduce the communication costs (time slots), we consider a partial communication design where in every round we only select a subset of nodes for broadcasting their information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Let p = [p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=', pN]T be a vector that contains the broadcast probabilities of the nodes with �N i=1 pi = K and K < N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' This means that in every communication round, K nodes are selected on average for broadcasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Therefore, the average number of transmission slots for completing J iterations is KJ < NJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Since in every iteration, some nodes are not scheduled for broadcasting their values, we compensate the missing infor- mation by using the biased compensation method proposed in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' The idea is to use a new mixing matrix W(t) in every round t, whose elements are given by wij(t) = � � � wijvj(t), if j ̸= i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' 1 − �N k=1,k̸=i wikvk(t), if j = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' (2) Here, vj(t) = 1 if node j is selected for broadcasting in the t-th communication round, and vj(t) = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' With our probabilistic scheduling approach, we have pi = E[vi(t)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Though the biased compensation method ensures convergence in mean square, in general the consensus value is not equal to the average of the initial values, since the new mixing matrix is only row stochastic but not necessarily doubly stochastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Heuristic Designs for Broadcast Probability Vector Intuitively, the broadcast probabilities should be determined in a way that can reflect the “importance” of the nodes in a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' As a first approach, we consider different methods of creating the broadcast probability vector based on commonly used metrics for network connectivity, such as degree, PageRank and betweenness centrality [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' 1) Degree-based method: Degree centrality of a node is the number of neighbors of the node: the degree of node j is dj = |Nj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' We can choose the broadcast probability of node i as pi = min{1, γdi} (3) where the constant γ is chosen such that �N j=1 pj = K, for a given value of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' 2) PageRank-based method: PageRank was first proposed to rank webpages using the hyperlink network structure of the web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' It can be used in any network as a measure of the node importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' The PageRank-based method gives us a probability vector p in the same way, but using the PageRank centrality of the nodes instead of their degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' 3) Betweenness-based method: Betweenness centrality of a node is defined as the fraction of shortest paths between pairs of other nodes that pass through it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Let bi represent the betweenness centrality of node i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Since bi is zero if node i has degree one, a small positive value is added to this metric, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=', the new score ˜bi of node i is: ˜bi = bi + β, (4) where β is a small positive number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' The probability vector p is then created in the same way as for the degree-based method, but using this new score instead of the node degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' 3 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Bias Correction With our probabilistic scheduling of broadcasting nodes in every round, the matrix W(t) is row stochastic with positive diagonals (as long as pi > 0, ∀i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' As a consequence of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content='1 in [10], the product �J t=0 W(t) converges almost surely, when J → ∞, to a random rank-one matrix, uαT , for some vector α = [α1, α2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=', αN]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' In general, a bias will be introduced, meaning that the consensus value will not be the average of the initial values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' There are several existing bias correction approaches in the literature of distributed consensus, such as the corrective consensus in [17], and the re-scaling method in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' The method proposed in [18] is not directly applicable in our system because of the time-varying communication topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' However, it is possible to correct the bias by using a pre-compensation method as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' For a given topology G = (N, E), a set of initial values {xi(0)}N i=1 and a scheduling probability vector p: 1) Run N consensus processes with probabilistic schedul- ing of broadcast transmissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' In the i-th consensus process, the initial value of every node is zero, except for node i, whose initial value is one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' In this case all nodes will converge to αi, since �∞ t=0 W(t)ei = uαT ei = αiu, where ei is the i-th column of the N × N identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' 2) Pre-compensate the initial values: for node i, the new initial value becomes ˜xi(0) = xi(0) αiN , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' 3) Run another consensus algorithm with initial node val- ues {˜x1(0), ˜x2(0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=', ˜xN(0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' This consensus process will converge to 1 N �N i=1 xi(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Importantly, the same seed must be used for the generation of (pseudo-)random numbers in the consensus mechanisms in Steps 1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' This can be achieved, for example, by using a source of shared randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Note that the pre-compensation weights computation (Step 1) before the actual consensus pro- cess (Step 3) requires extra communication and computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Specifically, it will consume JKN transmission slots, where J is the number of iterations in Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' However, this extra cost is incurred only once, since the pre-compensation weights can be reused for any number of future consensus processes, as long as the network topology and the probability vector remain the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Possible Extensions 1) Numerical Optimization of Broadcast Probabilities: As mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' II-B, for a fixed graph, we can optimize the second largest eigenvalue of the mixing matrix to achieve fast convergence as shown in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' However, following our approach with partial communication, W(t) is a time-varying random matrix whose elements are determined by the scheduling decision in every communication round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' It is generally difficult to analyze the eigenvalues of �∞ t=0 W(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Therefore, we focus on the expectation of W(t) and optimize the probability vector p by minimizing the second largest eigenvalue of E[W(t)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' The elements of E[W(t)] are: E[wij(t)] = � � � wijpj, if j ̸= i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' 1 − �N k=1,k̸=i wikpk, if j = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' (5) The optimization problem can be defined as: min p ρ(E[W(t)] − uuT /N) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' N � i=1 pi = K 0 ≤ pi ≤ 1, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=', N (6) for a given value of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' To the best of our knowledge, there is no tractable expres- sion that can characterize the relation between the second largest eigenvalue of E[W(t)] and the broadcast probability vector p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' It is possible to apply some derivative-free opti- mization method, such as Simultaneous Perturbation Stochas- tic Approximation (SPSA), for optimizing the broadcasting probabilities [19], [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' The convergence analysis of SPSA has been established in the literature with different assumptions on the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' However, due to the unknown structure of the objective function in (6), the convergence to the optimal solution in this case is not guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Another challenge of using this method for the eigenvalue optimization problem is that some broadcast probabilities can be zero depending on the network topology and the given K, which is undesirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' In practice, this can be handled by imposing a minimum value of the probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' 2) Alternative Bias Correction Methods: Another possible bias correction method is to apply the corrective consensus mechanism proposed in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' In that mechanism, a new set of auxiliary variables φij(t) is introduced to represent the amount of change that node i has made to its state value xi(t), due to the information received from its neighbor j at iteration t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Periodically, one corrective iteration takes place, where every node i transmits φij(t) to the corresponding neighbor j and computes ∆ij(t) = φij(t) + φji(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' This new set of variables ∆ij represents the bias accumulated in both directions and it is used to correct the values of xi and φij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Note that this bias correction method introduces extra communication overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' In every corrective iteration, �N i=1 di transmission slots are needed for the exchange of φij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Another issue with this method is that convergence is not guaranteed, even for a complete graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' (The convergence criteria given in [17] assume retransmissions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' SIMULATION RESULTS For simulations, we generate one instance of an Erdös-Rényi random graph with N = 100 nodes, that is undirected and connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' The initial values of the nodes are created by using a normal distribution with zero mean and unit variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Every plot is obtained by averaging 10 realizations with the same graph but different initial values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' The mixing matrix W = I − ϵL is computed for ϵ = 1 ∆(G)+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Broadcast Probability Vector Design First, we show the performance of our partial communica- tion design with different heuristic choices for the broadcast probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' We choose K = 80, which corresponds to 80% of the nodes scheduled for broadcasting their values in every communication round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' The results are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' We 4 observe a clear gain in terms of convergence speed (measured in transmission slots) with our partial communication design, especially with betweenness-based method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' However, the con- verged value is not equal to the average of the initial values, which implies a tradeoff between the convergence speed and the bias in the consensus value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' 0 1 2 3 4 5 10 4 10 2 100 Standard Deviation no_scheduling, K = 100 pagerank, K = 80 betweenness, K = 80 degree, K = 80 0 1 2 3 4 5 Transmission slots (×104) 10 2 100 RMSE Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Comparison of the standard deviation of the nodes values and its distance from true average (RMSE) for different heuristic methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Partial Communication with Bias Correction To deal with the bias introduced by our partial communica- tion design, we implement the pre-compensation mechanism introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' III-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' The results are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' We can see that by introducing a pre-compensation step before the actual consensus process, we can eliminate the bias in the consensus value while maintaining the advantage of our proposed design in reducing the communication costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' 0 2 4 6 8 10 12 Transmission Slots (×104) 10 11 10 7 10 3 RMSE 0 2 4 6 8 10 12 10 11 10 7 10 3 Standard Deviation no_scheduling, K = 100 betweenness pre-comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=', K = 80 betweenness, K = 80 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Performance with the proposed bias correction mechanism, using the betweenness-based probability vector design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Potential Improvement by Using SPSA Finally, we show the performance of our partial communi- cation design with the scheduling probabilities obtained with SPSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' 3, we compare the SPSA-based method and the 0 1 2 3 4 5 10 5 10 2 101 Standard Deviation no_scheduling, K = 100 betweenness, K = 80 SPSA, K = 80 0 1 2 3 4 5 Transmission slots (×104) 10 2 100 RMSE Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Comparison of the standard deviation of the nodes values and the RMSE for no scheduling, betweenness-based and SPSA-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' heuristic design of the probability vector using betweenness centrality, as detailed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' IV-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' We observe that with the SPSA-based method, we can achieve faster convergence as compared to the heuristic design, while keeping similar RMSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' This result is well supported by the fact that reducing the second largest eigenvalue of the expected mixing matrix improves convergence speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' CONCLUSION We proposed a partial communication design for distributed average consensus over wireless networks using probabilistic broadcast scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' A trade-off between convergence speed in terms of transmission slots and the bias in the consen- sus value was observed from simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' Several heuristic methods for assigning the node broadcast probabilities were proposed, as well as a pre-compensation mechanism for bias correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' We concluded that distributed consensus algorithms in wireless networks can benefit from partial communication in achieving consensus with reduced communication costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' As future work, an optimal selection of the number of broadcast- ing nodes per round could be investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFKT4oBgHgl3EQfEC1Y/content/2301.11714v1.pdf'} +page_content=' The possibility of 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Thakur b,∗, Chandra Prakash b, T.C. Goel a +a Department of Physics, Indian Institute of Technology, New Delhi-110 016, India +b Solid State Physics Laboratory, Lucknow Road, Timarpur, Delhi-110 054, India +Received 20 March 2003; received in revised form 20 May 2003; accepted 3 July 2003 +Abstract +Solid solution of 32 mol% of Lead Titanate in PMN–PT system has been prepared by columbite precursor method. Room temperature +X-ray diffraction study reveals the formation of perovskite phase with tetragonal structure. Dielectric measurements have been carried out +at different frequencies (0.1 kHz–1 MHz) as a function of temperature (RT to 235 ◦C). The phase transition was found to be of diffused +type. The polarization studies show ferroelectric nature of the material with a high value of remnant polarization (Pr) ∼21 �C/cm2 and +spontaneous polarization (Ps) ∼29 �C/cm2. Strain versus electric field (S–E) behavior shows piezoelectric nature of the material with high +value of maximum strain 0.14% at 60 kV/cm. Pyroelectric coefficient at room temperature has been found to be 3 × 10−2 �C/cm2 K. +© 2003 Elsevier Ltd and Techna Group S.r.l. All rights reserved. +Keywords: C. Dielectric properties; D. Perovskite; Diffuse phase transition; Relaxor +1. Introduction +Lead Magnesium Niobate (PMN), a prototype relaxor fer- +roelectric, demonstrates diffuse phase transition phenomena +and around 10 ◦C a quite high dielectric constant (∼=20,000) +[1]. The high value of dielectric constant, good voltage sta- +bility, excellent electrostrictive effects and lower sintering +temperature of PMN make it important for multilayer capac- +itors, actuators and electro-optic device applications [2,3]. +With PbTiO3 (PT), PMN forms a binary solution, (1 − +x)PMN–xPT, and the transition temperature of the system +increases with the increase in mol% of PT (Lead Titanate) +in the system [3]. 0.9PMN–0.1PT is a pronounced candi- +date to replace Barium Titanate (BT) in multilayer ceramic +capacitors because of its low sintering temperature as well +as higher dielectric constant at room temperature than that +of BT. This system also has a morphotropic phase boundary +(MPB) between 0.70PMN–0.30PT and 0.65PMN–0.35PT +compositions. The compositions near MPB of PMN–PT sys- +tem exhibit excellent piezoelectric properties, thus making +the material important for actuator and sensor applications +∗ Corresponding author. +E-mail address: omprakasht@hotmail.com (O.P. Thakur). +[2]. It has also been reported that compositions near MPB +of this system are showing good pyroelectric behavior [4] +but not much has been reported on both the pyro- and piezo- +electric properties of the same composition [5]. +Here, the PMN–PT (68:32) composition is prepared by +columbite precursor method and a systematic study of di- +electric, piezoelectric, pyroelectric and electromechanical +properties of PMN–PT (68:32) composition has been pre- +sented. +2. Experimental +Polycrystalline samples of PMN–PT (68:32) were pre- +pared by high temperature solid-state reaction via columbite +technique [6]. The starting materials were of oxide chem- +icals with purity better than 99% (all Aldrich). MgO and +Nb2O5 powders were ball milled in distilled water for 8 h +using ZrO2 balls as grinding medium. The slurry was dried +by heating at 80 ◦C on a hot plate with continuous stirring. +The dried powder was crushed in an agate mortar and cal- +cined in platinum crucible at 1200 ◦C for 4 h. This columbite +(MgNb2O6) was then crushed and sieved and the powder +was again recalcined for 4 h at 950 ◦C. TiO2 and 4 wt.% +excess PbO were mixed with columbite in stoichiometric +0272-8842/$30.00 © 2003 Elsevier Ltd and Techna Group S.r.l. All rights reserved. +doi:10.1016/j.ceramint.2003.07.003 + +ET SEVTERCERAMICS +INTERNATIONAL +ar ron/lnratel +eramir586 +P. Kumar et al. / Ceramics International 30 (2004) 585–589 +ratio and ball milled in distilled water using ZrO2 balls. The +final calcination was done at 700 ◦C for 4 h. This calcined +powder was again crushed, sieved and mixed with 4 wt.% +of polyvinyl alcohol (PVA) binder to impart the mechani- +cal strength to the green pellets. Pellets of 12 mm diameter +and 1.25 mm thickness were pressed with an applied pres- +sure of 10 t using uniaxial press. Binder was removed from +the green pellets by slowly heating the pellets up to 600 ◦C +and then holding for 1 h. The pellets were then sintered at +1250 ◦C for 4 h [7]. The electrical contacts were made by +coating silver paint on the flat surfaces of the sintered and +ground pellets. +X-ray diffraction (XRD) of the pellets was performed on +PW 3020 Philips type diffractometer using Cu K� radiation. +Dielectric constant (εr) and dielectric loss (tan δ) were calcu- +lated from the capacitance measurements using 4284A HP +LCR meter at different frequencies (0.1 kHz–1 MHz) as a +function of temperature (RT to 235 ◦C). Using Sawyer Tower +circuit, hysteresis (P–E) loop was taken with computer in- +terfaced Loop Tracer. The piezoelectric and electromechan- +ical parameters were measured in accordance with the 1961 +IRE standards on piezoelectric crystals [8]. Strain versus +electric field (S–E) behavior was taken using SS-50 Strain +Measurement System (Sensor Tec. Ltd., Canada). The sam- +ples were poled under corona discharge by applying electric +field of 6 kV for 0.5 h. The pyroelectric coefficient was de- +termined by measuring pyroelectric current (I) using Keith- +ley electrometer at a heating rate of 4 ◦C/min. The pyroelec- +tric coefficient (pi) was determined using the relation: pi = +(I/A)(dT/dt)−1, where I is the pyroelectric current (mea- +sured after repetition of three cycles), A is the electroded +area and dT/dt is the heating rate. +3. Results and discussion +Fig. 1 shows the room temperature XRD pattern of the +sintered sample. The diffraction pattern shows the intense +lines of perovskite phase. These diffraction lines were in- +dexed in different crystal systems and unit cell configura- +tions using a computer program package ‘Powdmult’. Out +of these a suitable tetragonal unit cell was selected for which +� �d(= dobs −dcal), where ‘d’ is inter-planer spacing, was +found to be minimum. The lattice parameters of the unit cell +were refined using least square fit method. The lattice pa- +rameters ‘a’ and ‘c’ are 3.9882 and 4.0549 Å, respectively, +with c/a, 1.016, and are in agreement with earlier reports +[9]. By using the relation [6] +Pyrochlore % = +� +Ipyro +(Iperov + Ipyro) +� +× 100, +pyrochlore phase was found to be ∼15%. +SEM photograph is illustrated in Fig. 2. It is found that +some grains are well developed up to the size of about 2 �m +(calculated by linear intercept method). A number of grains +Fig. 1. XRD of the sintered 0.68PMN–0.32PT. +and their grain boundaries are visible, along with a few pores +situated upon the grain boundaries. +Fig. 3 shows the variation of dielectric constant (εr) with +temperature at different frequencies (0.1 kHz–1 MHz). There +is a gradual increase in dielectric constant up to 150 ◦C and +this can be attributed to the dominance of interfacial po- +larization over dipolar polarization [10]. Strong dielectric +dispersion without any relaxor behavior has been observed +around 190 ◦C. The strong dispersion in dielectric constant +Fig. 2. SEM of the sintered 0.68PMN–0.32PT. + +INTENSITY +(arb. units +0(100) +30 +0(110) +0 (111) +40 +20 +(deg.) +0(002) +(200) +0 (201) +(210) +0(211) +0 (220) +Pyrochlore +Perovskite +->0(300)L- SE1 +EHT-20.0XV +MD-12 +PHOT0-18101 +10.0mP. Kumar et al. / Ceramics International 30 (2004) 585–589 +587 +50 +100 +150 +200 +250 +2500 +5000 +7500 +10000 +12500 +15000 +17500 +(5) +(4) +(3) +(2) +(1) +(1) 100Hz +(2) 1kHz +(3) 10kHz +(4) 100kHz +(5) 1MHz +εr +TEMPERATURE ( +oC) +Fig. 3. Temperature dependence of dielectric constant (εr) with tempera- +ture at different frequencies. +near phase transition temperature (Curie temperature) can be +due to a Debye-type hopping of defects or impurities over a +distribution of barriers. This can also be accounted due to the +possibility of marginally softening of lattice mode, which +on softening interacts increasingly with the defects and pro- +duce a frequency dependent relaxation [11]. Absence of re- +laxor behavior may be due to the large mol% of PT in the +system. +Fig. 4 shows the temperature variation of tan δ at different +frequencies (0.1 kHz–1 MHz). The temperatures of peak di- +electric loss and peak dielectric constant do not coincide up +to 100 kHz frequencies. Kramers–Kronig relation indicates +that this can be the consequence of temperature dependent +relaxation near Curie temperature [11]. +The nature of phase transition is ascertained by calculating +the degree of diffusion (γ). Fitting εr at T > Tmax in formula +[12] +ε−1 +r (T) = +1 +εr max. ++(T − Tmax)γ, +a graph between log(1/εr −1/εr max.) versus log(T −Tmax), +shown in Fig. 5, is plotted. From the slope of the graph value +50 +100 +150 +200 +250 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +(3) +(1) +(2) +(4) +(1) 100Hz +(2) 1kHz +(3) 10kHz +(4) 100kHz +(5) 1MHz +tanδ +TEMPERATURE (oC) +(5) +Fig. 4. Temperature dependence of dissipation factor (tan δ) with temper- +ature at different frequencies. +1.0 +1.1 +1.2 +1.3 +1.4 +1.5 +1.6 +1.7 +1.8 +-5.4 +-5.2 +-5.0 +-4.8 +-4.6 +-4.4 +-4.2 +-4.0 +-3.8 +LOG (1/εr-1/ εrmax.) +LOG (T-Tmax.) +Fig. 5. Variation of log(1/εr − 1/εr max.) vs. log(T − Tmax). +of ‘γ’ is calculated. Value of γ is found to be ∼2, which +suggests that the phase transition is of diffused type [12]. +The diffuse phase transition in the material may be due to the +structural disorder and compositional fluctuations in solid +solution. The value of εr and tan δ at room temperature and +transition temperature at 1 kHz are 2000, 0.025 and 16,500, +0.040, respectively. +Fig. 6 shows the polarization versus electric field (P–E) +behavior of 0.68PMN–0.32PT sample. This figure shows +that the material has good ferroelectric nature with Ps ∼ +29 �C/cm2 and Pr ∼ 21 �C/cm2, which are quite high as +compared to previous reported values [13]. At the maximum +applied field of 19 kV/cm, coercive field, Ec, is found to be +∼8.77 kV/cm. The high value of Ec as compared to earlier +reported ones [14] can be due to smaller grain size of the +material [15]. +The piezoelectric charge coefficient (d33), and the planar +electromechanical coupling coefficient (kp) are found to be +325 pC/N and 40%, respectively. These values are found to +be similar with the earlier reports (d33 ∼ 410 pC/N and kp ∼ +30%) [5]. Piezoelectric voltage constant (g33), calculated +using the relation g33 = d33/ε0εr, is found to be 7 Vm/N. +-25 +-20 +-15 +-10 +-5 +0 +5 +10 +15 +20 +25 +-30 +-20 +-10 +0 +10 +20 +30 +P(µC/cm +2) +E(kV/cm) +Fig. 6. Variation of polarization vs. electric field. + +588 +P. Kumar et al. / Ceramics International 30 (2004) 585–589 +-6 +-4 +-2 +0 +2 +4 +6 +-0.10 +-0.05 +0.00 +0.05 +0.10 +0.15 +STRAIN (%) +ELECTRIC FIELD (10kV/cm) +Fig. 7. Variation of strain with biaxial electric field. +The hysteresis plots between strain and electric field (S–E) +at room temperature with biaxial and uniaxial fields are +illustrated in Figs. 7 and 8. A typical butterfly loop, which +is a feature of piezoelectric system is observed for biaxial +field. A maximum strain of 0.14 and 0.18% is observed +for biaxial and uniaxial fields, respectively. The saturation +of strain in both cases was found around 60 kV/cm. The +hysteresis observed is due to the polarization reorienta- +tion [16] and confirms the piezoelectric nature. However, +pure PMN, being electrostrictive, does not exhibit strain +hysteresis. +The temperature variation of the pyroelectric coefficient, +pi, is shown in Fig. 9. The room temperature value of py- +roelectric coefficient is 3 × 10−2 �C/cm2 K and it increases +with the rise in temperature. The peak value of ‘pi’ is 65 × +10−2 �C/cm2 K which is slightly less than the earlier re- +ports (∼100 × 10−2 �C/cm2 K) [5]. The peak position is +found to be same in both pyroelectric coefficient and dielec- +tric behavior plots taken with the variation of temperature, +that confirms the phase transition is around 190 ◦C. Pyro- +-6 +-5 +-4 +-3 +-2 +-1 +0 +-0.05 +0.00 +0.05 +0.10 +0.15 +0.20 +STRAIN (%) +ELECTRIC FIELD (10kV/cm) +Fig. 8. Variation of strain with uniaxial electric field. +0 +50 +100 +150 +200 +250 +0 +10 +20 +30 +40 +50 +60 +70 +pi (10 +-2µC/cm +2K) +TEMPERATURE ( +oC) +Fig. 9. Variation of pyroelectric coefficient with temperature. +Table 1 +Pyroelectric figures of merits of the PMN–PT (68:32) composition at RT +Pyroelectric coefficient, pi (�C/cm2 K) +3 × 10−2 +Voltage figure of merit, Fv (V/cm2 J) +69.60 +Current figure of merit, Fi (nA cm/W) +12.32 +Materials figure of merit, FD (cm3/J)1/2 +0.0059 +electric figures of merit was calculated using specific heat +2.5 J/cm3 K and is listed in Table 1. +4. Summary +PMN–PT (68:32) system reveals perovskite phase with +tetragonal structure. It exhibits good ferroelectric character- +istics with a diffuse phase transition behavior. Addition of +PT in PMN near the vicinity of MPB makes the material +piezoelectric, which can be used for high power applica- +tions. High value of strain of this material can be exploited +in actuator applications while pyroelectric properties can be +used for bolometer applications. +References +[1] L.E. Cross, Ferroelectrics 76 (1987) 241. +[2] T.R. Shrout, J. Fielding Jr., Ultrason. Symp. (1990) 711. +[3] G.H. Haertling, J. Am. Ceram. Soc. 82 (1999) 797. +[4] J.-H. Park, B.-K. Kim, K.-H. Song, S.J. Park, Mater. Res. Bull. 30 +(1995) 435. +[5] G.B. Kim, S.W. Choi, Jpn. J. Appl. Phys. 39 (2000) 5552. +[6] S.L. Swartz, T.R. Shrout, Mater. Res. Bull. 17 (1982) 1245. +[7] Y.H. Chen, S. Hirose, D. Viehland, S. Takahashi, K. Uchino, Jpn. J. +Appl. Phys. 39 (2000) 4443. +[8] B. Jaffe, W. Cook, H. Jaffe, Piezoelectric Ceramics, Academic Press, +London, 1971. +[9] D.H. Suh, D.H. Lee, N.K. Kim, J. Eur. Ceram. Soc. 22 (2000) 219. +[10] L.L. Hench, J.K. West, Principles of Electronic Ceramics, Wiley, +New York, 1990. + +P. Kumar et al. / Ceramics International 30 (2004) 585–589 +589 +[11] M.E. Lines, A.M. Glass, Principles and Applications of Ferroelectrics +and Related Materials, Clarendon Press, Oxford, 1977. +[12] M. Kuwabara, S. Takahashi, K. Goda, K. Watande, Jpn. J. Appl. +Phys. 31 (1992) 3241. +[13] L.B. Kong, J. Ma, W. Zhu, O.K. Tan, Mater. Res. Bull. 37 (2002) +459. +[14] L.B. Kong, J. Ma, W. Zhu, O.K. Tan, J. Alloys Compd. 336 (2002) +242. +[15] C. Sakaki, B.L. Newalkar, S. Komarneni, K. Uchino, Jpn. J. Appl. +Phys. 40 (2001) 6907. +[16] K. Uchino, Ferroelectric Devices, Marcel Dekker, New York, 2000. + diff --git a/materials/content/tmp_files/1-s2.0-S0921452621006499-main.pdf.txt b/materials/content/tmp_files/1-s2.0-S0921452621006499-main.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..98c9fbea82e2dce31bc014244b3f79da959587fc --- /dev/null +++ b/materials/content/tmp_files/1-s2.0-S0921452621006499-main.pdf.txt @@ -0,0 +1,662 @@ +Physica B 625 (2022) 413490 +Available online 22 October 2021 +0921-4526/© 2021 Elsevier B.V. All rights reserved. +Electrostatic switch of magnetic anisotropy in the composite with +iron-metal granule and barium-titanate matrix +Bo Chen *, Zi-Run Li, Wen-Li Cui +Department of Physics, College of Science, North University of China, TaiYuan, China +A R T I C L E I N F O +Keywords: +Iron metal +Barium titanate +Magnetic anisotropy +Electric polarization +A B S T R A C T +This work investigates electrostatic switch of magnetic anisotropy in the composite with iron-metal granule and +barium-titanate matrix theoretically. As the spontaneous polarization is switched downward or upward in +barium-titanate matrix, the parallel or perpendicular magnetizing is preferred in iron-metal granule respectively. +This demagnetizing anisotropy originates from the long-range electrostatic interaction in matrix body, which +induces the inharmonic magnetic moment on granule surface. The effects of position bias and composite size are +delicately studied. The cross-section map displays distorted electric polarization in barium-titanate matrix, and +demagnetizing energy gradient in iron-metal granule. This work predicts the electrostatic switch of magnetic +anisotropy in metal/oxide composite, which would develop the integrated device and functional material. +1. Introduction +In magnetic granule, the magnetic anisotropy in an important and +notable property. The magnetic anisotropy is also called magneto- +crystalline and demagnetizing anisotropy. The magnetocrystalline +anisotropy originates from the spin-orbit-lattice interaction [1–3]. As +the magnetic moment is along the easy and hard orientation, the mag- +netocrystalline energy reaches minimum and maximum respectively. In +magnetic storage device, strong magnetocrystalline anisotropy is needed +to stabilize magnetic bit [4]. On the other hand, the demagnetizing +anisotropy is determined by demagnetizing field and magnetic pole +[5–7]. As granule is magnetized, the demagnetizing energy also reaches +minimum and maximum along the easy and hard orientation respec- +tively. In magnetic function material, the magnetic topological state is +altered by demagnetizing anisotropy remarkably [8]. +Both the magnetocrystalline and demagnetizing anisotropy are per- +manent in magnetic granule. To develop the integrated device and +functional material, the switch of magnetic anisotropy is needed. In +magnetic storage device, the magnetic moment is usually manipulated +by magnetic field, while the electrostatic manipulation of magnetic +moment would be much faster [9,10]. In magnetic function material, the +magnetic topological state is usually altered by electric current, while +the electrostatic alteration of magnetic topological state would consume +lower energy [11,12]. All these purpose can be achieved by the elec- +trostatic switch of magnetic anisotropy. +In the iron-metal/lead-zirconate-titanate composite, author’s previ- +ous work has studied the electrostatic switch of magnetic anisotropy +theoretically. As the electric polarization is switched upward or down- +ward by electrostatic field, the perpendicular or parallel magnetic +anisotropy is induced respectively [13,14]. As a typical dielectric ma- +terial, the spontaneous polarization and coercive field of lead zirconate +titanate (PbZrxTi1-xO3 i.e. PZT) is remarkable [15]. Thus, such electro- +static switch of magnetic anisotropy is reversible and nonvolatile. +However, the lead component of PZT is a pollution source. Moreover, +the fatigue behavior of PZT film is considerable, which would reduce the +device lifetime seriously [16,17]. +Compared to PZT, the barium titanate (BaTiO3 i.e. BTO) is a lead-free +dielectric material. Furthermore, the fatigue behavior of BTO is lighter +than that of PZT [18,19]. This work designs the composite of iron metal +(FM) granule and BTO matrix, then studies the coupling between mag- +netic anisotropy and electric polarization theoretically. Both the char- +acteristic of single granule, and the interaction of collective granules, +contribute the electrostatic switch of magnetic anisotropy in FM/BTO +composite. +2. Theory and calculation +The composite structure is illustrated in Fig. 1. The FM granule is +dispersed in BTO matrix, and the FM granule is formed by Fe, Co or Ni +metal. The top and bottom electrode is Au layer and SrRuO3 (SRO) +* Corresponding author. +E-mail address: BoChen@nuc.edu.cn (B. Chen). +Contents lists available at ScienceDirect +Physica B: Physics of Condensed Matter +journal homepage: www.elsevier.com/locate/physb +https://doi.org/10.1016/j.physb.2021.413490 +Received 14 August 2021; Received in revised form 11 October 2021; Accepted 18 October 2021 + +S5N 0921-606 +PHYSICA +B +CONDENSED MATTER +SdenceDirectEISEVTERPhysica B: Physics of Condensed Matter 625 (2022) 413490 +2 +substrate respectively. Half of the granule diameter and matrix thickness +is defined as R and t respectively. At room temperature, the BTO crys- +talline is tetragonal, and the spontaneous polarization is along (001) axis +[20]. As seen in Table 1, the (100)SRO and (100)BTO lattice constant is +close to each other, which prefers the (001) crystallizing of BTO matrix +on SRO substrate. Thus, the spontaneous polarization of BTO matrix is +along the out-of-plane orientation. +Near granule/matrix interface, the screening electron is accumulated +or depleted on FM granule surface, and the electric polarization is dis- +torted in BTO matrix. For isolated granule, the screening electron den- +sity is calculated by Thomas–Fermi model and spherical boundary +condition [13,14]. Considering the interaction of adjacent granules, the +screening electron density is rectified by perturbation method and mean +field approximation [13,14]. Then, the relation between electric polar- +ization and screening electron is set up by dielectric response model. +Finally, the distorted electric polarization in BTO matrix is displayed by +vector map. The detail is shown in author’s previous work [13,14]. +On FM granule surface, the screening electron is spin-split. Thus, +except the intrinsic magnetization, inductive magnetization is intro- +duced by screening electron [13,14]. Based on Table 2, the spin-split +rate of FM granule is calculated by Refs. [13,14]: +χ = +ρ↓ − ρ↑ +ρ↓ + ρ↑ + 4Jexρ↓ρ↑ +(1) +Correspondingly, the inductive magnetization is deduced through +magnetic exchange model. On granule surface, the inductive magneti- +zation induces surface demagnetizing field. In granule body, the +inductive magnetization induces volume demagnetizing field. The +relation between surface and volume demagnetizing field is set up by +spherical shell approximation [13,14]. Then the total demagnetizing +field is derived through numerical integration. Finally, the demagnet- +izing energy density in FM granule is displayed by contour map. The +detail is shown in author’s previous work [13,14]. +Using the above method, the demagnetizing energy FDM is proved a +function of electric polarization: +FDM = Fpara(P0)sin 2θM + Fperp(P0)cos 2θM +(2) +Here P0 = +25 and −25 μC/cm2 represents the downward and up- +ward spontaneous polarization in BTO matrix respectively. θM is the +angle between vertical orientation and magnetizing orientation in FM +granule. Obviously, the demagnetizing energy reaches extreme at θM = +0 and π/2, i.e. the perpendicular and parallel magnetizing. Thus, the +anisotropic energy △FDM is defined as the difference between perpen- +dicular and parallel demagnetizing energy: +ΔFDM = Fperp(P0) − Fpara(P0) +(3) +at △FDM > 0 and △FDM < 0, easy magnetizing is parallel and perpen- +dicular respectively. +3. Result and discussion +Under downward and upward polarization P0, the position depen- +dence of anisotropic energy △FDM is shown in Fig. 2. The size parameter +is R = 10 nm and t = 100 nm. In Fe/BTO composite, the parallel and +perpendicular magnetizing is stable under upward and downward po- +larization respectively. However, in both Co/BTO and Ni/BTO com- +posite, the downward and upward polarization prefers parallel and +perpendicular magnetizing respectively. This contrast is attributed to +the different spin-split rate between iron metals. As the electric polari- +zation is downward or upward in BTO matrix, the screening electron is +accumulated or depleted on FM granule surface respectively. The +inductive magnetization is proportional to screening electron density +and spin-split rate [13,14]. According to Table 2 and Equation (1), the +spin-split rate χ of Fe metal is negative, while that of both Co and Ni +metal is positive. Thus, under the same polarization, the inductive +magnetization on Fe granule surface is opposite to that on Co and Ni +granule surface. Comparing to Co/BTO and Ni/BTO composite, it’s +reasonable the magnetic anisotropy is different in Fe/BTO composite. +In both Fe/BTO and Co/BTO composite, the anisotropic energy +varies linearly with position bias. In Ni/BTO composite, the nonlinear +evolution of anisotropic energy is remarkable. An inflection point is +trended under downward polarization, while the extreme point is pre- +sent under upward polarization. This contrast is attributed to the +different magnetization between iron metals. Under downward or up- +ward polarization, the inductive magnetization is a perturbation to +intrinsic magnetization. Based on the perturbation theory, the larger +ratio of inductive to intrinsic magnetization, the higher level of pertur- +bation is reached. As seen in Table 2, the intrinsic magnetization of Fe +and Co metal is much larger than that of Ni metal. Thus, the perturbation +effect in both Fe/BTO and Co/BTO composite is mainly first-level, i.e. +the linear perturbation. However, the smaller intrinsic magnetization in +Ni/BTO composite leads to higher-level perturbation, i.e. the nonlinear +perturbation. +In all the composite, the anisotropic energy amplitude is increased +with volume ratio. This evolution is attributed to the electrostatic +interaction between adjacent granules. The screening electron on +Fig. 1. The granule/matrix structure of FM/BTO composite. +Table 1 +The matter parameter of BTO matrix [21,22], SRO substrate [13], and Au +electrode [13]. The lattice constant is labelled as c100 and c001. The δ, εr and P0 +represents screening depth, dielectric constant and spontaneous polarization +respectively. +c100 +c001 +δ +εr +P0 +BTO +0.399 nm +0.403 nm +– +50 +25 μC/cm2 +SRO +0.393 nm +– +0.60 nm +– +– +Au +0.407 nm +– +0.06 nm +– +– +Table 2 +The matter parameter of Fe, Co and Ni granule [23,24]. The state density is +labelled as ρ↑ and ρ↓. The Jex, δ and M0 represents exchange energy, screening +depth and intrinsic magnetization respectively. +ρ↑ +ρ↓ +Jex +Δ +M0 +Fe +0.87 eV-1 +0.24 eV-1 +2.40 eV +0.13 nm +1.71 mA/nm +Co +0.18 eV-1 +0.70 eV-1 +1.25 eV +0.15 nm +1.43 mA/nm +Ni +0.18 eV-1 +1.56 eV-1 +0.65 eV +0.09 nm +0.49 mA/nm +B. Chen et al. + +Au +Z +2R +SROPhysica B: Physics of Condensed Matter 625 (2022) 413490 +3 +granule surface induces long-range Coulomb interaction, which is +described through mean field approximation in this model [13,14]. On +isolated granule surface, the distribution of screening electron is har- +monic. However, as the Coulomb interaction occurs between adjacent +granules, the inharmonic distribution of screening electron is intro- +duced. Furthermore, the anisotropic energy is mainly associated with +inharmonic magnetization [13,14]. As the granule/matrix volume ratio +is increased, the Coulomb interaction between adjacent granules is +gradually enhanced. Thus, the larger anisotropic energy is reached at +higher volume ratio. +On the cross section of FM/BTO interface, the demagnetizing energy +and electric polarization are investigated delicately. As easy magne- +tizing is parallel, the demagnetizing energy density with distorted +electric polarization is displayed in Fig. 3. Here the cross section is along +the vertical direction in Fig. 1, and across the granule center. The +granule locates at the middle of matrix, i.e. z = 0. As easy magnetizing is +perpendicular, the demagnetizing energy density with distorted electric +polarization is also displayed in Fig. 4. +Outside granule/matrix interface, the electric polarization is dis- +torted from spontaneous polarization, i.e. the vertical polarization, +which has been reported by other works [25,26]. At volume ratio of +0.0%, the distorted polarization is concentrated on the bottom of each +granule, as shown in Figs. 3 and 4. However, as the volume ratio is +increased to 1.0%, the distorted polarization migrates to the top, as +shown in Figs. 3 and 4. In Fe/BTO and Co/BTO composite, the migration +is incomplete, i.e. light distortion is retained on the bottom. In Ni/BTO +composite, the migration is nearly complete. This contrast is attributed +to the different screening behavior between iron metals. As seen in +Table 2, the screening depth of Fe and Co is larger than that of Ni. In +theory, the screening depth of absolute conductor is zero. The screening +depth is smaller, the interaction between electric polarization and +screening electron is stronger. Thus, as the long-range Coulomb inter- +action is enhanced with increasing volume ratio, the migration of dis- +torted polarization is more distinct in Ni/BTO than Fe/BTO and Co/BTO +composite. +Inside granule/matrix interface, the demagnetizing energy density is +inhomogeneous, which has been reported by other works [27,28]. +Under parallel magnetizing, the gradient of energy density is mainly +along horizontal direction, as shown in Fig. 3. Increasing volume ratio +from 0.0% to 1.0%, the low-energy region is shirked seriously. Mean- +while, compared to Fe/BTO and Co/BTO composite, the high-energy +region is expanded much more in Ni/BTO composite. Under perpen- +dicular magnetizing, the vertical gradient of energy density is distinctly +shown in Fig. 4. From 0.0% to 1.0% ratio, the shirked high-energy re- +gion is remarkable. Meanwhile, the expanded low-energy region is much +more in Ni/BTO composite, than that in Fe/BTO and Co/BTO composite. +Fig. 2. Under downward and upward polarization, i. e. P0 > 0 and P0 < 0, the position dependence of anisotropic energy △FDM. V represents each granule volume. +The position bias z is between ± (t – R). The granule/matrix volume ratio η varies from 0.0% to 1.0%. +B. Chen et al. + +P.>0 +Po<0 +200 +Fe/BTO +Fe/BTO +150 +150 +(erg/cm" +100 +100 +50 +50 +0 +0 +-50 +0.00% +0.00% +DM +-50 +0.25% +0.25% +-100 +△F. +0.50% +0.50% +F +-100 +-150 +0.75% +0.75% +1.00% +-150 +1.00% +-200 +200 +Co/BTO +Co/BTO +150 +150 +100 +100 +50 +50 +0 +0 +-50 +0.00% +0.00% +-50 +0.25% +0.25% +-100 +△F. +0.50% +F +0.50% +-100 +-150 +0.75% +0.75% +1.00% +-150 +1.00% +-200 +200 +Ni/BTO +150 +Ni/BTO +150 +100 +100 +50 +50 +0 +0 +0.00% +0.00% +-50 +DM +-50 +0.25% +0.25% +-100 +AF +0.50% +0.50% +-100 +-150 +0.75% +0.75% +1.00% +-150 +1.00% +-200 +-1.0 +-0.5 +0.0 +0.5 +1.0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +z/(t-R) +z/(t-R)Physica B: Physics of Condensed Matter 625 (2022) 413490 +4 +On the whole, as volume ratio is increased, the long-range Coulomb +interaction results stronger evolution of demagnetizing energy in +Ni/BTO than Fe/BTO and Co/BTO composite. +The above both inside and outside behavior is in accordance with +Fig. 2. Coupling with each easy magnetizing, the spontaneous polari- +zation in Fe/BTO composite is opposite to that in Co/BTO and Ni/BTO +composite. Accompanying the weaker evolution of demagnetizing en- +ergy with volume ratio, the position dependence of anisotropic energy is +linear in both Fe/BTO and Co/BTO composite; while the stronger evo- +lution coexist with nonlinear dependence in Ni/BTO composite. +The size effect is notable in FM/BTO composite. As spontaneous +polarization is downward and upward, the size dependence of aniso- +tropic energy △FDM is shown in Fig. 5. Here the granule size R varies +from 5 to 15 nm, while the matrix size t varies from 50 to 150 nm. The +position bias is fixed as z = 0. +The size effect is similar in Fe/BTO and Co/BTO composite. As +perpendicular magnetizing is stable, i.e. △FDM < 0, the anisotropic +energy density △FDM/V is simultaneously enhanced with decreasing +granule size R and increasing matrix size t. As parallel magnetizing is +stable, i.e. △FDM > 0, the anisotropic energy density is monotonously +enhanced with decreasing granule size, while a broad peak appears with +increasing matrix size. Under both perpendicular and parallel +magnetizing, the energy density becomes stronger by raising volume +ratio. Furthermore, during the range of present granule and matrix size, +the energy density is unsaturated roughly. It reflects the linear pertur- +bation in Fe/BTO and Co/BTO composite. +However, the size effect is very different in Ni/BTO composite. Away +from minimum granule and maximum matrix size, i.e. R = 5 nm and t = +150 nm, the anisotropic energy density is suppressed sharply. Near +maximum granule and minimum matrix size, the anisotropic energy +density is nearly saturated. It indicates the nonlinear perturbation in Ni/ +BTO composite. Under upward polarization, the magnetic anisotropy is +unique, i.e. △FDM < 0 is kept. However, under downward polarization, +as granule size or matrix size is varied, the △FDM is switched between +positive and negative value. Thus, the downward polarization in- +troduces uncertain magnetic anisotropy in Ni/BTO composite. +4. Conclusion +In this work, the FM/BTO composite with granule/matrix structure is +designed. The coupling between magnetic anisotropy and electric po- +larization is studied theoretically. Under each spontaneous polarization, +the easy magnetizing is along parallel or perpendicular orientation, +which relies on the spin-split rate of FM metal. The position dependence +Fig. 3. As easy magnetizing is parallel, the demagnetizing energy density with distorted electric polarization near FM/BTO interface. In the up and down panel +separated by dash line, the volume ratio is 0.0% and 1.0% respectively. Outside granule/matrix interface, the color vector implies the deviation of electric polar- +ization from vertical orientation. Inside granule/matrix interface, the color area indicates the strength of demagnetizing energy density. (For interpretation of the +references to color in this figure legend, the reader is referred to the Web version of this article.) +B. Chen et al. + +Fe +Fe +Ni +Co +min +max +DM +DMPhysica B: Physics of Condensed Matter 625 (2022) 413490 +5 +of anisotropic energy implies the linear or nonlinear perturbation in FM/ +BTO composite. The magnetic anisotropy is enhanced with increasing +volume ratio, which originates from the long-range Coulomb interaction +between FM granules. In the cross-section map, the electric-polarization +distortion evolves simultaneously with magnetizing-energy distribution. +As size is varied, electric polarization induces unique magnetic anisot- +ropy in Fe/BTO and Co/BTO composite, while this coupling is uncertain +in Ni/BTO composite. +At room temperature, the cubic lattice constant of Fe and Ni is 0.29 +and 0.36 nm respectively, while the hexagonal lattice constant of Co is +0.25 and 0.41 nm [29]. In this work, the granule radius ranges from 5 to +15 nm. Obviously, the granule size is 10–50 times of lattice constant. As +a short-range interaction, the microscopic interaction on FM/BTO +interface only occurs within about one-lattice thickness, and the crys- +talline of FM/BTO interface should be well orientated [30]. Thus, the +long-range electrostatic and demagnetizing interaction is dominate in +present FM/BTO composite. If the granule size is less than 5 nm, the +microscopic interaction becomes more important, and this model should +be modified entirely. +Besides of demagnetizing anisotropy, the magnetocrystalline +anisotropy should be considered in FM granule. At room temperature, +the Fe and Ni metal displays cubic magnetocrystalline anisotropy, while +the Co metal displays hexagonal magnetocrystalline anisotropy [23]. +Without electric polarization, the magnetic moment fixes along easy +magnetocrystalline orientation. As electric polarization induces parallel +or perpendicular demagnetizing anisotropy, the easy magnetizing +orientation relies on the balance between demagnetizing and magne- +tocrystalline energy. It results the procession of magnetic moment be- +tween two different easy magnetizing orientations. This procession is +also called spin-nano oscillator, which could emit microwave [31,32]. +Further work will be carried. +Credit +Bo Chen: Conceptualization, Methodology, Writing, Funding acqui- +sition. Zi-Run Li and Wen-Li Cui: Data curation, Writing, Funding +acquisition. +Declaration of competing interest +The authors declare that they have no known competing financial +interests or personal relationships that could have appeared to influence +the work reported in this paper. +Fig. 4. As easy magnetizing is perpendicular, the demagnetizing energy density with distorted electric polarization near FM/BTO interface. In the up and down panel +separated by dash line, the volume ratio is 0.0% and 1.0% respectively. Outside granule/matrix interface, the color vector implies the deviation of electric polar- +ization from vertical orientation. Inside granule/matrix interface, the color area indicates the strength of demagnetizing energy density. (For interpretation of the +references to color in this figure legend, the reader is referred to the Web version of this article.) +B. Chen et al. + +Co +Ni +Fe +Fe +Ni +max +DM +DMPhysica B: Physics of Condensed Matter 625 (2022) 413490 +6 +Fig. 5. The size dependence of anisotropic energy △FDM. V represents each granule volume. The volume ratio is 0.0%, 0.5% and 1.0% in (a), (b) and (c) respectively. +In the left and right panel separated by dash line, the spontaneous polarization is downward and upward respectively. +B. Chen et al. + +Fe/BTO +Fe/BTO +40 +a +b +-10 +30 +(c) +-20 +20 +(C) +-30 +10 +(b +-40 +nnm +Co/BTO +Co/BTO +0 +(a) +40 +b +-15 +30 +(c) +-30 +(erg/cm3) +(c) +(erg/cm3) +20 +-45 +(b) +10 +-60 +P +Ni/BTO +Ni/BTO +0 +40 +(a) +(b) +-50 +0 +(C) +(b) +-100 +(a) +-40 +-150 +-200 +-80 +250 +-120Physica B: Physics of Condensed Matter 625 (2022) 413490 +7 +Acknowledge +This work is supported by the Shanxi Province Science Foundation +for Youths (201801D221143, 201901D211246, and 201901D211230); +the Scientific and Technological Innovation Programs of Higher Edu- +cation Institutions in Shanxi (2019L0535 and 2020L0289); and the +Science Foundation of North University of China (2017026 and +XJJ201905). +References +[1] I.S. Camara, C. Achkar, N. Liakakos, A. Pierrot, V. Pierron-Bohnes, Y. Henry, +K. Soulantica, M. Respaud, T. Blon, M. Bailleul, Enhanced magnetocrystalline +anisotropy in an ultra-dense array of air-exposed crystalline cobalt nanowires, +Appl. Phys. Lett. 109 (2016) 202406. +[2] C. 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Chen et al. + diff --git a/materials/content/tmp_files/3_PhysRevLett.115.037203.pdf.txt b/materials/content/tmp_files/3_PhysRevLett.115.037203.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..eaf4dc0294d91fcf146e79d509196d01cd0bbeee --- /dev/null +++ b/materials/content/tmp_files/3_PhysRevLett.115.037203.pdf.txt @@ -0,0 +1,695 @@ +Thermal Transport and Nonequilibrium Temperature Drop Across a Magnetic +Tunnel Junction +Jia Zhang,* Michael Bachman, Michael Czerner, and Christian Heiliger† +I. Physikalisches Institut, Justus Liebig University Giessen, Heinrich-Buff-Ring 16, 35392 Giessen, Germany +(Received 9 March 2015; published 15 July 2015) +In the field of spin caloritronics, spin-dependent transport phenomena are observed in a number of +current experiments where a temperature gradient across a nanostructured interface is applied. The +interpretation of these experiments is not clear as both phonons and electrons may contribute to thermal +transport. Therefore, it still remains an open question how the temperature drop across a magnetic +nanostructured interface arises microscopically. We answer this question for the case of a magnetic tunnel +junction (MTJ) where the tunneling magneto-Seebeck effect occurs. Our explanation may be extended to +other types of nanostructured interfaces. We explicitly calculate phonon and electron thermal conductance +across Fe=MgO=Fe MTJs in an ab initio approach using a Green function method. Furthermore, we are +able to calculate the electron and phonon temperature profile across the Fe=MgO=Fe MTJ by estimating the +electron-phonon interaction in the Fe leads. Our results show that there is an electron-phonon temperature +imbalance at the Fe-MgO interfaces. As a consequence, a revision of the interpretation of current +experimental measurements may be necessary. +DOI: 10.1103/PhysRevLett.115.037203 +PACS numbers: 75.76.+j, 31.15.E-, 63.20.kd, 85.80.Lp +In the 1990s, Johson and Silsbee [1] pioneered the +theoretical investigation of thermal effects in nanomagnetic +systems. However, it took until 2008, when Uchida et al. +discovered the spin Seebeck effect in NiFe films [2,3], for +the +spin-caloritronics +[4,5] +phenomenon +to +emerge. +Since then, interesting spin-caloritronic transport phenom- +ena +have +been +observed, +including +thermal +spin +injection +[6] +and +the +magneto-Peltier +effect +[7–9]. +Magnetothermoelectric phenomena, e.g., the tunneling +magneto-Seebeck effect, which occurs in magnetic tunnel +junctions (MTJs) and manifests the dependence of the +charge Seebeck coefficient on the magnetic alignment of +the magnetizations of the two layers sandwiching the thin +barrier layer, was demonstrated by experiment [10–12] +and explained by first-principles calculations [13,14]. +Furthermore, the thermal spin transfer torque has been +theoretically predicted [15,16] and corresponding experi- +ments are currently under way [17]. Thus, MgO MTJs may +become +potential +key +elements +for +spin-caloritronic +applications. +Many spin-caloritronic phenomena are driven by a +temperature gradient across an interface or a very thin +layer. However, at present the understanding of the temper- +ature drop and of the heat transport itself across an interface +is lacking. In this Letter we focus on MgO-based MTJs, but +our results can also at least qualitatively be applied to a +range of other systems. +To obtain the correct temperature drop across the barrier +in a MTJ, the appropriate value of the thermal conductance +κMTJ needs to be derived. In the analysis of experiments the +temperature across the MgO MTJs has been simply +simulated by COMSOL [10,11]. For κMTJ, a value was taken +which lies in between that for the thermal conductivity of +bulk MgO and that of the MgO thin film. This approach is +very unsatisfactory as these values differ by 1 order of +magnitude and the role of phonon modes in Fe or at the +Fe-MgO interface is completely ignored. +The phonon mean free path Λ of bulk MgO can be +estimated using kp ¼ 1=3CυΛ [18]. By taking experimen- +tal values in Table I, Λ for bulk MgO is around 6.4 nm. +Λ is larger than the typical MgO barrier thickness d +(between 0.6 and 2 nm) in MTJs. This clearly shows +that it is not appropriate to describe the thermal conduc- +tivity of the barrier by the bulk thermal conductivity of +MgO, and not even by the thin film value [19]. +Consequently, for a suitable description the phonon trans- +port across the whole Fe=MgO=Fe junction has to be +calculated explicitly. +Considering a MTJ it is not obvious how much the +tunneling electrons themselves contribute to thermal trans- +port. In Fe the thermal transport is dominated by electrons +whereas in MgO as an insulator the thermal transport is +determined by phonons. Moreover, what happens at the +interface? Are the electronic and phononic systems in +equilibrium, e.g., at the same temperature, at the interface? +If not, does it bear any consequences for the interpretation +of the experimental results? +In the following, we will answer these questions. We +will first describe the calculation of the phonon conduct- +ance κMTJ +p +and then that of the electron thermal conductance +κMTJ +e +. Eventually, we will define the overall conductance +κMTJ and calculate the temperature profile across the +Fe=MgO=Fe MTJs by estimating the electron-phonon +coupling. +PRL 115, 037203 (2015) +P H Y S I C A L +R E V I E W +L E T T E R S +week ending +17 JULY 2015 +0031-9007=15=115(3)=037203(5) +037203-1 +© 2015 American Physical Society + +The phonon transmission function and the resulting +contribution to the thermal conductance are calculated +by using the atomistic Green function method. The junction +is divided into three parts as shown in Fig. 1. The scattering +region consists of two Fe/MgO interfaces sandwiched by +two semi-infinite Fe leads. After constructing the Green +function, the phonon transmission function tpðεÞ and the +thermal conductance κp are calculated. Details are given +in Ref. [24]. +The Green function is calculated from the interatomic +force constants (IFCs). Thereby, we construct the Green +function of the whole MTJ from the IFCs of bulk Fe and the +IFCs of the scattering region. To obtain the IFCs we use the +density functional perturbation theory (DFPT) [25] imple- +mented in the ABINIT package [26]. We use Troullier- +Martins norm-conserving pseudopotentials [27] and the +GTH +(Goedecker-Teter-Hutter) +LDA +(Local +Density +Approximation) exchange-correlation functional [28]. To +verify the applicability of the method, we compute the +phonon band structure for bulk Fe (a ¼ 2.867 Å) and bulk +MgO (a ¼ 4.239 Å). For the self-consistent calculation of +bcc Fe (fcc MgO), we use an energy cutoff of 30 Ha +(34 Ha) and a k-point mesh of 16 × 16 × 16 (8 × 8 × 8). +We use a 8 × 8 × 8 q mesh for the dynamic matrix of both +materials. The calculated phonon dispersions together with +experimental values from the literature are shown in Fig. 2. +The comparison yields an excellent agreement. +To obtain the IFCs of the scattering region, we use a +supercell consisting of the MgO barrier with 4 monolayers +(ML) of Fe either side. The in-plane lattice constant is fixed +to a ¼ 2.867 Å (bulk Fe). The cell volume and atomic +coordinates are fully relaxed until the forces on all atoms +are smaller than 0.05 meV=Å. After the structural relax- +ation, self-consistent and DFPT calculations are conducted +according to different irreducible perturbations. For the +self-consistent calculations, we use a 10 × 10 × 2 k-point +mesh and an energy cutoff of 34 Ha. For DFPT calcu- +lations, we use a 5 × 5 × 1 q mesh with 6 inequivalent q +points in the Brillouin zone. There are in total 120, 150, +180, and 210 independent DFPT calculations for 3, 5, 7, +and 9 monolayers of MgO, respectively. +The phonon density of states (PDOS) projected on each +atom in the scattering region is shown in Fig. 3(a). The +PDOS for the Fe layer near the MgO interface is signifi- +cantly different from the bulk PDOS of Fe due to the +interface phonon bonding modes between Fe MgO. This +pronounced interface phonon peak is located at 8.5 meV. +Inside the MgO barrier there are several phonon peaks. +However, compared with the bulk phonon modes of MgO, +the peak positions inside the MgO barrier are shifted in +energy. The phonon modes at the Fe-MgO interface and +inside the MgO barrier will have an impact via the electron- +phonon interaction and can be observed in inelastic electron +tunneling spectra [31]. For example, an inelastic electron +tunneling spectrum peak at 80 meV in the d2I=dV2 curve +has been attributed to the Mg-O surface phonon modes in +MgO [31], and this energy value is close to the largest peak +position present in our projected PDOS. +To construct the IFCs of the whole junction we take away +from the interface the IFCs of bulk Fe. Our calculated tpðεÞ +TABLE I. +Experimental thermal properties of bulk Fe and MgO. +Thermal property at 300 K +bcc Fe +fcc MgO +Phonon thermal conductivity kp (Wm−1K−1) +17.55 [20] +49.9 [21] +Electron thermal conductivity ke (Wm−1K−1) +62.75 [20] +� � � +Volumetric specific heat C (106 Jm−3 K−1) +� � � +3.345 [22] +Average phonon group velocity (m=s) +� � � +7000 [23] +Phonon mean free path (nm) +� � � +6.39 +FIG. 1 (color online). +Sketch of a Fe=MgOð3 MLÞ=Fe MTJ +used for phonon transmission calculation. The scattering region +for the supercell calculation is indicated. +FIG. 2 (color online). +Phonon dispersions of (a) bcc Fe and +(b) fcc MgO along high symmetry directions calculated by +ABINIT (color lines). The dotted data are experimental values +taken from Refs. [29,30]. Lines with different colors correspond +to different phonon branches. +PRL 115, 037203 (2015) +P H Y S I C A L +R E V I E W +L E T T E R S +week ending +17 JULY 2015 +037203-2 + +ThermalcurrentJ +T+△T +scattering region +Fe +0 +Mg +Fe2 +Fe4 +Bulk L +Mg01 Mg02 Mg03 +BulkRfor different MgO thicknesses are shown in Fig. 3(b). The +cutoff energy of tpðεÞ is 38 meV and corresponds to the +cutoff frequency of bulk Fe. The general shape of tpðεÞ is +almost independent of the MgO thickness. Minor shifts of +the peak positions are visible, which may originate from +quantized phonon modes within the barrier. Using tpðεÞ we +calculate κMTJ +p +[24]. Its temperature dependence is shown in +Fig. 3(c). There is a slight difference for different MgO +thicknesses. At 300 K, κMTJ +p +is found to be on the order +of 108 Wm−2 K−1. +The screened-KKR (Korringa-Kohn-Rostoker) Green +function method [13] is adopted to calculate the electronic +transmission teðεÞ, which depends on the relative magnetic +orientation of the Fe leads to each other. Details of the +method are described in Refs. [13,14]. κMTJ +e +is obtained +from teðεÞ [32,33]. κMTJ +e +for parallel or antiparallel mag- +netic alignment are shown in Fig. 3(d). Since the electrons +are tunneling through the MgO, κMTJ +e +decreases exponen- +tially with increasing barrier thickness. Even at the smallest +considered MgO thickness of 3 monolayers, κMTJ +e +at 300 K +is 1 order of magnitude smaller than κMTJ +p +. +Using κMTJ +p +and κMTJ +e +we calculate the temperature profile +in Fe=MgO=Fe MTJs by solving a one-dimensional ther- +mal transport equation. We simultaneously consider the +electron and phonon transport, which carry the heat flux +across the junction as well as the electron-phonon inter- +action in the Fe leads. It is reasonable to neglect the magnon +contribution to the thermal transport since the magnon +transport across the junction is assisted by electron tunnel- +ing through electron-magnon interaction, which is at least 1 +order of magnitude smaller than the electron and phonon +transport process. +The energy balance equations in the Fe leads are [34] +kFe +e +d2TeðzÞ +dz2 +− GephðTe − TpÞ ¼ 0; +kFe +p +d2TpðzÞ +dz2 +þ GephðTe − TpÞ ¼ 0; +ð1Þ +where kFe +e +and kFe +p are the electron and phonon thermal +conductivities in Fe, respectively. For these quantities we +take the bulk values, which are listed in Table I. Geph is the +electron-phonon interaction factor, which is determined by +Geph ¼ πkBλhω2iDðεFÞ [35]. kB is the Boltzmann constant, +λ is the electron-phonon mass enhancement parameter, +DðεFÞ is the electron density of states at the Fermi energy, +and hω2i ≈ θ2 +D=2 is the second moment of the phonon +spectrum defined by McMillan [36], where θD is the Debye +temperature. Using the corresponding parameters listed in +Table II, Geph of Fe is 9.925 × 1017 Wm−3 K−1. This +value for Fe is comparable but slightly smaller than that +of +nickel +(10.5 × 1017 Wm−3 K−1) +and +platinum +(10.9 × 1017 Wm−3 K−1) [35]. +The solutions of Eq. (1) for the left (L) and right (R) Fe +layers are +TLðRÞ +e;p ðzÞ ¼ BLðRÞ +e;p þ CLðRÞ +e;p z þ DLðRÞ +e;p eþð−Þðz=lFeÞ; +ð2Þ +where the coefficients are given by +BLðRÞ +e +¼ BLðRÞ +p +; +CLðRÞ +e +¼ CLðRÞ +p +; +DLðRÞ +e +¼ − kFe +p +kFe +e +DLðRÞ +p +: +Thus, the last term in Eq. (2) accounts for a electron- +phonon imbalance at the Fe=MgO interface. Consequently, +lFe ¼ +ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi +1 +Geph +kFe +e kFe +p +kFe +p þ kFe +e +s +; +1 +l2 +Fe +¼ 1 +l2e +þ 1 +l2p +ð3Þ +is +the +characteristic +length +of +this +imbalance +and +l2e;p ¼ κFe +e;p=Geph. We have to calculate 6 coefficients, +BLðRÞ +e +, CLðRÞ +e +, and DLðRÞ +e +, by using the following 6 boundary +conditions, +FIG. 3 (color online). +(a) Projected phonon density of states +(PDOS) for a supercell with 8 ML Fe and 3 ML MgO. The atom +index is the same as in Fig. 1. The PDOS for bulk Fe and MgO is +shown for reference. (b) The phonon transmission as a function of +energy for Fe=MgOð3–9 MLÞ=Fe MTJs. (c) The phonon and +(d) electron thermal conductance of Fe=MgOð3–9 MLÞ=Fe MTJs +as a function of temperature. +TABLE II. +Parameters used for calculating the electron-phonon +interaction factor of Fe. +λ +θD (K) +DðεFÞ (states/Ha) +Geph (Wm−3 K−1) +0.243 [37] +470 [38] +23.02 [37] +9.925 × 1017 +PRL 115, 037203 (2015) +P H Y S I C A L +R E V I E W +L E T T E R S +week ending +17 JULY 2015 +037203-3 + +dTLe;pðzÞ +dz +���� +z¼−d=2 +¼ dTRe;pðzÞ +dz +���� +z¼d=2 +; +κMTJ +e;p +� +TLe;p +�−d +2 +� +− TRe;p +�d +2 +�� +¼ −κFe +e;p +dTRe;pðzÞ +dz +���� +z¼d=2 +; +TLe ðz ¼ −LÞ ¼ TL; +TRe ðz ¼ LÞ ¼ TR; +where the last two conditions are some given temperatures +at the left and right of the MTJ, which has a total thickness +of 2L. +To illustrate our result, we show in Fig. 4 the electron +and phonon temperature profiles for a Feð10 nmÞ= +MgOð9 MLÞ=Feð10 nmÞ +MTJ +with +a +temperature +difference +of +1 +K. +The +consequence +of +different +electron and phonon temperatures near the MgO=Fe +interface +is +significant +especially +for +the +definition +and evaluation of thermoelectric physical quantities. For +example, +the +Seebeck +coefficient +S +is +defined +as +S ¼ ΔV=ΔTe, where ΔTe is the electron temperature drop +since the Seebeck effect is a consequence of electron +transport. +Although we have a nonequilibrium situation, it might be +useful to define an effective temperature drop ΔTeff, which +can be used to define a κMTJ ¼ q=ΔTeff, where q ¼ +−kFe +e ½dTRe ðzÞ=dz� − kFe +p ½dTRpðzÞ=dz� is the total thermal +current through the junction. Note that q is conserved +whereas the electron and phonon thermal current alone are +not due to the imbalance. +Following Ref. [34] we define Teff by linear extrapola- +tion of the temperature from the equilibrium towards the +MgO barrier (see Fig. 4). For the case shown in Fig. 4, we +calculate κMTJ ¼ 1.681 × 108 Wm−2 K−1. This value can +be used in a simple network model to estimate the electron +temperature drop by calculating ΔTeff. To evaluate the +error in this estimation, we calculate +Te − Teff +Teff − Tp +¼ kFe +p +kFe +e +: +ð4Þ +This means that if kFe +e ≫ kFe +p , which is the case for most +metals, Teff is closer to Te than to Tp. If this is not the case +or a more precise knowledge of ΔTe is needed, one has to +solve appropriate transport equations, i.e., Eq. (1), for the +whole junction, but taking the first-principles values for +κMTJ +e +and κMTJ +p +. Moreover, the effect of the imbalance will +be larger if the phonon interface conductance is larger. This +effect is shown in the inset of Fig. 4, where we assume a 10 +times larger κMTJ +p +than the first-principles value, which +corresponds, e.g., to different materials. Note that even if +the phonon temperature drop across the barrier decreases, +the drop in the electron temperature will remain. +κMTJ is of particular importance for the interpretation of +measurements. In experiments a Seebeck voltage is mea- +sured and a temperature drop is estimated using diffusion +models. As stated earlier, often a value of bulk or thin film +value of MgO is used for κMTJ. In Fig. 5 we plot our +calculated κMTJ in comparison to bulk and thin film values +of MgO for different MgO thicknesses. Our results are +almost independent of the MgO thickness. As an example, +the value used in Ref. [10] is indicated by the green star. +This implies that the estimated Seebeck coefficients in +Ref. [10] may be too high. +In addition, we can calculate the thermoelectric figure of +merit ZT ¼ S2GT=κMTJ by calculating the Seebeck coef- +ficient S and the electrical conductance G. Because the +electrons have to tunnel, G decreases exponentially with +increasing barrier thickness and so does the value of ZT. S +is almost independent of barrier thickness with a value of +about 20 μV=K. Even for a very thin barrier of 3 ML MgO, +we obtain a very small ZT value of about 10−3 at 300 K. +FIG. 4 (color online). +The electron temperature Te, the phonon +temperature Tp, and the linear extrapolation of electron temper- +ature Teff profiles across a Fe=MgOð9 MLÞ=Fe MTJ with the +temperature at the left side (L ¼ −10 nm) TL ¼ 301 K and right +side (L ¼ 10 nm) TR ¼ 300 K. For this system, lFe ¼ 3.72 nm. +The inset shows the temperature profiles assuming a 10 times +larger κMTJ +p +than the first-principles value. +FIG. 5 (color online). +The junction thermal conductance as a +function of MgO barrier thickness in this work (black) compared +to bulk MgO (red dashed line) and thin film MgO (blue dashed +line). The value used by Walter et al. [10] is shown as a green star. +PRL 115, 037203 (2015) +P H Y S I C A L +R E V I E W +L E T T E R S +week ending +17 JULY 2015 +037203-4 + +This value decreases to 10−7 for 9 ML of MgO. +Consequently, MTJs are at the moment not promising +for thermoelectric applications. +In conclusion, we show that there is an imbalance of the +electron and phonon temperature at nanomagnetic inter- +faces. This leads to different temperature drops for elec- +trons and phonons and one has to be cautious interpreting +corresponding experimental results. Nevertheless, in the +Fe=MgO=Fe MTJ studied, we observed a large interface +resistance for the phonons. As a consequence, a large +drop for the electron temperature also exists, which is +responsible for the high Seebeck voltages observed in +experiments. But even for MTJs with low phonon +interface resistances, the drop of the electron temperature +remains large due to the strong imbalance of electron +and phonon temperature. For a MTJ with metallic leads we +define an effective thermal conductance by defining an +effective temperature. The corresponding values given in +Fig. 5 provide approximate values of the conductance +of the MTJ (barrier plus interfaces), which may prove +useful in simple models. However, if more reliable values +of the temperature drops are needed, a calculation +of the coupled electron and phonon system is necessary. +In any case, for the correct quantitative description first- +principles calculations are a strict requirement due to the +coherent transport across the nanomagnetic interface. +We thank Professor M. Münzenberg, Professor M. +Verstraete, and Professor P. J. Klar for useful discus- +sions +and +acknowledge +support +from +Deutsche +Forschungsgemeinschaft (SPP 1538 via Grant No. HE +5922/4-2). We acknowledge support within the LOEWE +program of excellence of the Federal State of Hessen +(project initiative STORE-E). This project was supported +by the Laboratory of Materials Research (LaMa) of JLU. +*Jia.Zhang@exp1.physik.uni‑giessen.de +†Christian.Heiliger@physik.uni‑giessen.de +[1] M. Johnson and R. H. Silsbee, Phys. Rev. B 35, 4959 +(1987). +[2] K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. +Ando, S. Maekawa, and E. Saitoh, Nature (London) 455, +778 (2008). +[3] J. Xiao, G. E. W. Bauer, K. C. Uchida, E. Saitoh, and S. +Maekawa, Phys. Rev. B 81, 214418 (2010). +[4] G. E. W. Bauer, A. H. MacDonald, and S. Maekawa, Solid +State Commun. 150, 459 (2010). +[5] G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Nat. Mater. +11, 391 (2012). +[6] J. C. Le Breton, S. Sharma, H. Saito, S. Yuasa, and R. +Jansen, Nature (London) 475, 82 (2011). +[7] M. Hatami, G. E. W. Bauer, Q. F. Zhang, and P. J. Kelly, +Phys. Rev. B 79, 174426 (2009). +[8] J. Flipse, F. L. Bakker, A. Slachter, F. K. Dejene, and B. J. +van Wees, Nat. Nanotechnol. 7, 166 (2012). +[9] J. Flipse, F. K. Dejene, D. Wagenaar, G. E. W. 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(Wiley, +New York, 1986). +PRL 115, 037203 (2015) +P H Y S I C A L +R E V I E W +L E T T E R S +week ending +17 JULY 2015 +037203-5 + diff --git a/materials/content/tmp_files/6_clark2002.pdf.txt b/materials/content/tmp_files/6_clark2002.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..109041e1c9a54a54340817168d448ebce5e2b162 --- /dev/null +++ b/materials/content/tmp_files/6_clark2002.pdf.txt @@ -0,0 +1,701 @@ +Materials Transactions, Vol. 43, No. 5 (2002) pp. 881 to 886 +Special Issue on Smart Materials-Fundamentals and Applications +c⃝2002 The Japan Institute of Metals +Magnetostrictive Properties of Galfenol Alloys Under Compressive Stress +Arthur E. Clark1, Marilyn Wun-Fogle2, James B. Restorff2 +and Thomas A. Lograsso3 +1Clark Associates, Adelphi, MD 20783, USA +2Naval Surface Warfare Center, Carderock Division, Code 645, W. Bethesda, MD 20817, USA +3Ames Laboratory, Ames, IA 50011, USA +Fe–Ga alloys, in which the α-Fe structure is maintained, are rich sources of high strength, low cost magnetostrictive alloys for transducer +and vibration reduction applications. Although the magnetostriction of Fe itself is very low, when a relatively small fraction of the Fe atoms are +replaced by Ga, the magnetostriction, (3/2)λ100, increases greatly. Until recently, the highest magnetostriction was found with the replacement +of Fe by Al (Alfenol). In this paper, we present our measurements of magnetostriction on Fe1−xGax, 0.13 ≤ x ≤ 0.24, (Galfenol). With the +substitution of 19% Ga for Fe in Fe1−xGax, a 12-fold increase in magnetostriction to ∼ 400 ppm occurs, even though Ga is non-magnetic. In +these alloys, the saturation magnetizations remain high, Ms ∼= 1.7 T, and the Curie temperatures are far above room temperature, TC ∼= 700◦C. +In most alloys studied, the magnetostrictions and magnetizations are fully saturated in fields less than 24 kA/m, even under compressive stresses +>100 MPa. For x = 0.24 (near Fe3Ga), an anomalous increase in magnetostriction with temperature occurs with a peak magnetostriction above +room temperature. Small additions of Ni and Mo to the binary Fe–Ga alloys decrease the room temperature value of λ100. +(Received September 20, 2001; Accepted December 6, 2001) +Keywords: magnetostriction, Galfenol, iron-gallium alloys +1. +Introduction +It was recently recognized that a substantial increase in the +magnetostriction, (3/2)λ100, of Fe occurs with the substitution +of small amounts of Ga for Fe.1,2) This is true as long as the +bcc α-Fe phase is maintained.3) This phase is not the equilib- +rium phase in Fe1−xGax at room temperature for x > 0.15.4) +By rapid quenching into water from temperatures ≥800◦C, +the disordered α-Fe structure can be extended to larger values +of x and the magnetostriction further increased.3) The phe- +nomenal increase in Fe magnetostriction with Ga is remark- +able, since Ga is non-magnetic. +This paper is divided into two parts. In the first part we +present: (1) the field dependence of the magnetostriction, +(3/2)λ100, of Fe0.81Ga0.19 under high compressive stresses +at room temperature and (2) the temperature dependence of +(3/2)λ100 from −269◦C to 42◦C. Magnetostriction measure- +ments show that quenching from 800◦C and 1000◦C increases +the magnetostriction over 40% to values ∼ 400 ppm at room +temperature and >420 ppm at −269◦C. This slight increase +in magnetostriction as the temperature is decreased from 315 +to −269◦C is consistent with the small increase of magneti- +zation reported over the same temperature range.3,5) We find +for Fe0.81Ga0.19 that the magnetostriction has a well-behaved +temperature dependence and an intrinsic magnetostriction +that is very large. For Fe0.76Ga0.24 (near Fe3Ga), this is no +longer true. In the second part of the paper we report the +effect of small amounts of Ni and Mo on the saturation mag- +netostriction constants, λ100 and λ111, of the Fe–Ga alloys. +In all cases the large positive magnetostriction constant, λ100, +decreases in value. +2. +Sample Preparation +To prepare single crystal samples of Fe1−xGax, as-cast in- +gots containing Ga (99.999% pure) and Fe (99.99% pure) +were inserted into alumina crucibles and heated to 1650◦C. +The ingot/crucible was stabilized for 1 h and then withdrawn +at a rate of ∼ 2 mm/h. Following crystal growth, the alloys +were annealed at 1000◦C for 168 h and furnace cooled at a +rate of 10◦C/min. The large single crystals were oriented and +rods (∼ 2.5 cm × 0.6 cm dia.) and discs (∼ 0.3 cm × 0.6 cm +dia.) of the proper crystalline directions were extracted. Sam- +ples were examined for homogeneity and Fe/Ga ratio. Ini- +tial magnetostriction and magnetization measurements were +made on these furnace-cooled alloys. Following these mea- +surements, some alloys were then reheated to 800◦C and +1000◦C in evacuated quartz tubes for 1 h and finally rapidly +cooled by quenching into a water bath. +3. +Experimental Methods +A conventional dead-weight apparatus was used to ap- +ply compressive loads to the Fe1−xGax alloys at room tem- +perature.6) Magnetic fields up to 80 kA/m were applied to +the samples from a solenoid energized by a constant cur- +rent source. Magnetic hysteresis loops were calculated from +the emfs generated by a small pick-up coil surrounding the +center of the samples. Displacements as a function of mag- +netic field at compressive stresses up to 120 MPa were de- +termined from the output of three linear variable differen- +tial transformers (LVDT’s). See Fig. 1. In order to obtain +the intrinsic (T =0) values of the saturation magnetostriction, +measurements were made in high fields from room tempera- +ture to −269◦C on oriented single crystals. Disc samples of +the appropriate alloys were affixed with special low temper- +ature non-magnetoresistive strain gages (Kyowa K-19-1S1). +(The temperature dependence of the gage factors was taken +from Gersdorf.7)) The small disc samples were mounted on +a rotating fixture and inserted into a liquid He cryostat po- +sitioned between the poles of a large electromagnet. Strains +along the [100] direction were measured as a function of an- + +882 +A. E. Clark, M. Wun-Fogle, J. B. Restorff and T. A. Lograsso +iron end +pieces + lead +weights +(stress) +pickup coil +table +H coil +sample +LVDT s +, +Fig. 1 +Apparatus for measuring magnetization and magnetostriction as a +function of compressive stress. +H +Strain gage aligned +along [100] direction +Fig. 2 +Configuration for measuring saturation magnetostriction constants. +Special low temperature Kyowa K-19-1S1 strain gages were used for low +temperature measurements. +gle relative to the applied field in magnetic fields of 400, +800, 1200, and 1600 kA/m and fit to the expression ∆l/l = +λγ,2 cos2 θ + λγ,4 cos4 θ where λγ,2 = (3/2)λ100. See Fig. 2. +4. +Magnetostriction of Binary Fe1−xGax Alloys +To date, the largest magnetostriction is found in a sample of +Fe0.81Ga0.19 rapidly quenched into water from 800◦C. Figure +3 illustrates the dependence of magnetostriction, (3/2)λ100, on +Ga concentration. For samples with x = 0.17, the α-Fe phase +is near equilibrium at room temperature, the magnetostriction +is about 300 ppm, and the effect of cooling rate on the sam- +ple is minimal. For x = 0.24, the magnetostriction is about +270 ppm, and again the effect of cooling rate on the mag- +netostriction is very small. However, between these values of +x, a large cooling-rate dependent magnetostriction peak ap- +pears. For x = 0.19, and x = 0.21, rapid quenching from +800◦C greatly improves the magnetostriction over furnace- +cooled alloys. For x = 0.19, the improvement is ∼ 40%. +The room temperature magnetostriction of quenched (disor- +dered) Fe1−xGax (x ∼ 0.19) exceeds that of all magnetostric- +tive 3d transition metals, such as Co, permendur, and Alfenol. +This is fascinating since Ga is non-magnetic, i.e. the entire +magnetostriction arises from Fe in the diluted α-Fe structure. +0.0 +0.1 +0.2 +0.3 +0.4 +100 +150 +200 +250 +300 +350 +400 +Fe1-xGax +(3/2) +l00 (x 10 +-6) +x +Fig. 3 +Room temperature magnetostriction of Fe1−xGax +alloys (■) +quenched into water from 800◦C, (△) furnace cooled at 10◦/min. Mea- +surements were made at 1600 kA/m. The dotted line is a guide for the eye +through the data for the quenched alloys. For concentrations of x ≥ 0.17 +the furnace cooled alloys may be multiple phase. +Because of the sharp non-linear increase in magnetostriction +above that of Fe in the disordered α-Fe–Ga alloys it is be- +lieved that the magnetostriction is not due to conventional +magnetoelastic effects but due to the onset of short range +order and the presence of asymmetric clusters of Ga atoms +along [100] directions in the α-Fe structure.3) For very low +values of x there are few clusters. For large values of x near +0.25, it becomes difficult if not impossible to form the disor- +dered bcc structure since the alloys greatly prefer the ordered +DO3 or B2 states. Very significantly the measurements of λ111 +for Fe1−xGax and Fe1−xAlx show little or no change with x, +in contrast to the great change in λ100.1,2,8) +To determine the intrinsic value of the magnetostriction +in the Fe–Ga alloys, measurements of λ100 were made from +room temperature to near absolute zero. (Note that the tem- +perature dependence of λ100 of pure Fe is unusual, possess- +ing two magnetostriction peaks, a small one between −173◦C +and 27◦C and a larger one near the Curie temperature.9)) The +magnetization of Fe1−xGax, 0 < x < 0.20, was found to +be simple and decreases only a few percent with temperature +from −268◦C to 27◦C.3,5) The satisfying result of a similar +small decrease in the magnetostriction over the same temper- +ature range is illustrated in Fig. 4. Here the magnetostric- +tions, (3/2)λ100, of bcc Fe and rapidly cooled Fe0.81Ga0.19 +are compared. Note the expanded scales. The large value +of (3/2)λ100 is intrinsic and not the result of an anomalous +temperature dependence. Angular dependences of the strains +of Fe0.81Ga0.19 at −269◦C and 22◦C are illustrated in Fig. 5. +The curves are excellent fits to λγ,2 cos2 θ dependence where +λγ,2 = (3/2)λ100. It should be pointed out that the satisfying +agreement between the magnetization and magnetostriction +temperature dependences was not observed for Fe0.76Ga0.24. +For this composition, while the magnetization still exhibits +a small increase with decreasing temperature from 27◦C to +−269◦C, the magnetostriction loses nearly half of its value +over the same range. See Fig. 6. Both furnace-cooled and +rapidly quenched samples were tested and revealed the same +temperature dependence. This anomalous behavior, which + +Magnetostrictive Properties of Galfenol Alloys Under Compressive Stress +883 +0 +50 +100 +150 +200 +250 +300 +350 +30 +35 +380 +390 +400 +410 +420 +430 +Fe0.81Ga0.19 +Fe +(3/2) +100 (x 10 +-6) +Temperature, T/K +Fig. 4 +Temperature dependence of the magnetostriction of quenched +Fe0.81Ga0.19 and Fe. Data for Fe0.81Ga0.19 was taken at 1200 kA/m. Satu- +ration magnetostriction for Fe was taken from Ref. 9. +0 +50 +100 +150 +200 +250 +300 +350 +400 +0 +50 +100 +150 +200 +250 +300 +350 +400 +450 +Fe0.81Ga0.19 +(3/2) +100 (x 10 +-6) +Angle + -269 C + +22 C +Fig. 5 +Angular dependence of the magnetostriction of Fe0.81Ga0.19 at +−269◦C and room temperature under an applied field of 1200 kA/m. +yields a peak in magnetostriction above room temperature, is +not understood. The normal temperature dependence for the +magnetostriction was again observed for the larger Ga con- +centration of x = 35%.10) +Because of the large magnetization (∼ 1.7 T) of these al- +loys, the magnetic fields required to achieve the large magne- +tostriction, even under large compressive stresses are small. +Figures 7 and 8 illustrate the dependence of the magnetostric- +tion vs. magnetic field for various compressive stresses up +to ∼ 95 MPa for furnace cooled Fe0.83Ga0.17 and quenched +Fe0.81Ga0.19 alloys, respectively. The 17% alloy saturates at a +slightly lower magnetic field and has a lower saturation mag- +netostriction. The magnetostriction increases by 18% as the +α-Fe structure is extended from 17% to 19% Ga. The satura- +tion magnetization for the 19% sample is slightly smaller than +the 1.75 T value for the 17% sample. However in all cases, +fields less than 32 kA/m are required to effectively achieve +saturation magnetostriction at stresses up to ∼ 100 MPa. The +effect of rapid quenching in obtaining the large strains in +Fe0.81Ga0.19 is shown in Fig. 9. In this figure are compared +150 +200 +250 +300 +0 +50 +100 +150 +200 +250 +300 +1.20 +1.25 +1.30 +1.35 +(a) + Furnace cooled + Quenched from 730 C +(3/2) +100 (x 10 +-6) +(b) + +Magnetization, M/T +Temperature, T/K +Fig. 6 +(a) Anomalous magnetostriction of Fe0.76Ga0.24 furnace-cooled at +∼ 10◦C per hour (■) and quenched into water from 730◦C (▲), and (b) +magnetization of furnace-cooled Fe0.76Ga0.24, between −269◦C and 22◦C. +Note the anomalous rise in magnetostriction with temperature near room +temperature is accompanied by a normal small decrease in magnetization. +-32 +-24 +-16 +-8 +0 +8 +16 +24 +32 +-2.0 +-1.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +1.5 +0 +50 +100 +150 +200 +250 +300 + + Magnetic Field, H/kA m +-1 +(b) +Applied Stress: + 10.3 MPa (1.5 ksi) + 27.5 MPa (4.0 ksi) + 50.5 MPa (7.3 ksi) + 73.5 MPa (10.6 ksi) + 96.5 MPa (14.0 ksi) +Magnetization, M/T +Fe0.83Ga0.17 +(a) +Magnetostriction (x 10 +-6) +Fig. 7 +Room temperature magnetic field dependences of (a) magnetostric- +tion along the [100] direction and (b) magnetization along the [100] direc- +tion of Fe0.83Ga0.17 for various compressive stresses. +magnetostriction curves vs field for stresses of 20 MPa and +50 MPa before and after quenching. A remarkable 30% in- +crease in magnetostriction is observed for H ≥ 32 kA/m. +5. +Magnetostriction of Fe–Ga–Ni and Fe–Ga–Mo Alloys +In this section we explore the effect of small amounts +of Ni on the magnetostriction of Fe1−xGax (x += +0.11 +and x = 0.16) alloys. +We have shown earlier that while + +884 +A. E. Clark, M. Wun-Fogle, J. B. Restorff and T. A. Lograsso +-32 +-24 +-16 +-8 +0 +8 +16 +24 +32 +-2.0 +-1.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +1.5 +0 +50 +100 +150 +200 +250 +300 +350 +400 +(b) + +Magnetization, M/T +Magnetic Field, H/kA m +-1 + 5.8 MPa (0.8 ksi) + 20.5 MPa (3.0 ksi) + 35.3 MPa (5.1 ksi) + 50.0 MPa (7.3 ksi) + 64.7 MPa (9.4 ksi) + 79.5 MPa (11.6 ksi) + 94.2 MPa (13.7 ksi) +Fe0.81Ga0.19 +(a) +Magnetostriction (x 10 +-6) +Fig. 8 +Room temperature magnetic field dependences of (a) magnetostric- +tion along the [100] direction and (b) magnetization along the [100] direc- +tion of Fe0.81Ga0.19 for various compressive stresses. +λ100 increases dramatically with substitutions of Ga into +bcc Fe, the small negative value of λ111 of bcc Fe re- +mains almost unchanged.1,2) Thus the magnetostriction of +the highly magnetostrictive Fe–Ga alloys is very anisotropic: +λ100/λ111 ∼= −10. Bozorth (using A. Schulze’s data), has +inferred that small percentages of Ni when added to Fe sub- +stantially reduces |λ111| and decreases the absolute magne- +tostrictive anisotropy.11) In Fig. 10, we compare the magne- +tostrictions along the [100] direction for Fe0.86Ga0.11Ni0.03 +and Fe0.814Ga0.16Ni0.026 furnace-cooled alloys. +For the +Fe0.86Ga0.11Ni0.03 sample, the decrease in the magnetostric- +tion inferred from the Ni-free sample is severe (∼ 40%). +For the Fe0.814Ga0.16Ni0.026 sample, the decrease is not as +large (∼ 15%). Our measurements show an increase of λ111 +from −16 ppm for the Ni-free sample of Fe0.87Ga0.131) to +∼ 0 for Fe0.811Ga0.162Ni0.127.12) Thus, while the magnitude +of λ111 is reduced by the addition of Ni, λ100 is also re- +duced. The substitution of small amounts of Ni for Fe also re- +duces the strains available under various compressive stresses. +Figure 11 illustrates the field dependence of the magnetostric- +tion Fe0.83Ga0.135Ni0.035 under various stresses up to 122 MPa. +This figure can be compared to Fig. 7 for the Ni free Fe–Ga +alloy. +We have also examined the effect of small Mo substitu- +tions for Fe in the Fe–Ga alloys. Hall has shown that λ100 +of bcc Fe increases moderately with the addition of 2%–4% +Mo.13) Figure 12 compares the magnetostrictions (3/2)λ100 +and (3/2)λ111 for Fe0.84Ga0.13Mo0.03 and Fe0.85Ga0.10Mo0.05, +respectively. +Here, while λ100 is again slightly reduced, +(3/2)λ111 increases in negativity to −36 ppm. This increase +-64 +-48 +-32 +-16 +0 +16 +32 +48 +64 +0 +50 +100 +150 +200 +250 +300 +350 +4000 +50 +100 +150 +200 +250 +300 +350 +400 +Stress = 20 MPa + + +Magnetostriction (x 10 +-6) +Magnetic Field, H/kA m +-1 + Fe0.813Ga0.187 furnace cooled + from 1000 C at 10 C/min. + Fe0.813Ga0.187 quenched from + 1000 C into water. + Fe0.81Ga0.19 furnace cooled + from 1000 C at 10 C/min. + Fe0.81Ga0.19 quenched from + 800 C into water. +(b) +(a) +Stress = 50 MPa +Fig. 9 +Room temperature magnetostriction along the [100] direction vs. +magnetic field of the Fe–Ga alloy with ∼ 19%Ga before and after quench- +ing for compressive stress of (a) 50 MPa and (b) 20 MPa. +0 +50 +100 +150 +0 +100 +200 +300 +400 +0 +50 +100 +150 +200 +250 +(a) +Fe0.814Ga0.16Ni0.026 +Fe0.86Ga0.11Ni0.03 +(b) +(3/2) +100 (x 10 +-6) +Angle +Fig. 10 +Angular dependence of room temperature magnetostriction along +the [100] direction at H = 1200 kA/m for (a) Fe0.86Ga0.11Ni0.03 and (b) +Fe0.814Ga0.16Ni0.026 furnace-cooled alloys. +in the negativity of λ111 is unprecedented. Fe–Ga–Mo has a +large negative anisotropy of approximately −5 (Note, how- +ever, because of the difficulty of synthesizing identical com- +positions of the [100] and [111] samples, the alloy com- +positions for the [100] and [111] strains are slightly differ- +ent.). The values of magnetostriction and magnetization un- +der compressive stresses for Fe.84Ga0.13Mo0.03 are illustrated +in Fig. 13. The saturation magnetizations are above 1.7 T in +all alloys. Compare Figs. 7, 11, and 13. + +Magnetostrictive Properties of Galfenol Alloys Under Compressive Stress +885 +20 +40 +60 +80 +100 +120 +140 +-32 +-24 +-16 +-8 +0 +8 +16 +24 +32 +-2.0 +-1.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +1.5 +Magnetostriction (x 10 +-6) + Magnetic Field, H/kA m +-1 +(a) +(b) +Fe0.83Ga0.135Ni0.035 + +Magnetization, M/T + 5.8 Mpa + 20.4 MPa + 49.7 MPa + 79.0 MPa + 108.3 MPa + 122.3 MPa +2.0 +Fig. 11 +Room temperature magnetic field dependences of (a) mag- +netostriction along the [100] direction and (b) magnetization of +Fe0.83Ga0.135Ni0.035 for various compressive stresses. +Fig. 12 +Angular dependence of the room temperature magnetostriction (a) +along the [100] direction for Fe0.84Ga0.13Mo0.03 and (b) along the [111] +direction for Fe0.85Ga0.10Mo0.05 for H = 1200 kA/m. Note: λ100 > 0; +λ111 < 0. +6. +Summary +Fe1−xGax, in its simple bcc structure, exhibits magne- +tostrictions, (3/2)λ100’s, as high as 395 ppm at room temper- +ature, much larger than common Fe and all other known 3d +transition metal alloys. Rapid quenching increases the solu- +bility of Ga in bcc Fe and thus the magnetostriction increases +with increasing x for 0.17 < x ≤ 0.19. Our observed max- +imum magnetostriction occurs in samples of ∼ 19% Ga in +Fe that were rapidly quenched into water from 800◦C. For +50 +100 +150 +200 +-32 +-16 +0 +16 +32 +-2.0 +-1.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +1.5 +(b) +(a) + +Magnetostriction (x 10 +-6) + Magnetic Field, H/kA m +-1 +Fe0.84Ga0.13Mo0.03 + +Magnetization, M/T + 5.6 MPa + 34.2 MPa + 48.4 MPa + 77.0 MPa + 91.3 MPa + 119.8 MPa +2.0 +Fig. 13 +Room temperature magnetic field dependence of (a) magnetostric- +tion along the [100] direction and (b) magnetization of Fe0.84Ga0.13Mo0.03 +for various compressive stresses. +x ≤ 0.19, we observed a normal (small) decrease in magne- +tostriction with temperature from −269◦C to 22◦C. On the +other hand, for both quenched and furnace-cooled samples of +x = 0.24 (near Fe3Ga), we find an anomalous increase in +magnetostriction over the same temperature range. Thus, a +peak in the magnetostriction occurs above room temperature. +This in not consistent with the characteristic normal decrease +in magnetization with temperature. λ111 remains negative at +room temperature in all binary and ternary alloys reported to +date. The addition of small amounts of Mo to the Fe–Ga alloy +increases the magnitude of λ111 while the addition of Ni de- +creases the magnitude of λ111 to near zero. +7. +Acknowledgments +This work was supported by the U.S. Office of Naval Re- +search, the Carderock Division of the Naval Surface Warfare +Center’s In-house Laboratory Independent Research Program +sponsored by the Office of Naval Research administered un- +der Program Element 0601152N, and the Office of Basic En- +ergy Sciences, Materials Sciences Division, of the U.S. De- +partment of Energy under Contract No. W-7405-ENG-82. +REFERENCES +1) A. E. Clark, J. B. Restorff, M. Wun-Fogle, T. A. Lograsso and D. L. +Schlagel: IEEE Trans. on Magn. 36 (2000) 3238–3240. +2) A. E. Clark, M. Wun-Fogle, J. B. Restorff, T. A. Lograsso, A. R. +Ross and D. L. Schlagel: Proc. Actuator 2000 Conference, (Bremen, +Germany, June 19–21, 2000). +3) A. E. Clark, M. Wun-Fogle, J. B. Restorff, T. A. Lograsso and J. R. +Cullen: IEEE Trans. on Magn. 37 (2001) 2678–2680. + +200 +(a) +[100Fe..Ga +Mo.. +10"5) +0.13 +0.03 +150 +100 +50 +0 +(b) +Mo +40 +20 +0° +100° +2009 +300° +400° +Angle886 +A. E. Clark, M. Wun-Fogle, J. B. Restorff and T. A. Lograsso +4) See for example: Binary Alloy Phase Diagrams, ed. in chief T. B. +Massalski, second edition, (ASM International, Materials Park, OH, +USA, 1990). +5) N. Kawamiya, K. Adachi and Y. Nakamura: J. Phys. Soc. Japan 33 +(1972) 1318–1327. +6) M. Wun-Fogle, J. B. Restorff, A. E. Clark and J. F. Lindberg: Proc. +Actuator 98 Conference, (Bremen, Germany, June 17–18, 1998). +7) R. Gersdorf: On Magnetostriction of Single Crystal of Iron and Some +Dilute Iron Alloys, Ph. D. Dissertation (University of Amsterdam, +1961). +8) R. C. Hall: J. Appl. Phys. 30 (1960) 816–819. +9) E. Tatsumoto and T. Okamoto: J. Phys. Soc. Japan 14 (1959) 1588. +10) J. R. Cullen, A. E. Clark, M. Wun-Fogle, J. B. Restorff and T. A. +Lograsso: J. Magn. and Magn. Matls. 226 (2001) 948. +11) See Bozorth, R. M., Ferromagnetism, D. Van Nostrand Company, +Princeton, N. J. (1959) p. 667. +12) J. B. Restorff, M. Wun-Fogle, A. E. Clark, T. A. Lograsso, A. R. Ross, +and D. L. Schlagel: Proceedings of the 46th Annual Conference on +Magnetism and Magnetic Materials, (Seattle, Washington, Nov. 12–16, +2001). +13) R. C. Hall: J. Appl. Phys. 31 (1960) 1037–1038. + diff --git "a/materials/content/tmp_files/MXene\345\261\217\350\224\275.pdf.txt" "b/materials/content/tmp_files/MXene\345\261\217\350\224\275.pdf.txt" new file mode 100644 index 0000000000000000000000000000000000000000..492a62755f740860705d367565b94901ac7d555e --- /dev/null +++ "b/materials/content/tmp_files/MXene\345\261\217\350\224\275.pdf.txt" @@ -0,0 +1,1235 @@ +Beyond Ti3C2Tx: MXenes for Electromagnetic +Interference Shielding +Meikang Han, Christopher Eugene Shuck, Roman Rakhmanov, David Parchment, Babak Anasori, +Chong Min Koo, Gary Friedman, and Yury Gogotsi* +Cite This: ACS Nano 2020, 14, 5008−5016 +Read Online +ACCESS +Metrics & More +Article Recommendations +* +sı +Supporting Information +ABSTRACT: New ultrathin and multifunctional electromag- +netic interference (EMI) shielding materials are required for +protecting electronics against electromagnetic pollution in the +fifth-generation networks and Internet of Things era. Micro- +meter-thin Ti3C2Tx MXene films have shown the best EMI +shielding performance among synthetic materials so far. Yet, +the effects of elemental composition, layer structure, and +transition-metal arrangement on EMI shielding properties of +MXenes have not been explored, despite the fact that more than +30 different MXenes have been reported, and many more are +possible. Here, we report on a systematic study of EMI +shielding properties of 16 different MXenes, which cover single- +metal MXenes, ordered double-metal carbide MXenes, and +random solid solution MXenes of M and X elements. This is the largest set of MXene compositions ever reported in a +comparative study. Films with thicknesses ranging from nanometers to micrometers were produced by spin-casting, spray- +coating, and vacuum-assisted filtration. All MXenes achieved effective EMI shielding (>20 dB) in micrometer-thick films. The +EMI shielding effectiveness of sprayed Ti3C2Tx film with a thickness of only ∼40 nm reaches 21 dB. Adjustable EMI shielding +properties were achieved in solid solution MXenes with different ratios of elements. A transfer matrix model was shown to fit +EMI shielding data for highly conductive MXenes but could not describe the behavior of materials with low conductivity. This +work shows that many members of the large MXene family can be used for EMI shielding, contributing to designing ultrathin, +flexible, and multifunctional EMI shielding films benefiting from specific characteristics of individual MXenes. +KEYWORDS: MXene, two-dimensional, electromagnetic interference shielding, conductivity, film +T +he arrival of commercial fifth generation (5G; 450 +MHz52 GHz) networks and a significant increase in +the number of wireless Internet of Things (IoT) +devices working across diverse frequency ranges made +communication stability and security without electromagnetic +interference (EMI) a critical requirement.1 Specifically, for +flexible wearable devices and miniaturized electronics with +complex architectures, such as electronic skins, sensors, +Bluetooth components, etc., it has become more challenging +to confront the selective jamming in different frequency +ranges.2 +Conventional metals (Ag, Cu, Ni, and others) provide +excellent barriers against electromagnetic waves, but metal foils +cannot be used in tiny microelectronic devices, and deposition +of thin metal films onto uneven device surfaces is not an easy +task.3,4 In addition, EMI shielding effectiveness (SE) of metal +coatings is limited by the skin depths of metals, decreasing +efficiency of ultrathin metal coatings and limiting their use in +wearable and portable electronic devices.5 Graphene, which +has low density, high strength, and excellent chemical stability, +was considered until recently as the most promising alternative +to replace metals for EMI shielding.6,7 In 2016, Ti3C2Tx (Tx +represents the surface termination groups) MXene films were +demonstrated to have outstanding EMI shielding capability at +micrometer-scale thickness, outperforming both metals and +carbon materials (graphene, carbon nanotubes, and carbon +fibers).8 Since then, tremendous efforts focused on optimizing +the EMI shielding performance of Ti3C2Tx and manufacturing +of Ti3C2Tx aerogels, Ti3C2Tx/polymer composites, and +Ti3C2Tx/carbon hybrids.9−14 MXenes are produced by a +Received: +February 14, 2020 +Accepted: +March 12, 2020 +Published: March 12, 2020 +Article +www.acsnano.org +© 2020 American Chemical Society +5008 +https://dx.doi.org/10.1021/acsnano.0c01312 +ACS Nano 2020, 14, 5008−5016 +Downloaded via TONGJI UNIV on December 16, 2020 at 03:03:42 (UTC). +See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles. + +TiyNb. +CTx +TisC2T, +60 +Nb.V, +TisCNT, +EMI SE (dB) +Ti,CT, +40 +CT +Mo,TiC,T +20 +Nb,CT +Mo,Ti2C3Tx +Nb4C,Tx +0 +100 +101 +102 +103 +104 +Conductivity (S cm-1ACNANO +Defining nanesdience +cnt nonteoinology +Plasmonic photocatalysis +withAu@Ptcore-shell +nanocrystals +COoVD19:Acallforphysical +stientistsandengineers +Fishigelatinfilms for flexible +electroluminescentdevicesscalable selective etching in acidic solutions, which leads to +−O, −OH, and −F surface terminations making them +solution-processable and available for various coating manu- +facturing techniques, including spray-coating, spin-casting, +inkjet printing, dip-coating, and interfacial assembly.15−18 In +addition, these functionalized surfaces facilitate adhesion and +bonding to substrates. All the above advantages indicate that +MXene is an alternative to the conventional metals and carbon +materials for EMI shielding. However, few studies have +explored the EMI shielding properties of MXenes other than +Ti3C2Tx, although more than 30 kinds of MXenes with +different compositions and structures have been published.19 +MXenes are typically synthesized from MAX phases, which +have the general formula Mn+1AXn (n = 1−4).20 There are +three common MAX phase structures, M2AX, M3AX2, and +M4AX3, where M represents the early transition metal, A is an +A-group element of group 13 to 15 of the periodic table, and X +is carbon and/or nitrogen.21,22 Consequently, MXenes +commonly have three structures: M2XTx, M3X2Tx, and +M4X3Tx, as shown in Figure 1. On the basis of the different +M and X elements, they can also be sorted as monometal +MXenes, double-metal (M′ and M″) MXenes, and double-X +(C and N carbonitrides) solid solution MXenes. Double-metal +MXenes exist in two different forms, depending on the +distribution of M′ and M″ metals: (1) solid solution MXenes, +where the M-sites are randomly occupied by two kinds of +metal atoms, and (2) ordered MXenes, where the M′ and M″ +elements are separated into different layers (Figure 1). +Currently, most MXenes show active surfaces and good +dispersion in water and some organic solvents due to the acid- +etching process.23 Therefore, the MXene family is a promising +platform for multifunctional EMI shielding coatings/devices. +Despite the fact that the EMI shielding properties of +Ti3C2Tx have been explored extensively, little is known about +the effects of layer structure, elemental composition, or metal +element arrangement on EMI shielding performance. Here, we +synthesized 16 different MXenes, comprising various elemental +compositions and structures that include the three primary +classes of MXenes, including monometal MXenes, ordered +double-metal MXenes, and random solid solution MXenes +(Figure 1). The EMI shielding performance of MXene films +with the thicknesses from nano to microscale was investigated. +Fitting of the results for different MXenes with a transfer +matrix model provides some insights into the relationships +between their EMI shielding properties and frequency/ +thickness/electrical conductivity. The produced very large set +of EMI shielding capability data for the MXene family enables +the rational design of EMI shielding coatings using different +MXenes. +RESULTS AND DISCUSSION +Sixteen different MAX phases, including mono-M (Ti2AlC2, +V2AlC2, and Nb2AlC2) and solid solution M2AX (TiyNb2−yAlC +and NbyV2−yAlC (y = 0.4, 0.8, 1.2, and 1.6)), M3AX2 (Ti3AlC2, +Ti3AlCN, and Mo2TiAlC2), and M4AX3 (Nb4AlC3 and +Mo2Ti2AlC2), were initially synthesized. All synthesis con- +ditions were summarized in Table S1. X-ray diffraction (XRD) +patterns of all MAX phases are shown in Figure S2. +Correspondingly, 16 different MXenes were obtained by the +selective removal of the Al layers from MAX phase precursors +using different etching and delamination processes. Figure 1 +shows the freestanding MXene films prepared via vacuum- +assisted filtration from the colloidal MXene solutions. MXenes +have widely differing optical properties, which can be visually +observed in the freestanding films.17,24 These films represent +the primary MXene categories: (A) mono-M MXenes. Ti2CTx +(green), Nb2CTx (gold), V2CTx (bronze), Ti3C2Tx (purple), +and Nb4C3Tx (dark gray); (B) double-X MXene. Ti3CNTx +(violet black); (C) double-M MXenes. Mo2TiC2Tx (silver +gray), Mo2Ti2C3Tx (silver), TiyNb2−yCTx and NbyV2−yCTx. Of +these, Mo2TiC2Tx and Mo2Ti2C3Tx are ordered with the +arrangement of Mo−Ti−Mo and Mo−Ti−Ti−Mo, respec- +tively.25 TiyNb2−yCTx and NbyV2−yCTx are random solid +solutions whose colors change based on their specific chemical +compositions. All the elemental ratios for TiyNb2−yCTx and +NbyV2−yCTx are based on compositions of solid solution MAX +phases, assuming no transition-metal loss during etching. +Complete delamination of all MXenes was confirmed by XRD +patterns of the MXene films (Figure S3). All of them show a +prominent (002) diffraction peak of MXene at 2θ ≈ 5.3−7.3°. +For all systems where the MAX is converted to MXene, the Al +layer is removed and replaced with the surface terminations, in +addition to water and intercalants between the layers, leading +to enlarged d-spacing compared to MAX phase precursors.26 +Hence, the observed d-spacing values are related to both the +thickness of the MXene flakes (n) and the intercalants. For +example, the (002) peaks d-spacing of Ti2AlC, Ti3AlC2, and +Nb4AlC3 is 6.7 Å (2θ = 13.1°), 9.2 Å (2θ = 9.60°), and 12.0 Å +Figure 1. MXenes synthesized in this work. Atomic structures +viewed from the [110] zone axis for three types of MXenes +(M2XTx, M3X2Tx, and M4X3Tx, where T is shown as a bridging +oxygen on the surface). They also can be sorted into four different +kinds: mono-transition-metal MXenes (1M), solid solution on M- +sites (SS-M), solid solution on X-sites (SS-X (C and N)), and +ordered double-transition-metal MXenes (2M). Digital images of +vacuum-filtrated freestanding films from different MXene colloidal +solutions. “1M” MXene films: M2CTx (Ti2CTx, Nb2CTx, and +V2CTx), Ti3C2Tx, and Nb4C3Tx; SS-M MXene films: TiyNb2−yCTx +and NbyV2−yCTx (y = 0.4, 0.8, 1.2 and 1.6); eight solid solutions on +M-sites MXenes with different ratios of metal elements were +synthesized in this work, here Nb1.2V0.8CTx film is as an example; +“2M” MXene films: Mo2TiC2Tx and Mo2Ti2C3Tx; “SS-X” MXene +film: Ti3CNTx. All the film diameters are 47 mm. +ACS Nano +www.acsnano.org +Article +https://dx.doi.org/10.1021/acsnano.0c01312 +ACS Nano 2020, 14, 5008−5016 +5009 + +M +Mono-transition metal MXenes (1M) +Solid solution on M-sites +Solid solution on X-sites (SS-X) +(SS-M) +Ordered double-transition metal MXenes (2M) +M' +M" +C +N +Tx (O, OH, F) +Ti2CTx +Nb2CTx +TisC2Tx +Ti,CNTx +Nb4C3Tx +1M +1M +1M +SS-X +1M +V2CTx +Nb1.2Vo.8CT +Mo2TiC2T, +Mo2Ti2C3Tx +1M +SS-M +2M +2M(2θ = 7.36°), respectively. After conversion of these MAX +phases into MXenes, the (002) peak d-spacing of Ti2CTx, +Ti3C2Tx, and Nb4C3Tx increased to 12.1 Å (2θ = 7.28°), 12.6 +Å (2θ = 7.00°), and 16.6 Å (2θ = 5.32°), respectively. The d- +spacing expansion from M2XTx, M3X2Tx to M4X3Tx can be due +to the increase in MXene flake thicknesses with increasing n, as +well as the presence of intercalants, such as tetramethylammo- +nium hydroxide (TMAOH), used to delaminate MXene, and +water molecules. +To investigate the EMI shielding performance of different +MXenes, spin-casting, spray-coating, and vacuum-assisted +filtration were used to fabricate MXene films of various +thickness. For MXene films with a thickness of ∼2 nm, we used +a 130 μm thick glass slide with negligible EMI SE as substrate +(Figure S4) and deposited Ti2CTx and Ti3C2Tx films by spin- +casting. Spray-coating was used to fabricate less than 150 nm +Ti2CTx and Ti3C2Tx films. Using vacuum-assisted filtration of +the colloidal MXene solutions, ∼1−15 μm thick MXene films +were prepared. All vacuum-filtered MXene films were free- +standing and flexible, which is attributed to the two- +dimensional (2D) morphology of the delaminated MXene +flakes (Figure 2a,b). All MXenes investigated here have similar +2D morphologies with varying lateral dimensions. As an +example, Ti2CTx flakes and films are presented in Figure 2a−d. +Both spray-coated films and filtered films show a similar +aligned layered structure in cross section (Figure 2c,d). It is +noteworthy that the spray-coated films have better (more +planar) stacking order than the filtered films owing to the +layer-by-layer drying process. This was further confirmed by +surface roughness analysis of spin-cast and spray-coated films +using optical and SEM images (Figure S5). The density of +spray-coated Ti3C2Tx films was ∼3.8 g cm−3, while that of +filtered Ti3C2Tx films was ∼2.7 g cm−3. The more uniform +stacked layers lead to a higher film density (better interflake +contact), which results in a higher electrical conductivity. +Figure 2e−g shows the total EMI SE (SET) values for all +MXene films of similar thicknesses (5 ± 0.3 μm) in the X-band +(8.2−12.4 GHz). All exhibit a quasilinear frequency-dependent +behavior whereby the SET decreases with increasing frequency. +This indicates that all studied MXenes have a similar +frequency-dependent conductive behavior. When the film +thickness is ∼5 μm, the total EMI SE values of all MXenes, +except Nb2CTx and Mo2TiC2Tx, exceed 20 dB, which means +more than 99% shielding efficiency in the whole X-band +Figure 2. Frequency-dependent EMI shielding performance of different MXenes. (a) In-plane ([002] zone axis) TEM image of a typical +MXene flake (Ti2CTx as an example). (b) Cross section TEM image of a double-layer T2CTx MXene flake. (c) SEM image of the cross +section of a spray-coated Ti2CTx film on glass substrate showing an aligned layered structure. (d) SEM image of the cross section of a +vacuum-filtered freestanding Ti2CTx film showing the well-aligned layers. (e) EMI shielding effectiveness of different MXene (M2XTx, +M3X2Tx, and M4X3Tx) films (5 ± 0.3 μm thick) in the 8.2−12.4 GHz range. EMI shielding effectiveness of solid solution (f) TiyNb2−yCTx and +(g) NbyV2−yCTx MXene films (5 ± 0.3 μm thick), showing the controllable change of EMI SE with the chemistry. (h) The average EMI SE +(SER, SEA, and SET) of different MXene films (5 ± 0.3 μm thick) in the 8.2−12.4 GHz range, showing the reflection and absorption +contributions. +ACS Nano +www.acsnano.org +Article +https://dx.doi.org/10.1021/acsnano.0c01312 +ACS Nano 2020, 14, 5008−5016 +5010 + +e +70 +50 +I SE (dB) +60 +TisC2Tx +40 +Ti,CT +30 + Ti1.6Nbo.4CTx +Ti,CNTx +EMI + Ti1.2Nbo.8CTx +50 +20 + Tio.8Nb1.2CTx +Ti,CTx +Tio.4Nb1.6CTx +500nm +(dB) +10 +Nb,CTx +40 +8 +9 +10 +1112 +13 +EMI SE +V,CTx +Frequency (GHz) +g +onm +Mo,Ti2CTx +30 +40 +Nb4CTx +(dB) +100nm +20 +V,CT +EMI SE +30 +Mo,TiC,T, + Nbo.4V1.6CT, +-Nb0.8V1.2CT, +20 +Nb1.2Vo.8CT, +10. +Nb,CTx + Nb1.6Vo.4CT, +um +10 +Nb,CT, +8 +9 10 11 12 13 +8 +9 +10 +11 +12 +13 +Frequency (GHz) +Frequency (GHz) +h +70 +SER +SEA +SET +M3X2Tx +60 +M2XTx +B 50 +Ti,Nb2-yCT, +Nb,V, +VCT +E 40 +S +M.X,T +30 +20 +10. +0 +人+ +人+ +C +C +C +C? +C3 +CS +Nb2 +Lo? +NbA +Ti2 +K1o! +Mo2 +Mo2(Figure 2e). It was observed that Ti-based MXenes (Ti2CTx, +Ti3CNTx, and Ti3C2Tx) with SET values above 40 dB have +better EMI shielding performance than other MXenes. In +particular, the performance of solid-solution MXenes +(TiyNb2−yCTx and NbyV2−yCTx) was compared with mono- +M M2CTx MXenes (Ti2CTx, Nb2CTx, and V2CTx), as shown +in Figure 2f,g. For TiyNb2−yCTx, their SET values are between +Ti2CTx and Nb2CTx, and they increase with the atomic ratio of +Ti. Similarly, the SET values of NbyV2−yCTx increase with the +amount of V. These trends indicate that the EMI shielding +properties of solid-solution MXenes are tunable by controlling +the M′/M″ ratios. We further studied the reflection and +absorption behavior of different MXenes. The average EMI SE +(SET, SER, and SEA) values in the X-band for different MXenes +with a similar thickness (5 ± 0.3 μm) is shown in Figure 2h. +MXenes with higher SET exhibit higher SER and SEA values +simultaneously, indicating that MXenes with higher EMI +shielding capability have a higher contribution from EM wave +reflection. Typically, the SET, SER, and SEA values for +TiyNb2−yCTx and NbyV2−yCTx decrease with increasing +amounts of Nb. For M3X2Tx MXenes, the average SET, SER, +and SEA of Ti3C2Tx reach 57.4, 17.5, and 39.9 dB, respectively, +while Ti3CNTx values are 49.7, 17, and 32.7 dB, respectively. +We synthesized Ti3C2Tx and Ti3CNTx using the same etching +(HF and HCl) and delamination (LiCl) methods. It suggests +that the mono-X MXene has better EMI shielding performance +than the double-X MXene (carbonitride). For ordered double- +M MXene, Mo2TiC2Tx, whose outer metal layers are Mo, +shows a much lower EMI shielding capability, in comparison +with Ti3C2Tx. +Because EMI shielding properties are sensitive to the +thickness of the shielding layers, we measured the EMI +shielding performance of all MXene films over a wide range of +thicknesses and applied a model based on the transfer matrix +method to validate them. Figure 3a illustrates the continuum +multilayer structure with a plane EM wave (far-field) at normal +incidence to single MXene layers with interlayered free space +(details of calculations are presented in Supporting Informa- +tion). Generally, Simon’s formula is the most common +approach to verify the EMI shielding performance for highly +conductive materials with thicknesses above the skin depth (d +≫ δ).27 However, it is not applicable to nanometer- and sub- +micrometer-thick MXene films. The advantage of the transfer +matrix modeling for ultrathin shielding layers, compared to +Simon’s formula that neglects multiple reflection between the +layers, is shown in Figure S1. In absence of a model capturing +the physics behind interactions of atomically thin materials and +EM waves,28 this looks like the best option available today. +Figure 3b shows typical nanometer-thick Ti3C2Tx (dark +green) and Ti2CTx (dark purple) films fabricated by spray +coating and transparent single-layer MXene films that were +spin-cast on glass substrates. The thickness of single-layer +Ti3C2Tx and Ti2CTx flakes was measured using atomic force +microscopy (AFM, Figure S6). The film thickness was +calculated based on the transmittance and the absorption +coefficients of Ti3C2Tx and Ti2CTx films at 550 nm wavelength +(details in Figure S7).29,30 The visible transmittance and color +change with the increasing film thickness of Ti3C2Tx and +Ti2CTx films are shown in Figure S8. Figure 3c shows the EMI +SET values of various nanometers-thick Ti3C2Tx and Ti2CTx +films at 10 GHz. As the Ti3C2Tx films thickness increases from +∼2 to 137 nm, the SET value increases from 1.4 to 33 dB. It is +highly significant that at ∼40 nm film thickness, the SET value +reaches 21 dB, indicating that more than 99% of the EM waves +Figure 3. Thickness-dependent EMI shielding performance of different MXenes. (a) Schematic illustration for transfer matrix model +simulating the interaction between the incident EM wave and MXene layers at normal incidence. (b) Digital images of spin-cast single-layer +MXene films (Ti3C2Tx and Ti2CTx), and spray-coated MXene films on glass substrates. (c) Simulated and experimental EMI SE values of +spin-cast and spray-coated films (Ti3C2Tx and Ti2CTx) with different thicknesses (<150 nm) at 10 GHz. (d) Simulated and experimental +EMI SE values of vacuum-filtered MXene (M2XTx, M3X2Tx, and M4X3Tx) films with thicknesses from ∼1 to 15 μm at 10 GHz. Simulated and +experimental EMI SE values of solid solution (e) NbyV2−yCTx and (f) TiyNb2−yCTx MXene films with different thicknesses at 10 GHz. The +electrical conductivity values for the simulation are listed (c−f). +ACS Nano +www.acsnano.org +Article +https://dx.doi.org/10.1021/acsnano.0c01312 +ACS Nano 2020, 14, 5008−5016 +5011 + +a +n1 +n2 +nj +nm +Et. +E +Et +Incident wave +Transmittedwave +E +Ei +E2 +Reflected wave +dMXene +d air +b +Spin-casting +Spray-coating +d 80 +e +Transfermatrix simulation +50 +1000 S cm +TisC2Tx +*10! +8500 S cm +70. +TisCNTx +250 S cm +tDrexel +ntDrexel +150 S cm +V2CTx +Ti,CTx +30 +90 s cm1 +E +Nbo.4V1.6CTx +V2CT, +60 s cm +60 +2700 S cm +S +.20 +Nbo.8V1.2CT, +M +25 S cm +TisC2Tx +1600 S cm +E 10 +Nb1.2Vo.8CTx +tDrexel +atDrexel +50 +Nb1.6Vo.4CTx +1000 S cm +0 +(dB) +Transfer matrix simulation +Nb2CTx +0 2 4 6 8 10121416182022 +SE +40 +250 S cm +Thickness (μm) +c +40 +Transfer matrix simulation +30 +60 +90 S cm +TisC2Tx +60 S cm +50 +00 +30 +Ti,CT, +14000 S cm-1 +Scm +20 +040 +600Scm +25 S cm +Ti2CTx +E20 +830 +110Scm +Ti1.6Nbo.4CTx +10 +Mo2Ti2C3Tx +S +M +10 +1600 S cm-1 +Ti1.2Nbo.8CTx +Nb4C,Tx +M20 +25Scm +10 +Tio.gNb1.2CTx +Mo2TiC2Tx +0 +0. +Tio.4Nb1.6CTx +Nb,CT, +0- +Transfer matrix simulation +Nb2CTx +0 +50 +100 +150 +200 +0 2 4 6 8 101214 161820 +0 2 4 6 8 10121416182022 +Thickness (nm) +Thickness (um) +Thickness (um)have been intercepted. For Ti2CTx films, at 94 nm film +thickness, the SET value reaches 13 dB, which is equivalent in +shielding to a ∼14 nm thick Ti3C2Tx film. The gray line shows +the simulated results using the transfer matrix model with the +electrical conductivity. The modeling results agree sufficiently +well with the experimental data. The SET values of Ti3C2Tx +and Ti2CTx films decrease with the increasing frequency in the +X-band (Figure S9a,b). Both SEA and SER values increase with +the increasing thickness (Figure S9c,d). +We further measured the total EMI SE values of all the +vacuum-filtered MXene films with thicknesses ranging from ∼1 +to 15 μm. As shown in Figure 3d, the SET values of all the films +show a nonlinear monotonic increase with the thickness, in +fairly good agreement with the simulated results from the +electrical conductivity values (gray line). This statistical result +demonstrates that the transfer matrix model is more suitable +for predicting the EMI shielding properties of thin films, as +opposed to Simon’s formula, which predicts a linear relation- +ship between thickness and EMI SE. Significantly, except +Nb2CTx, all studied MXene films achieved SET > 20 dB below +10 μm thickness, including M2XTx (Ti2CTx and V2CTx), +M3X2Tx (Ti3C2Tx, Ti3CNTx, and Mo2TiC2Tx), and M4X3Tx +(Nb4C3Tx and Mo2Ti2C3Tx). In particular, Ti3C2Tx, Ti3CNTx, +Ti2CTx, V2CTx, and Mo2Ti2C3Tx films show SET > 20 dB +below 2.5 μm film thickness. Compared to Nb2CTx, Nb4C3Tx, +which has a similar elemental composition, shows a higher +EMI SE at all film thicknesses. The same trend was found for +Ti2CTx and Ti3C2Tx. It indicates that higher conductivity and +EMI shielding can be obtained by increasing the number of +layers in the MXene structure. The EMI shielding properties of +two kinds of solid solution MXenes (NbyV2−yCTx and +TiyNb2−yCTx) were also measured and modeled at different +thicknesses. Compared to pure Ti2CTx, V2CTx, and Nb2CTx, +the SET values of NbyV2−yCTx monotonically increase with the +V content, likewise with TiyNb2−yCTx and Ti content (Figure +3e, f). Additionally, both show excellent EMI shielding +capabilities; TiyNb2−yCTx and NbyV2−yCTx films can achieve +SET > 20 dB below 4 and 8 μm film thicknesses, respectively. +This suggests that, in addition to Ti3C2Tx, many other MXenes +can be used as thin EMI shielding coatings, although the +performance of Ti3C2Tx is still the best, possibly due to the +optimized synthesis process that minimized the concentration +of defects and maximized the conductivity compared to initial +reports.13,14 This is important, as MXenes offer a variety of +colors and mechanical, electrical, and other properties, +enabling development of multifunctional EMI shielding films +and coatings. Furthermore, the EMI shielding performance is +demonstrated to be tunable by control of the chemistry of +solid-solution MXenes over a broad range of thicknesses and +compositions. The frequency-dependent EMI SE values of all +MXene films with different thicknesses can be found in Figure +S10. Notably, the electrical conductivity of the sprayed +Ti3C2Tx films is ∼14 000 S cm−1, compared to the vacuum- +filtered films (∼8500 S cm−1). This is attributed to better +alignment and denser packing of MXene layers in the sprayed +films (Figure S5). +To further understand the correlation between the electrical +conductivity and EMI SE of MXene films, we measured the +electrical conductivity of all vacuum-filtered MXene films using +the four-point probe method. As shown in Figure 4a, Ti-based +MXenes (Ti3C2Tx, Ti3CNTx, Ti2CTx, and Ti1.6Nb0.4CTx), +along with V2CTx, exhibit electrical conductivity higher than +1000 S cm−1, while Nb-based MXenes have relatively low +electrical conductivity. It is noteworthy that the conductivity of +MXenes varies with different synthesis processes and +precursors. For the solid solution MXenes studied, the +electrical conductivity of TiyNb2−yCTx and NbyV2−yCTx films +decreases with the increasing Nb content, consistent with the +change of EMI shielding performance. This indicates that both +the electrical conductivity and EMI shielding capability of +MXenes are adjustable via their chemical composition. +We further simulated the total EMI SE of MXene films with +a thickness of 5 μm using the transfer matrix modeling at the +frequency of 10 GHz and compared the modeled values to +experimentally measured electrical conductivity and SET. As +shown in Figure 4b, for MXenes with an electrical conductivity +of more than 100 S cm−1, the measured values are in close +agreement with the simulated results (gray line), showing a +nonlinear increase of SET with the increasing conductivity. +However, for MXenes with an electrical conductivity of less +than 100 S cm−1, the measured SET values are larger than +predicted. This mismatch between the experimental results and +theoretical calculations implies that the electrical conductivity +is not the only factor responsible for EMI shielding +performance of MXenes. On one hand, our theoretical +calculations using both the transfer matrix model and Simon’s +formula do not take into account dielectric polarization effect. +The dielectric interactions between the incident EM wave and +MXene layers are barely detected for those MXenes with +higher conductivity, since the strong interaction between +Figure 4. Conductivity-dependent EMI shielding performance of different MXenes. (a) The electrical conductivity of different vacuum- +filtered MXene films. (b) Comparison of EMI SE values between transfer matrix simulation and vacuum-filtered MXene films with a +thickness of ∼5 μm at 10 GHz, showing the relationship between EMI SE and the electrical conductivity of MXenes. +ACS Nano +www.acsnano.org +Article +https://dx.doi.org/10.1021/acsnano.0c01312 +ACS Nano 2020, 14, 5008−5016 +5012 + +a +b +-8570 +70 + cm-1) +TisC2Tx +9000 +M'2CTx +2712 +Transfer matrix simulation +M,M"2-yCTx +60 +Ti,Nb2-yCTx +M'sX2Tx +1610 +Ti,CNTx +Nb,V2-yCTx +S + 2700 +1370 +EMI SE (dB) +50 +Conductivity ( +M'4C,Tx +M'2M"2C,Tx +Ti2CTx +40 +1350 +998 +867 +V2CTx +30 +900 +518 +349 +Mo2TiC2T, +227 +Mo2Ti2C,Tx +108 +20 +Nb2CTx +450 +5 +5 +Nb4C3Tx +4 +5 +5 +10 +0 +- +Nb2CTx +Nb1.6Vo.4CTx +Mo2TiC2Tx +Tio.4Nb1.6CTx +Nb4C,Tx +Nbo.8V1.2CTx +Mo2Ti2C,Tx +V2CTx +Ti2CTx +TisC2Tx +Ti1.2Nbo.8CT, +TisCNT, +0 +1 +10 +100 +1000 +10000 +Conductivity (S cm-1)abundant free electrons of highly conductive MXene and +incident EM wave dominates the shielding behavior. However, +for MXenes with relatively low electrical conductivity, such as +Nb2CTx, the contribution of dielectric interactions between +EM wave and the materials to EM wave dissipation is not +negligible. MXenes have abundant surface groups and point +defects caused by the acid-etching process, as well as adsorbed +tetramethylammonium (TMA) ions between the layers.31,32 +They may experience strong polarization accompanied by +relaxation loss in the altered EM field, contributing to EM +wave absorption. On the other hand, both the transfer matrix +model and Simon’s formula do not take into account the +difference between AC conductivity and DC conductivity for +MXenes in the X-band. The frequency-dependent conductivity +behavior of different MXenes at gigahertz frequencies is not +known, although AC conductivity usually asymptotically +approaches the DC conductivity for metals and carbon +materials.33−35 Further investigation is required to quantify +the dielectric properties and AC conductivities of MXenes for +better understanding of the interaction between EM waves and +MXenes. Also, physical models accounting for nanometer-scale +heterogeneity of assembled MXene films should be developed. +We summarized the progress of metal-based hybrids and +carbon materials for EMI shielding in recent years and +highlighted the characteristic interval for the MXene films with +∼2 nm to 15 μm thickness based on the measured EMI +shielding properties of different MXenes (details in Table S3). +As depicted in Figure 5, it is apparent that most metal-based +hybrids and carbon materials require characteristic layer +thickness of ∼0.1−3 mm to achieve useful EMI SE values, +even though some progress has been made in developing thin +shielding layers using graphene and metal-based hybrids.36,37 +By contrast, MXenes are effective EMI shields over a broad +range of thicknesses, with a minimum thickness of only ∼40 +nm sufficient to achieve 20 dB EMI SE. Furthermore, higher +EMI shielding performance can be obtained through further +optimization of MAX phase and MXene synthesis, and through +modification of the surface chemistry. Many of the MXenes +studied in this work were not even previously reported, and no +optimization of synthesis and conductivity was done for the +majority of materials in this study. MXenes not only show +superiority with respect to thickness and the feasibility of +different coating techniques but also offer a variety of +candidates for manufacturing multifunctional EMI shielding +coatings coupled with the optical, mechanical, and other +properties of different MXenes. It is promising for design of +gradient multilayers with tunable EMI shielding properties +using the various combinations of different MXenes. More +broadly, the fact that numerous MXenes with different +compositions and structures are still unexplored shows the +possibility to extend the EMI shielding interval of MXenes +toward higher EMI shielding effectiveness and thinner layers. +CONCLUSIONS +In summary, 16 MXenes with different compositions and +M2XTx, M3X2Tx, and M4X3Tx structures have been synthe- +sized, delaminated, and used to make films of varying +thickness. This is the largest set of MXene compositions +compared in any single study, which demonstrates a wide +variety of MXene properties and ways to control them. These +samples offer an insight into effects of structure and +composition on properties of MXenes. We have shown that +the investigated MXene films, ranging from 40 nm to several +micrometers in thickness, are promising as thin EMI shielding +coatings. A transfer matrix model has been shown to fit the +EMI shielding performance of MXenes, at least for the most +conducting members of the MXene family. Since MXenes have +a wide variety of useful properties, these results also build a +platform for developing multifunctional materials with tunable +EMI shielding properties using various MXenes. +METHODS +Materials. For synthesis of the MAX phase precursors, Ti (99.5%, +−325 mesh, Alfa Aesar), Al (99.5%, −325 mesh, Alfa Aesar), V +(99.5%, −325 mesh, Alfa Aesar), Nb (99.99%, −325 mesh, Beantown +Chemical), AlN (98%, 10 μm, Aldrich), C (graphite, 99%, −325 +mesh, Alfa Aesar), TiC (99.5%, typically 2 μm, Alfa Aesar), and Mo +(99.9%, −250 mesh, Alfa Aesar) powders were used. For top- +ochemical synthesis to MXene, hydrochloric acid (HCl, 36.5−38%, +Fisher Chemical), hydrofluoric acid (HF, 48.5−51%, Acros +Organics), lithium chloride (LiCl, 99%, Acros Organics), lithium +fluoride (LiF, 98.5%, Alfa Aesar), and tetramethylammonium +hydroxide (TMAOH, 25 wt %, Acros Organics) were used directly +without further purification. +Synthesis of MAX Powders. Prior to synthesis, all precursor +powders (Table S1, Supporting Information) were mixed with 10 mm +zirconia balls in a 2:1 ball/powder ratio. The mixture was placed into +plastic jars, then ball milled at 50 rpm for 18 h. The powder mixture +was then transferred to alumina crucibles, which were placed into a +high-temperature furnace (Carbolite Gero). Ar was flowed through +the furnace for 1 h prior to heating, then was continually flowed +through the furnace during synthesis. For all samples, the heating rate +was 3 °C min−1. Depending on the chemistry and composition, +different temperatures and holding times were used (details in Table +S1). After they cooled, the samples were milled using a TiN-coated +milling bit, then were sieved to less than 75 μm. +Synthesis of MXenes. The synthesis conditions of 16 MXenes +are summarized in Table S2 in Supporting Information. The synthesis +procedures in detail for different MXenes are as follows. +Synthesis of Ti3C2Tx, Ti2CTx, and Ti3CNTx. Ti3C2Tx, Ti2CTx, and +Ti3CNTx were synthesized by the selective etching of the +corresponding MAX phase powders (Ti3AlC2, Ti2AlC, and Ti3AlCN) +with HF and HCl. Typically, 12 mL of HCl, 2 mL of HF, and 6 mL of +deionized (DI) water were mixed first. After that, 1 g of MAX powder +was added to the solution, and then the mixture was kept stirring for +24 h at room temperature. After the etching was done, the reacted +Figure 5. Performance of different materials toward EMI shielding. +A comparison of the total EMI SE thickness for the MXene family +with metal-based hybrids and carbon materials (graphene, carbon +nanotubes, carbon fibers, and others). The gray dashed line is the +baseline for the commercial EMI shielding materials (SET > 20 +dB). A direct comparison of the electrical conductivity and EMI +shielding properties of different MXenes with other materials from +the literature over the last five years can be found in Table S3. +ACS Nano +www.acsnano.org +Article +https://dx.doi.org/10.1021/acsnano.0c01312 +ACS Nano 2020, 14, 5008−5016 +5013 + +140- +Predicted region +MXenes +120- +Metal-based hybrids +Graphene +100 +(dB) +Carbon nanotubes +Other carbon materials +SE( +80- +EMI +60. +40 +20 +O +0:0:0800 +0. +10-3 +10-2 +10-1 +100 +101 +102 +103 +Thickness (um)solution was centrifuged at 3500 rpm for 2 min. This washing process +was repeated until the pH value is greater than 6. The centrifuged +sediment was added into a solution of LiCl in DI water with a +concentration of 20 mg mL−1. The mixture was stirred for 4 h at room +temperature. After that, the solution was centrifuged at 3500 rpm for +10 min. The centrifugation was repeated, until the supernatant +became black. The black solution was collected and centrifuged at +7500 rpm for 3 min. The final supernatant was used for the +preparation of MXene films. +Synthesis of Mo2TiC2Tx, Mo2Ti2C3Tx, and Nb4C3Tx. Mo2TiC2Tx, +Mo2Ti2C3Tx, and Nb4C3Tx were synthesized by the selective etching +of the corresponding MAX phase powders (Mo2TiAlC2, Mo2Ti2AlC3, +and Nb4AlC3) with HF. Typically, 1 g of MAX phase powder was +added into 20 mL of HF with stirring at 50 °C. The etching times for +Mo2TiAlC2, Mo2Ti2AlC3, and Nb4AlC3 were 48 h, 96 h, and 7 d, +respectively. After they were etched, the reacted solution was washed +with DI water through centrifugation (3500 rpm, 2 min) several +times, until the pH value was greater than 6. After that, the +centrifuged sediment was added into a solution with 0.5 g of TMAOH +and 20 mL of DI water and stirred for 12 h at room temperature. The +mixture was centrifuged several times with DI water at 9000 rpm for +10 min, until the pH value is less than 8. At last, the solution was +centrifuged at 3500 rpm for 10 min. The black supernatant was the +dispersion of delaminated MXene flakes in water. +Synthesis of TiyNb2−yCTx. TiyNb2−yCTx (y = 0.4, 0.8, 1.2, and 1.6) +was synthesized by the selective etching of TiyNb2−yAlC powders with +a mild method (LiF + HCl). Typically, LiF was dissolved in a mixture +of 5 mL of DI water and 15 mL of HCl. After that, 1 g of TiyNb2−yAlC +powder was added to the etchant solution gradually and stirred for 48 +h at 35 °C. Following the reaction, the solution was centrifuged with +DI water at 3500 rpm for 2 min. This washing procedure was repeated +several times, until the pH was greater than 6. Finally, the delaminated +TiyNb2−yCTx was obtained when the stable black dispersion formed. +The supernatant was attained by centrifugation at 7500 rpm for 3 +min. +Synthesis of NbyV2−yCTx. NbyV2−yCTx (y = 0, 0.4, 0.8, 1.2, 1.6, and +2) was synthesized by the selective etching of the corresponding +NbyV2−yAlC powders with HF. The etching and delaminated +processes are the same as those of Mo2TiC2Tx, except the etching +temperature is 35 °C. +Fabrication of MXene Films. The glass slides (24 × 40 × 0.13 +mm, Corning) were used as the substrate for spin-casting and spray- +coating. Before depositing MXene flakes, the substrates were cleaned +with a bath sonication in ethanol and DI water sequentially and then +dried with compressed air. After that, the substrates were cleaned by +oxygen plasma with a power of 100 W for 5 min at a gas flow of 3 +sccm. +Spin-Casting. The ∼2 nm thick MXene films (Ti3C2Tx and +Ti2CTx) were prepared using spin-casting method. Typically, the +colloidal solution with a concentration of 1 mg mL−1 was spin-cast +onto the substrate with spin rate of 1000 rpm for 30 s, followed by +5000 rpm for 10 s. +Spray-Coating. The MXene films (Ti3C2Tx and Ti2CTx) with the +thickness of less than 150 nm were prepared using spray-coating +method. Typically, the colloidal solution with a concentration of 1 mg +mL−1 was sprayed onto the substrate manually. The spray flow was +controlled to avoid droplets agglomerating on the substrate. Air flow +from a dryer above the substrate provided sufficiently fast drying after +each spray. +Vacuum-Assistant Filtration. The freestanding MXene films were +prepared by vacuum-filtering the colloidal MXene dispersion on a +polypropylene film with a thickness of 25 μm (3501 Coated PP, +Celgard LLC) and followed by drying in vacuum. The concentration +of the colloidal MXene dispersion was calculated based on the weight +of the film and the filtered volume of the dispersion. +All films were dried in a vacuum oven at 70 °C for 12 h before +testing. +Characterization. The morphology and structure of MXene +flakes and films were observed using a three-dimensional (3D) laser +scanning confocal microscopy (Keyence, VK-X1000), scanning +electron microscopy (SEM; Zeiss Supra 50VP), and transmission +electron microscopy (TEM; F-30, FEI-Tecnai). The thickness of +MXene flakes was measured by atomic force microscopy (AFM; +Multimode 8, Bruker) with a Si tip (Budget Sensors Tap300Al-G; f 0 = +300 kHz, k = 40 N m−1) in a standard tapping mode in air. XRD +patterns were recorded with Ni-filtered Cu Kα radiation (λ = 1.54 Å; +Miniflex, Rigaku) operated at 40 kV and 15 mA. The electrical +conductivity of MXene films was measured by a four-point probe +instrument (ResTest, Jandel Engineering Ltd.) with a probe distance +of 1 mm. The thickness of vacuum-filtrated films was measured by a +micrometer with 0.1 μm accuracy. UV−Vis spectroscopy was +performed from 300 to 1000 nm (Evolution 201, Thermo Scientific, +10 mm path length quartz cuvette), and the transmittance value at +550 nm was used to quantify the thickness of the spin-cast and spray- +coated films. Scattering parameters of the films were measured using a +vector network analyzer (8720ES, Agilent) with a WR-90 rectangular +waveguide in the frequency range of 8.2−12.4 GHz. +ASSOCIATED CONTENT +* +sı Supporting Information +The Supporting Information is available free of charge at +https://pubs.acs.org/doi/10.1021/acsnano.0c01312. +Additional experimental conditions and results, includ- +ing the tables of synthesis conditions for MAX phases +and MXenes; transfer matrix modeling; XRD patterns of +MAX phases and MXenes; surface morphologies of +MXene films; AFM images of MXene flakes; the +thickness calculation of MXene films, and corresponding +UV−vis spectra; EMI SE of all MXene films. Table on +the comparison of EMI shielding performance of various +materials (PDF) +AUTHOR INFORMATION +Corresponding Author +Yury Gogotsi − A. J. Drexel Nanomaterials Institute and +Department of Materials Science and Engineering, Drexel +University, Philadelphia 19104, Pennsylvania, United States; +orcid.org/0000-0001-9423-4032; Email: gogotsi@ +drexel.edu +Authors +Meikang Han − A. J. Drexel Nanomaterials Institute and +Department of Materials Science and Engineering, Drexel +University, Philadelphia 19104, Pennsylvania, United States; +orcid.org/0000-0003-3309-988X +Christopher Eugene Shuck − A. J. Drexel Nanomaterials +Institute and Department of Materials Science and Engineering, +Drexel University, Philadelphia 19104, Pennsylvania, United +States; +orcid.org/0000-0002-1274-8484 +Roman Rakhmanov − A. J. Drexel Nanomaterials Institute and +Department of Materials Science and Engineering and +Department of Electrical and Computer Engineering, Drexel +University, Philadelphia 19104, Pennsylvania, United States +David Parchment − A. J. Drexel Nanomaterials Institute and +Department of Materials Science and Engineering, Drexel +University, Philadelphia 19104, Pennsylvania, United States +Babak Anasori − A. J. Drexel Nanomaterials Institute and +Department of Materials Science and Engineering, Drexel +University, Philadelphia 19104, Pennsylvania, United States; +Department of Mechanical and Energy Engineering, Integrated +Nanosystems Development Institute, Purdue School of +Engineering and Technology, Indiana University−Purdue +University Indianapolis, Indianapolis 46202, Indiana, United +States; +orcid.org/0000-0002-1955-253X +ACS Nano +www.acsnano.org +Article +https://dx.doi.org/10.1021/acsnano.0c01312 +ACS Nano 2020, 14, 5008−5016 +5014 + +Chong Min Koo − Materials Architecturing Research Centre, +Korea Institute of Science and Technology, Seoul 02792, Korea; +orcid.org/0000-0002-8674-9236 +Gary Friedman − Department of Electrical and Computer +Engineering, Drexel University, Philadelphia 19104, +Pennsylvania, United States +Complete contact information is available at: +https://pubs.acs.org/10.1021/acsnano.0c01312 +Author Contributions +M.H., D.P., and R.R. synthesized MXenes and fabricated +MXene films. C.S. and B.A. synthesized MAX phases. M.H. +performed the electromagnetic measurement and analyzed the +data. G.F. and C.K. contributed to the prediction modeling. +M.H. wrote the manuscript, with input from all coauthors. Y.G. +initiated the study and supervised the work. +Notes +The authors declare no competing financial interest. +ACKNOWLEDGMENTS +We thank Dr. N. Kurra, Dr. X. Xiao, and Dr. M. Shekhirev for +the MXene film morphology studies, as well as M. Anayee for +computer programming. We are grateful to Murata Manu- +facturing Co., Ltd, Japan, for providing equipment used for +testing EMI shielding properties of MXenes. C.K. acknowl- +edges the financial support from the National Research +Foundation +of +Korea +(2017R1A2B3006469 +and +2019M3D1A2014004). +REFERENCES +(1) Médard, M. Is 5 Just What Comes after 4? Nat. Electron. 2020, 3, +2−4. +(2) Huang, W.; Zhou, J.; Froeter, P. J.; Walsh, K.; Liu, S.; Kraman, +M. D.; Li, M.; Michaels, J. A.; Sievers, D. J.; Gong, S.; Li, X. 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Sci. 2019, 54, 7165−7179. +ACS Nano +www.acsnano.org +Article +https://dx.doi.org/10.1021/acsnano.0c01312 +ACS Nano 2020, 14, 5008−5016 +5016 + diff --git "a/materials/content/tmp_files/\344\270\215\345\220\214\346\236\201\345\214\226\346\215\237\350\200\227\346\250\241\345\274\217\345\220\270\346\263\242.pdf.txt" "b/materials/content/tmp_files/\344\270\215\345\220\214\346\236\201\345\214\226\346\215\237\350\200\227\346\250\241\345\274\217\345\220\270\346\263\242.pdf.txt" new file mode 100644 index 0000000000000000000000000000000000000000..91331d017f8f3b50559ccac1ab43a4aa0ea9ecbd --- /dev/null +++ "b/materials/content/tmp_files/\344\270\215\345\220\214\346\236\201\345\214\226\346\215\237\350\200\227\346\250\241\345\274\217\345\220\270\346\263\242.pdf.txt" @@ -0,0 +1,1406 @@ +www.afm-journal.de +© 2022 Wiley-VCH GmbH +2112294 (1 of 10) +ReseaRch aRticle +Synergistic Polarization Loss of MoS2-Based Multiphase +Solid Solution for Electromagnetic Wave Absorption +Zhenguo Gao, Zhenhui Ma, Di Lan, Zehao Zhao, Limin Zhang, Hongjing Wu,* and +Yanglong Hou* +Given tunable hybridization structures in solid solutions, fascinating electro- +magnetic (EM) properties can be achieved for regulating EM wave (EMW) +absorption. Herein, a novel metal–organic cooperative interactions method is +proposed to manipulate the vacancy, interstitial, substitutional, and heteroin- +terface structures in molybdenum disulfide (MoS2) solid solution simultane- +ously, thence meeting the synergistic polarization loss on various point and +face sites. Assisted by the coordination between Cu2+ and polydopamine (PDA), +the effect of Cu modification on MoS2 is highly improved, which further lead to +polarization loss on S vacancy, interstitial Cu, substitutional N, and heteroint- +erface between carbon and MoS2. Contributing to the synergetic effect among +multiple polarizations, the Cu/C@MoS2 solid solution exhibit ultrahigh EMW +absorption performance, of which EMA with twice PDA delivers the effective +absorption bandwidth of 7.12 GHz and minimum reflection loss of −48.22 dB +(2.5 mm). The energy attenuation of Cu/C@MoS2 improved almost 266.7% +and 222.2% than C@MoS2 and Cu@MoS2, respectively. Finally, this work +reveals the structural dependency of solid solution materials of EMW absorp- +tion and establishes an entirely new polarization loss model. +DOI: 10.1002/adfm.202112294 +Z. G. Gao, D. Lan, Z. Zhao, L. Zhang, H. J. Wu +MOE Key Laboratory of Material Physics and Chemistry under +Extraordinary +Northwestern Polytechnical University +Xi’an 710072, China +E-mail: wuhongjing@mail.nwpu.edu.cn +Z. Ma +Department of Physics +Beijing Technology and Business University +Beijing 100037, China +Y. Hou +Beijing Key Laboratory for Magnetoelectric Materials and Devices +(BKLMMD) +Beijing Innovation Center for Engineering Science and Advanced +Technology (BIC-ESAT) +School of Materials Science and Engineering +Peking University +Beijing 100871, China +E-mail: hou@pku.edu.cn +The ORCID identification number(s) for the author(s) of this article +can be found under https://doi.org/10.1002/adfm.202112294. +protection, civilian electronic information +devices, military stealth weapons, etc.[1] +Great efforts have been devoted to pre- +pare dielectric loss type EMAs thanks to +its complex loss mechanism, especially +the multiple polarization relaxation loss.[2] +Conventionally, the dipoles among what- +ever “points” or “faces” of all polar dielec- +tric materials undergo inelastic thermal +motion under the external alternating +Electromagnetic (EM) wave (EMW), which +will induce the attenuation of EM energy +by thermal dissipation.[3] Unfortunately, +as an essential prerequisite for theoretical +analysis of polarization loss mechanism, +the construction of various polarization +models with corresponding EM respon- +sive properties for EMW absorption have +not been investigated comprehensively. +In order to study the nature of polari- +zation phenomenon in depth, it has been +investigated by some classical experi- +ments and theories. Typically, polarization +will be occurred in the forms of “orientation–deorientation” of +polarization sites or “aggregation–dispersion” of charges on +whatever kinds of heterostructures.[4,5] Afterward, the repeated +and periodical reaction of polarization centers under incident +EM field could result in energy forms transformation (EM- +thermal) and energy attenuation (thermal).[6,7] Thence, the +EMW could be attenuated during the polarization relaxation +progress. In general, the polarization process during EMW +absorption occurs not only on face–face (heterointerface) and +point–point (dipole), but also on point–face heterogeneous +sites. The polarization taking place on face–face and point– +point sites was typically regarded as interfacial polarization and +dipoles polarization, respectively, while interaction between +point and face can also initiate polarization relaxation.[8] +However, the respective and simultaneous manipulation of +polarization characteristic toward different kinds of polariza- +tion sites for controllable polarization loss has not been sum- +marized. In view of the discussion above, it is of extremely +indispensable to design a series of research medium materials +with tunable polarization sites structures, thus exploring the +polarization loss mechanism. +The solid solutions have been paid ever growing attention +as promising candidates for EMAs. Typically, there are three +types of solid solutions: interstitial solid solutions, substitu- +tional solid solutions, and vacancy solid solutions.[9] For the +1. Introduction +Electromagnetic wave absorption materials (EMAs) are espe- +cially attractive owing to universal application in human health +Adv. Funct. Mater. 2022, 2112294 + +Q +Check for updateswww.afm-journal.de +www.advancedsciencenews.com +2112294 (2 of 10) +© 2022 Wiley-VCH GmbH +interstitial solid solutions, solute elements (guest elements) +always enters the solvent lattice interstitially.[10] While for +the substitutional solid solutions, a part of solvent elements +(parent elements) should be replaced by solute elements.[11] +Additionally, the replacement requires the similar atomic +radius and comparable electronegativity between solvent +and solute elements according to Hume-Rothery rules.[12] +For vacancy solid solutions, they are usually resulted from +the missing occupancy on some sites.[13] Moreover, all of the +three type of solid solutions can induce “point” site for polari- +zation. In this case, on the one hand, abundant point–point +and point–face polarization centers could be built if the solid +solution is formed by 2D materials. On the other hand, a mul- +tiphase solid solution hybrid will therefore build face–face +polarization site. +Representatively, molybdenum disulfide (MoS2) is one +of the most typical 2D transition-metal dichalcogenides +stacked by S–Mo–S layers by van der Waals interactions.[14] +More importantly, MoS2 is greatly available to form solid +solution with tunable hybrid structure. According to crystal- +field theory, it has been studied that the incompletely filled +Mo4d orbital in octahedral symmetry give rise to narrower +band gap than that in hexagonal symmetry.[15] Therefore, +much attention has been paid to rearrange the S atoms +in 2H-MoS2 bulks to render octahedral coordination. For +instance, it is confirmed that alkali metals and some transi- +tion metals can be effective interstitial metals to promote +the coordination conversion, of which extra electrons can be +donated into d orbital of Mo, leading to numerous polariza- +tion centers in interstitial solid solution.[16] For substitution, +S atoms can be easily lost and replaced by some atoms with +similar atomic hybrid structure under violently thermal +motion, such as sp2 N atom.[17] In this case, taking account +the MoS2 multiphase solid solution into the research object, +it is greatly feasible to study the effective polarization loss +via synergistic regulation of face–face, point–point, and +point–face polarization. +Herein, we analyzed the polarization loss mechanism +through a series of MoS2-based solid solution with tunable +interstitial, substitutional, and vacancy structure fabricated +by a novel metal–organic cooperative interaction method for +the first time. Simultaneously, the synergistic effect of point– +point, point–face, and face–face polarization on EMW absorp- +tion was analyzed systematically. First of all, the EM proper- +ties of defected MoS2, Cu intercalated MoS2 (Cu@MoS2), and +carbon modified MoS2 (C@MoS2) were analyzed to study +the point–point, point–face, and face–face polarization loss, +respectively. Afterward, the metal–organic cooperative inter- +actions between Cu and polydopamine (PDA) led to the con- +struction of Cu intercalated MoS2 with carbon modification +(Cu/C@MoS2), which achieved synergistic manipulation of +these three types of polarization. Finally, assisted by charge +migration performance of semiconductors according to the +electrochemical analysis, this research built the polarization +models for MoS2 and even solid solution structural EMAs. In +this case, this work not only provides a reliable case for fabri- +cating EMAs with tunable polarization sites but also performs +as a valuable reference for theoretical analysis about polariza- +tion loss mechanism. +2. Result and Discussion +2.1. Preparation of MoS2-Based Multiphase Solid Solutions +To achieve the controllable manipulation of interstitial, substi- +tutional, vacancy and heterointerface structures in one kind of +solid solution, 2H-MoS2 is employed as crude material in this +work. Briefly, a novel metal–organic cooperative interactions +method was utilized to fabricate MoS2-based multiphase solid +solution (Figure 1). As is well known, PDA has been regarded +as a kind of typical great adhesive polymer, attributed to abun- +dant chemically active groups, such as hydroxyl, amino, and +imino groups.[18] Additionally, due to advisable atomic radii of +Cu (0.71 Å) as well as the strong coordination ability between +PDA and Cu2+,[19] the Cu2+, MoS2, and PDA were employed +as the guest element, parent element, and auxiliary adhesive, +respectively. +The interstitial, substitutional, and vacancy structures pri- +marily appeared in Cu intercalation, N substitution, and S +defects, respectively. In detail, Cu was inserted into the MoS2 +films by electrostatic interaction to build a point–face hybrid +structure (Cu@MoS2). The PDA was also induced into the +MoS2 layers but was further triggered to polymerize (PDA@ +MoS2) and form a 2D carbon, thus building a carbon–MoS2 +face–face hybrid structure (C@MoS2). Moreover, a part of S +atoms were substituted by N atoms during high-temperature +annealing process by thermal motion, which introduced point +modification on 2D surface. In order to eliminate the interfer- +ence of hybrid materials, pure MoS2 was utilized to study the +effect of point–point hybrid structure by introduce S vacancy.[20] +To analyze the synergistic effect of all kinds of polarization +sites above, they were combined into multiphase solid solution +(Cu/C@MoS2) with interstitial, substitutional, vacancy, and het- +erointerface structures. First, Cu2+ can be effectively embedded +in MoS2 favoring by the strong electrostatic force of the Cu2+–O +(PDA) coordination bond, leading to the construction of Cu2+ +interstitial MoS2 solid solution (Cu/PDA@MoS2). Then, the +high-temperature annealing of Cu/PDA@MoS2 not only pro- +moted the replacement of S by N in MoS2, which formed the +substitutional solid solution, but also introduced vacancy in +solid solution via losing S occupancy in MoS2 unit cells. Addi- +tionally, for further determination of the effect of PDA assis- +tance, tailoring mass ratios (0.5, 1.0, and 2.0) of PDA were +utilized in PDA@MoS2 and Cu/PDA@MoS2, as more PDA +always induced more interstitial and carbon phase. In this case, +diverse heterogeneous structure could be built and combined +by multiple “point” (vacancy, interstitial, and substitutional) +and “face” (MoS2, carbon) sites. +2.2. Effect of Vacancy, Interstitial, and Substitutional Solid +Solution Structure on EMW Absorption +Above all, the vacancy, interstitial, and substitutional struc- +tures in MoS2, Cu@MoS2, and C@MoS2 were characterized to +analyze the effect of relative polarization properties on EMW +absorption. As is shown in Figure 2a,b, no other character- +istic peaks can be observed except MoS2 [P63/mmc] (PDF, card +no. 37-1492) in X-ray diffraction (XRD) patterns of all these +Adv. Funct. Mater. 2022, 2112294 + +www.afm-journal.de +www.advancedsciencenews.com +2112294 (3 of 10) +© 2022 Wiley-VCH GmbH +samples,[21] which can be indispensable prerequisites for ana- +lyzing polarization sites. Beforehand, it is worthwhile to claim +that there are not only 2H-MoS2 but also 1T-MoS2. In general, +there is only thermodynamically stable 2H-MoS2 in nature +mines, while 1T-MoS2 can only be produced artificially by +either physical or chemical approaches.[22] The difference in S +atoms arrangement make the electrical conductivity of 1T-MoS2 +107 higher than 2H-MoS2, so that the coexistence of these two +kinds of coordination atoms in solvent MoS2 will be able to reg- +ulate the semiconductor properties. +The vacancy, interstitial, and substitutional structures were +first determined by the XRD patterns and X-ray photoelectron +spectroscopy (XPS) spectra. The pure MoS2 was taken into +account for vacancy analysis. The absolutely pure crystalline +structure confirmed the definite role of defect polarization in +pure MoS2. In XPS spectra (Figure S2a,d, Supporting Infor- +mation), the Mo 3d and S 2p spectra of pure MoS2 displayed a +typical 2H-MoS2 structure, of which the peaks at 232.3, 229.2, +163.3, and 162.0 eV can be associated with 2H Mo 3d3/2, 2H Mo +3d5/2, 2H S 2p1/2, and 2H S 2p3/2, respectively,[23] indicating that +vacancy had almost no influence on coordination structure of +MoS2. In detail, the quantified vacancy structure of pure MoS2 +was analyzed by XRD Rietveld refinement (Figure S14a and +Table S3, Supporting Information). After calculating the loss +ratio of 4f occupancy by S atoms, it is illustrated that there were +almost 35.3% vacancies in pure MoS2 lattice (Table S4, Sup- +porting Information). +For interstitial solid solution, the Cu intercalation was +detected in Cu@MoS2 by XPS and XRD analysis. On the one +hand, the appearance of Cu peak of Cu@MoS2 at about 395 eV +in whole region XPS spectra (Figure S1, Supporting Informa- +tion) also verified the modification of Cu ions in MoS2.[24] As +is shown in Figure 2d, both the Cu 2p1/2 and Cu 2p3/2 spectra +can be deconvoluted into three parts of Cu2+ (954.9, 934.8 eV), +Cu+ (951.9, 932.7 eV), and satellite (963.7, 940.0 eV).[25] On the +other hand, there were no diffraction signals of Cu phase in +XRD pattern of Cu@MoS2, while the (004) diffraction peak +in XRD pattern of Cu@MoS2 shifted to lower Bragg position +compared with that of distinct MoS2.[26] These results sug- +gested the successful intercalation of Cu in MoS2, which also +indicated that the MoS2 layers spacing was expanded by the Cu +ions along with the c direction. Additionally, the appearance of +Cu+ proved the reduction of Cu2+, which may be stimulated +by the lost electrons of S atoms in MoS2, therefore leading to +the transformation of S coordination. Thence, the 1T Mo 3d3/2 +(231.7 eV), 1T Mo 3d5/2 (228.5 eV), 1T S 2p1/2 (162.8 eV), and 1T +S 2p3/2 (161.6 eV) further confirmed the formation of 1T MoS2 +(Figure S2b,e, Supporting Information).[27] +For substitutional solid solution and heterointerface struc- +ture, the crystalline and chemical structures of C@MoS2 were +Figure 1. Schematic illustration of preparation of MoS2-based multiple phase solid solution. +Adv. Funct. Mater. 2022, 2112294 + +NH2 +-OH +Rearrangement +Polymerization +Oxidation +HO +HO +OH +OH +HO +OHHO +OH +DA +Indole +PDA +DA +PDA +Cu2+ +Carbon +Ⅱ +Carbon +Nitrogens +Cu2+ +DA +Pyrolysis + +Substitutional +Interstitial +Interstitial +Bulk MoS, +Cu@Mos +PDA@MoS, +C@Mos, +II +Carbon +Cu2+ +DA +Coordination +Pyrolysis +Interstitial +Interstitial +Substitutional +Bulk MoS2 +PDA@MoS2 +Cu/PDA@MoS2 +CulC@MoS2www.afm-journal.de +www.advancedsciencenews.com +2112294 (4 of 10) +© 2022 Wiley-VCH GmbH +investigated. It can be deduced by the XRD patterns (Figure 2b) +that the degree of preferred orientation along stacking direc- +tion (002) was decreased by the increasing PDA. Furthermore, +the interlayer distance (d) was effectively expanded, whereas +the bulk MoS2 was exfoliated to thin planes and the number +of layers enhanced gradually according to the Debye–Scherrer +equation (Figure 2c and Table S1, Supporting Information).[28] +The characteristic XPS peak of SC bond at 163.8 eV further +determined the carbon modification in C@MoS2.[29] Besides +the carbon phase modification introduced above, the PDA also +attained the substitutional solid solution by the S replacement +with N. As is shown in Figure 2d, the XPS peaks at 398.9, 399.9, +Figure 2. XRD patterns of a) MoS2, Cu@MoS2, and b) C@MoS2. c) Variation of layers distance and number of layers of C@MoS2. d) XPS Cu 2p spectra +of Cu@MoS2 and XPS N 1s spectra of C@MoS2. HRTEM, SAED images, and line profiles of e) MoS2, f) Cu@MoS2, and g) C@MoS2 2.0, respectively. +h) HAADF and corresponding elemental mapping images of C@MoS2 2.0. i) Schematic structures of polarization centers in MoS2, Cu@MoS2, and +C@MoS2, respectively. j) Atomic occupancy of Mo, S, and N of C@MoS2. k) Schematic unit cell structures obtained with Rietveld Refinement of C@ +MoS2 0.5. ε′ and ε″ values of l) MoS2, Cu@MoS2, and m) C@MoS2. 3D RL values of n) MoS2, o) Cu@MoS2, p) C@MoS2 0.5, q) C@MoS2 1.0, and +r) C@MoS2 2.0, respectively. +Adv. Funct. Mater. 2022, 2112294 + +6.3 +10 +a +b +c +d +C@Mos, +2p1/2 +Cu 2p +(002) +(103) +Sat +(002) +-d +Intensity +Sat! +0.5 +Intensityla.u. +Intensityla.u. +6.2 - +Cu2+ +Cu+ +Cu+cu+ +Cu@Mos +(103) +(105) (008) +Mos, +967 +957 +947 +937 +927 +1.0 +(004) +Number +C@Mos, +N 1s +6.1 +/a. +R-NH +2.0 +Intensity +Mo3p +Cu@Mos, +EN- +Mo-N +MoS, #37-1492 P63/mmc +Mos, #37-1492 P63/mmc +-- Number of layers +NH +LELE +Mo3p +6.0- +10 +30 +50 +70 +90 +10 +30 +50 +70 +90 +0.5 +2 +405 +400 +395 +390 +20/° +20/° +Molar ratio of PDA/MoS +Binding energy I eV +MoS2 +Cu@Mos +f +9 +C@Mos, +e +Single crystal +Polycrystal +Single crystal +MoS,(002) +MoS,(002) +10.1/nm +101/nm +10 1/nm +1.84/3=0.613nm +1.89/3=0.63nm +1.87/3=0.623nm +5 nm +5 nm +5 nm +3 +Position / nm +Position / nm +Position / nm +HAADF +h +S +Face--Face +Point--Point +Point--Face +Cu +C +S +100 nm +100 nm +Mo: +Mo +C +Mo +Mos, +Mos. +Vacancy(s) +Mo-Vacancy(S) +Cu+/Cu2+-MoS +Face2 C-MoS +100nm +100'nm +0.25 +Vacancy +4.0- +k +Cu@Mos +4.0- +m +C@MoS, +IMoIS +N +0.20- +N +co 3.6- + 3.5- +Mos, +3.0. +3.2 +2 +0.6- +—0.5 +Cu@Mos, +1.0 +0.05- +Mos. +co 0.4 +C +OOODDOOODDOODDD +0.00 +0.2+ +0.5 +1.0 +2.0 +b +2 +6 +10 +14 +18 +2 +6 +10 +14 +18 +Molar ratio of PDA/MoS, +FrequencylGHz +Frequency/GHz +p +n +0 +r +MoS2 +Cu@Mos +0 +C@MoS, 0.5 +C@MoS, 1.0 +C@MoS,2.0 +0 +0 +-10.00 +ect +4 +-6 +0-30 +-30.30 +-6.75 +-7.27 +9~0 +-5.55 +-5.83 +f,=2.48GHZ +.6 +-8 +2.7mmwww.afm-journal.de +www.advancedsciencenews.com +2112294 (5 of 10) +© 2022 Wiley-VCH GmbH +and 401.5 eV can be assigned to the pyridinic (N), pyrrolic +(NH), and graphitic (RNH2) groups respectively inheriting +from the PDA.[30] More importantly, the characteristic peak of +Mo–N at 397.8 eV confirmed the construction of N coordination +in MoS2 lattice by substituting the S occupancy.[31] The XRD +Rietveld refinement of C@MoS2 was performed to study the N +substitution effect of PDA in details (Figure S4 and Table S2, +Supporting Information). As is shown in Figure 2j, the N occu- +pancy appeared in C@MoS2, with increasing trend by the PDA +amount, which indicated that the N of PDA can successfully +substitute the S atoms on 4f site of P63/mmc MoS2. Addition- +ally, all the schematic unit cell structures obtained with Rietveld +Refinement were displayed in Figure 2k and Figure S18, Sup- +porting Information, where the 4f sites were composed of S, +N, and vacancy simultaneously in C@MoS2 and Cu/C@MoS2. +The scanning electron microscope (SEM) and transmission +electron microscope (TEM) images were utilized to further con- +firm all kinds of solid solution structures. As is shown in Fig- +ures S5–S7, Supporting Information, the initial MoS2 exhibited +bulk structure, whereas the films were effectively exfoliated in +Cu@MoS2 and C@MoS2. What’s more, the profile of Cu and +N in energy dispersive X-ray (EDX) images further proved the +Cu and N modification in Cu@MoS2 and C@MoS2, respec- +tively. The high-resolution TEM (HRTEM) images revealed +the lattices deformation on whatever points and faces. For +vacancy solid solution, the MoS2 showed typical regular single +crystal structure with the (002) interplanar spacing of 0.613 nm +(Figure 2e).[32] Notably, the intricate light and dark bright spots +on lattice fringes may be resulted from the contrast variation +caused by defects. For interstitial solid solution, the TEM image +in Figure 2f revealed that the interstitial Cu2+ could expanded +the (002) interplanar spacing to 0.630  nm. For substitutional +solid solution, the carbon also increased the (002) interplanar +spacing (0.623 nm), and the TEM image further illustrates the +introduction of the carbon layer (Figure 2g). It can be observed +in high-angle annular dark field (HAADF) and EDX elemental +mapping images (Figure 2h) that the derivative carbon was not +only covered on the surface, but also embedded in MoS2, which +was in well correspondence with the XPS and XRD results. +Additionally, the corresponding selected-area electron diffrac- +tion (SAED) profiles indicated that the Cu2+ made the MoS2 +into polycrystalline structure, while carbon diffused the diffrac- +tion profile.[33] +The effect of S vacancy, interstitial Cu, substitutional N, and +carbon phase hybrid on EM properties was studied by the com- +plex EM parameters (complex permittivity [εr  =  ε″  −  jε″] and +permeability [μr = μ″ − jμ″]), of which real part (ε′, μ′) and imag- +inary (ε″, μ″) determine the energy storage and attenuation per- +formance in terms of dielectric and magnetic, respectively.[34] +As is shown in Figure  2l,m, both Cu and carbon enhanced +the ε′ and ε″ values, but the effect of Cu is more significant. +According to the free electron theory ε +σ +πε +′′ = +⋅ +� +�� +� +�� +2 +1 +0 +f +,[35] results +indicated that interstitial Cu initiated considerable sources for +free electrons in MoS2. In this way, the EMW absorbability of +Cu@MoS2 was improved dramatically, whose reflection loss +(RL) value reached −30.30 dB (2.7 mm) with an effective absorp- +tion bandwidth (fE, RL←10  dB) of 2.48  GHz (Figure  2o and +Figure S9, Supporting Information). However, no other EMW +reached an RL value of −10 dB (Figure 2n,p–r and Figure S8, +Supporting Information). The EMW absorption mechanism +was analyzed on the basic of dielectric (tanδε) and magnetic +loss tangent (tanδμ) values first (Figure S10, Supporting Infor- +mation).[36] It can be generalized that better absorption perfor- +mance always accompanied with higher tanδε, indicating that +the EMW absorption was mainly attributed to the dielectric +loss.[37] +In this case, three kinds of heterogeneous structure models +can be built to analyze corresponding polarization performance +according to traditional polarization theory. As is illustrated +in Figure  2i schematically, the MoS2 with S vacancy, Cu@ +MoS2, and C@MoS2 formed typical point–point (Mo-vacancy +[S]), point–face (Cu–MoS2), and face–face (carbon–MoS2) het- +erojunctions, respectively. Through local defect adjustment, S +vacancy caused typical defect polarization in MoS2. The dielec- +tric property difference between carbon matrices and MoS2 +films induced interfacial polarization. The electron cloud struc- +ture on the interface between Cu ions and MoS2 as well as their +relative space position would also undergo regular deformation +under the applied EM field, thereby realizing the corresponding +polarization phenomenon. +2.3. Synergistic Effect of Vacancy, Interstitial, and Substitutional +Solid Solution Structure on EMW Absorption +The simultaneous manipulation of vacancy, interstitial, substi- +tutional, and heterointerface structure was performed through +Cu/C@MoS2. The fundamental crystalline and chemical states +were investigated by XRD patterns, Raman, and XPS spectra +at first. As is shown in Figure 3a, the XRD patterns confirmed +that the basic crystalline symmetry of Cu/C@MoS2 still showed +similar structure with instinct MoS2. Besides, the gradually +decreasing preferred orientation along stacking direction (002) +verified the splitting effect of Cu and carbon. In Raman spectra +(Figure 3b), two visible Raman absorption at 405 and 379 cm−1 +could be ascribed to the out-of-plane (A1g) and in-plane (E12g) +vibration of MoS2 films, respectively.[38] It is noticeable that +the intensity ratio of A1g/E12g decreased with increasing PDA, +which is attributed to the pronounced A1g vibration and +exposed MoS2 edge sites.[39] Additionally, the increasing Raman +scattering intensity revealed that the molecular polarizability +in MoS2 was improved by the Cu and carbon synergistically.[40] +According to the classic dielectric polarization theory, although +the response frequency of molecular polarization is much +higher than the gigahertz frequency range, there is no effec- +tive relaxation loss, and the molecular polarization enhance- +ment may also increase the dielectric response ability, thereby +increasing the polarization loss. Similarly, the Raman scattering +intensity of 1T-MoS2 also behaved in the same trend, indicating +that the carbon promoted the conversion of S coordination by +interstitial Cu, although carbon itself did not have this effect. +In well coincidence with the Raman analysis, the XPS Mo 3d +and S 2p spectra further determined the generation of 1T-MoS2 +(Figure S14, Supporting Information). In addition, the XPS O +1s peaks located at 530.3, 531.8, and 533.2 eV can be associated +with carbonyl, hydroxyl, and oxygen in lactone or anhydride +groups, respectively.[41,42] It is noteworthy that the molar ratio +Adv. Funct. Mater. 2022, 2112294 + +www.afm-journal.de +www.advancedsciencenews.com +2112294 (6 of 10) +© 2022 Wiley-VCH GmbH +of oxygen species with higher oxidation states were increased +with the PDA (Figure S16, Supporting Information), which +indicated that increasing amount of PDA stimulated the charge +outflowing from MoS2 under the catalysis of Cu. +For detailed vacancy, interstitial, substitutional, and het- +erointerface structure analysis in Cu/C@MoS2 solid solution, +XRD Rietveld refinement, and HRTEM were further investi- +gated.[43] The Rietveld refinement results for the XRD patterns +were summarized in Figure S17 and Table S3, Supporting +Information. For vacancy structure, the lattice vacancies were +reflected by the atomic occupancies in MoS2 unit cells. As is +shown in Figure 3c, the occupancy of Mo hardly changed, while +occupancy of S first increased and then decreased with the +increasing PDA. This phenomenon revealed that the instinct +MoS2 was more willing to generate S vacancies, whereas the +interstitial and covered PDA performed a role in preventing the +loss of S. Nevertheless, excessive carbon modification caused +S to move out of the lattice, forming S–C interaction simulta- +neously (Figure S14, Supporting Information). For interstitial +structure, the interstitial structure was studied relying on the +stacking structures of MoS2 films (Figure  3d and Table S5, +Supporting Information). The Cu ions without PDA had the +strongest splitting effect on MoS2 films, while the introduction +of PDA reduced this effect. Otherwise, the number of MoS2 +layers decreased with increasing PDA, which can be attributed +to the metal–organic cooperative interactions. For substitu- +tional structure, the substitutional structure in MoS2 solid solu- +tion was further analyzed based on the S replacement on 4f site +by N in P63/mmc space group unit cell. It can be observed in +Table  S3, Supporting Information, that the N (4f) occupancy +Figure 3. a) XRD patterns and b) Raman spectra of Cu/C@MoS2. c) Atomic occupancy of Mo, S, and N and d) variation of layers distance and number +of layers of Cu/C@MoS2. HRTEM, SAED images, and line profiles of e) Cu/C@MoS2 0.5, f) Cu/C@MoS2 1.0, and g) Cu/C@MoS2 2.0. h) Schematic +illustration of vacancy, interstitial, substitutional, and multiphase structure in MoS2 solid solution. i) ε′ and ε″ values of Cu/C@MoS2. 3D RL values of +j) Cu/C@MoS2 0.5, k) Cu/C@MoS2 1.0, and l) Cu/C@MoS2 2.0, respectively. +Adv. Funct. Mater. 2022, 2112294 + +0.25 +6.19 +(002) +Cu/C@MoS, +b +IMo lIs +NI +0-Numberof layers +a +2HA, +(103) +6.18 +0.20 +0.5 +Intensity I a.u. +Intensityla.u. +6.17 +A.E2=1.78 +1.0 +2.0 +1.0 +AE2-1.36 +6.15 +2.0 +0.5 +Ag/E2=1.25 +0.05 +MoS,#37-1492 P63/mmc +Mos, +Au +A1g/E2=1.24 +6.14 +0.00 +10 +30 +50 +70 +90 +800 +600 +400 +200 +0 +0.5 +1.0 +2.0 +0.5 +2 +Molar ratio of PDA/MoS2 +20/° +Raman shift / cm +Molar ratio of PDA/MoS2 +Cu/c@MoS.0.5 +Cu/C@MoS, 1.0 +Cu/C@MoS, 2.0 +e +g +MoS,(101) +MoS. (004) +10.1/nm +101/nm +.101/nm +2.63A +3.37 A +3.10/8=0.263nm +2.02/8=0.337nm +3.37/7=0.339nm +5nm +5nm +5nm +2 +3 +Position / nm +Position/nm +Position / nm +h +Vacancy +Interstitial +6 +-0.5 Cu/C@MoS2 +1.0 +5 +3 +Vacancy(C) +Interstitial(Cu) +2- +Cu/c@Mos, +-6.75 +1- +Vacancy(S) +Interstitial(C) +2 +6 +10 +14 +18 +Gap +FrequencylGHz +Carbon +Nitroge +k +0 +0 +0 +-2 +-10.00 +Mo +-20 +Mos, +-4 +-30 +-40 +-7.27 +-48.22 +Substitutional(N) +Carbon +f.=7.12GHZ +Substitutional +Multiphase +3.1mm +Solid solutionwww.afm-journal.de +www.advancedsciencenews.com +2112294 (7 of 10) +© 2022 Wiley-VCH GmbH +was exactly increased with the PDA modification. The SEM +(Figures S17–S19, Supporting Information) and HRTEM +images (Figure  3e–g) proved the Cu and PDA modification +again. It is obviously shown in SEM images that the MoS2 +films were gradually separated with PDA, where the Cu and N +modification was confirmed by corresponding EDX elemental +mapping images. The HRTEM images with more and more +random line profiles and expanded (004) interplanar spacing +showed that the lattices were distorted with increasing PDA. +According to the solid solution structure analysis, the syn- +ergistic polarization properties were studied based on the EM +properties of Cu/C@MoS2. It is presented in Figure 3i that both +the ε′ and ε″ values of Cu/C@MoS2 were increased with the +enhancement of PDA, especially the Cu/C@MoS2 2.0, which +suggested the dramatic improvement of dielectric polarization. +Additionally, the tanδε showed almost the similar regularity +(Figure S24, Supporting Information), revealing that the EM +attenuation optimization was primarily attributed to the die- +lectric loss.[44] Herein, it is notable that the RL of all C@MoS2 +samples displayed almost the same values above −10 dB, which +indicated that the conductive loss cannot effectively improve +the EMW absorption performance in MoS2-based EMAs. In +this case, the conductive loss of Cu/C@MoS2 by carbon derived +from the same PDA with C@MoS2 was also negligible. There- +fore, the EMW absorption was primarily attributed to the polar- +ization loss rather than conductive loss. Finally, the Cu/C@ +MoS2 2.0 obtained excellent EMW absorption performance +(Figure 3l and Figure S23, Supporting Information), whose RL +value reached −48.22  dB (thickness  =  2.5  mm, fE  =  4.8  GHz) +and the widest fE reached 7.12 GHz (thickness = 3.1 mm).[45] Fur- +thermore, tunable amount of PDA modification was employed +to analyze the synergistic effect of these three kinds of solid +solution structures on EMW absorption. When PDA was in +shortage, the minimum RL values only reached −6.75 (Cu/C@ +MoS2 0.5) and −7.27 dB (Cu/C@MoS2 1.0), indicating that the +synergistic effect dramatically improved the polarization loss. +2.4. Construction of Polarization Models for +Effective EMW Absorption +The effect of simultaneous regulation of multiple polariza- +tion mechanism was analyzed based on the whole MoS2 solid +solution systematically. Above all, it can be typically studied +by Debye relaxation curves (Figure S25, Supporting Informa- +tion) that more obvious Cole–Cole semicircles can be observed +due to Cu and carbon modification, determining the improved +polarization relaxation process.[46] Herein, the polarization +states were further investigated according to the semiconduc- +tors characteristic via electrochemical detection. Indeed, the +redox charge transfer reaction in MoS2 solid solution was natu- +rally attributed to the adsorption and desorption of Na+, while +an electron would enter the MoS2 layer and combine with the +hole,[47] finally leading to the polarization response on various +point and face sites. Above all, it can be observed in I–V charac- +teristics in Figure 4a–c that all the solid solutions transformed +into typical p-type semiconductors after whatever Cu or PDA +treatment, indicating that the charge migration was primarily +attributed to the holes (Figure 4e,f).[48] In details, both Cu and +carbon modification decreased the Tafel slopes, which was +mainly ascribed to the improving carrier behaviors. Addition- +ally, it can be found that the overpotential increased gradually +with the PDA, revealing that the electrode polarization of MoS2 +solid solution was more easily to be triggered in the assistance +of carbon phase. Electrochemical impedance spectra further +determined that the synergistic effect of vacancy, interstitial, +substitutional, and heterointerface structure in solid solution +decreased charge transfer resistance (Rct) (Figure  4d and +Figure S28, Supporting Information),[49] suggesting higher car- +rier mobility in Cu/C@MoS2. +To sum up, the contribution of vacancy, interstitial, substitu- +tional, and heterointerface on polarization loss and even EMW +absorption was analyzed according to the structural and elec- +trochemical research. I) Vacancies cannot generate effective +polarization loss relying on only their own contribution, since +the minimum RL value of pure MoS2 with S vacancy was only +−5.55 dB, which verified the invalid effect of point–point hybrid +structure. II) The modification of Cu was the main source of +polarization loss, which led to the minimum RL value of Cu@ +MoS2 reaching −30.30 dB. In other words, the point–face struc- +ture formed by introducing point hybrid outside the polar sur- +face will promote effective polarization loss. III) Comparing the +EMW absorption performance of C@MoS2 and Cu/C@MoS2, +it can be inferred that the heterogeneous interface between +carbon matrices and MoS2 films also cannot induce energy +loss through the polarization relaxation, which is in line with +the classical dielectric polarization theory. Additionally, similar +to the role of vacancy, the point–point interaction formed by N +substitution cannot optimize the EMW absorption effectively, +indicating that the dipoles polarization usually cannot lose EM +energy alone. IV) Fortunately, the regulation of carbon dramati- +cally improved the interaction between Cu and MoS2, while a +greater charge delocalization and migration can be formed to +cause energy loss. As is shown in Figure 4h, the ε′ and ε″ values +of Cu/C@MoS2 were increased exponentially with the enhance- +ment of carbon, especially the ε′ and ε″ value of Cu/C@MoS2 +was 1.36 and 4.16 times higher than MoS2, while those of C@ +MoS2 almost unchanged. In this case, it can be concluded that +quadra-tunable heterogeneous structures in Cu/C@MoS2 solid +solution formed a great synergistic effect among vacancy, inter- +stitial, substitutional, and heterointerface structures. Finally, +as is shown in Figure 4g, a polarization loss model of Cu/C@ +MoS2 can be built, in which the synergistic polarization was +induced among point–point, point–face, and face–face hetero- +geneous sites constructed by S vacancy, interstitial Cu, substi- +tutional N, and heterointerface between carbon and MoS2, +thereby leading to excellent polarization loss. +Finally, the optimization mechanism of EMW absorption +performance by synergistic effect among multiple polarization +loss was further determined by transmission coefficients (T) +and absorption coefficients (A).[50] According to the scattering +parameters in Figure S29, Supporting Information, it can be +calculated that the remaining energy (T) after one time EMW +penetration was dramatically reduced with the enhancement of +PDA in Cu/C@MoS2, while that in C@MoS2 have almost did +not changed (Figure 4i and Table S7, Supporting Information). +Moreover, the energy attenuation (A) of Cu/C@MoS2 improved +almost 266.7% and 222.2% than C@MoS2 and Cu@MoS2 +Adv. Funct. Mater. 2022, 2112294 + +www.afm-journal.de +www.advancedsciencenews.com +2112294 (8 of 10) +© 2022 Wiley-VCH GmbH +respectively, which indicate that the synergistic effect among +various point–point, point–face, and face–face sites on wher- +ever vacancy, interstitial, substitutional, and heterointerface +structures in MoS2 solid solution effectively improved the polar- +ization loss. Additionally, the as-prepared MoS2 solid solution +have been confirmed to be best EMAs compared to various of +representative MoS2 hybrids reported recently (Figure  4j and +Table S8, Supporting Information), such as defects containing +MoS2,[51] phase transformation MoS2,[52] metal/carbon/RGO/ +semiconductors composites.[53–64] In this case, this work can +Figure 4. I–V characteristics for a) MoS2 and Cu@MoS2, b) C@MoS2, and c) Cu/C@MoS2. d) Nyquist plots of MoS2 and Cu/C@MoS2. e) Molecular +hybrid orbital of MoS2. f) Mechanism of semiconductor formation. g) Schematic representation of polarization models in multiple phase MoS2 solid +solution. h) Average values for ε′ and ε″ of all samples. i) Average values for T and A of all samples. j) Comparison of EMW absorbability of representa- +tive MoS2 EMAs with various solid solutions structures. +Adv. Funct. Mater. 2022, 2112294 + +e +12 +e +xz yz +2H +p-type semiconductor +d +x2-y? xy +-2 +4d +8 +-4 +log(i/A) +-6 +4 +x?-y² z? +0.5 +MoS, +01.0 +Cu@Mos +4d. +Mos +-8. +0.2.0 +-0 +-0.8-0.6-0.4-0.2 +8 +16 +24 +32 +xy xz +-1.0 +0.0 +0 ++yz +Potential / V +z' +Cu2+ +Microcurrent +b +h +Cu+ +C@Mos, +-2 +Holes +p-type semiconductor +e +-3 +Semiconductor +log(i/A) +-4 +0.5 +-5 +1.0 +-6. +2.0 +2H-S coordination +1T-S coordination +-1.0 +-0.8-0.6-0.4-0.2 +0.0 +Potential / V +g +Face1 (Carbon) +c +Point. +Face +-1. +Cu/C@Mos, +p-type semiconductor +-2 +da +Point (Cu) +-3 +F +log(i/A) +ac +Polarization +-4 +MoS, +model +0.5 +cu +-5 +1.0 +Face2 (MoS2) +-6. +2.0 +-1.0-0.8-0.6-0.4-0.2 +0.0 +Multiple phase solid solution +Potential / V +！ +h +0.4 +55 +C@Mos, +0.7 +C@Mos, +This work +0.4 +Culc@Mos, +-50 +14 +13 +12 +1 +02 +0.2 +0.3 +-45 +43 +0.6- +dB +100 +5 +2 +A +T +-40 +16 +11 +2 +13 +4.5 +Cu/C@MoS, +-0.2 +R +-35 +8 +0.5 +e" +Cu/C@Mos +4.0 +-30 +4V +C@MoS2 +3.5 +0.4 +0.1 +-25+ +0 +0.5 +1.0 +2.0 +0 +0.5 +1.02.00 +0.5 +1.0 +2.0 +3 +5 +6 +7 +7 +Molar ratio of PDA/MoS, +Molar ratio of PDA/MoS, +f_ / GHzwww.afm-journal.de +www.advancedsciencenews.com +2112294 (9 of 10) +© 2022 Wiley-VCH GmbH +not only provide a great methodological guidance on design +solid solution type EMAs, but also further revealed the syner- +gistic polarization loss mechanism. +3. Conclusion +In summary, Cu/C@MoS2 solid solutions with controllable S +vacancy, Cu interstitial, N substitutional, and carbon/MoS2 het- +erointerface structures were prepared by a novel metal–organic +cooperative interactions method. Results indicated that the +polarization loss was mainly attributed to the Cu–MoS2 point– +face interaction, while the introduction of point–point and +face–face interaction initiated strong synergistic effect, thus +boosting excellent EMW absorption performance. Particularly, +Cu/C@MoS2 2.0 was regarded as most satisfactory EMA with +minimum RL of −48.22 dB and fE of 7.12 GHz. The absorbed +energy during one time of EMW penetration improved almost +266.7% and 222.2% than C@MoS2 and Cu@MoS2, respectively. +Finally, it can be concluded that the synergistic polarization in +MoS2 solid solution primarily optimized the dielectric loss by +improving the response activity of charge carriers (holes) and +the interaction with point (Cu) and face (carbon) hybrid. In this +case, this work provides new models for polarization loss mech- +anism analysis systematically and shed new light on the design +of solid solution structural EMAs. +4. Experimental Section +Preparation of PDA@MoS2: A novel metal–organic cooperative +interactions method was utilized to synthesize MoS2-based multiphase +solid solutions. First, commercial 2H-MoS2 was pretreated by sonication. +After standing for 20  min, the MoS2 was recollected from the upper +suspension by centrifugation. Then, 1 g of recollected MoS2 and 0.5 g +dopamine hydrochloride was stirred and dispersed in 200 mL deionized +water. After 12 h of stirring, suitable amount of tris(hydroxymethyl) +aminomethane was added into the system above to keep a stable pH +of aqueous solution around 8.5. Then the polymerization was kept for 6 +h, and the PDA@MoS2 was obtained by filtration with washing for three +times by deionized water. Simultaneously, different mass ratio (0.5, 1.0, +and 2.0) between dopamine hydrochloride and MoS2 was performed +to obtain PDA@MoS2 0.5, PDA@MoS2 1.0, and PDA@MoS2 2.0, +respectively. +Preparation of Cu@MoS2 Precursor and Cu/PDA@MoS2: First, +equal quality of PDA@MoS2 0.5 and CuCl2·2H2O were dispersed in +200 mL deionized water with 6 h of stirring. Then, Cu/PDA@MoS2 0.5 +was obtained with filtration and vacuum drying at 60 °C for 12 h. Similarly, +Cu@MoS2 precursor, Cu/PDA@MoS2 1.0, and Cu/PDA@MoS2 2.0 +could be fabricated with the same process but with MoS2, PDA@MoS2 +1.0, and PDA@MoS2 2.0, respectively. +Preparation of Cu@MoS2, C@MoS2, and Cu/C@MoS2: All the +precursors were annealed at 600  °C for 4 h at Ar atmosphere. This +way, C@MoS2 0.5, C@MoS2 1.0, and C@MoS2 2.0 were obtained from +PDA@MoS2 0.5, PDA@MoS2 1.0, and PDA@MoS2 2.0, respectively. +Similarly, Cu/C@MoS2 0.5, Cu/C@MoS2 1.0, and Cu/C@MoS2 2.0 were +obtained from Cu/PDA@MoS2 0.5, Cu/PDA@MoS2 1.0, and Cu/PDA@ +MoS2 2.0, respectively. For comparison, the pure MoS2 and Cu@MoS2 +were also treated by 600 °C annealing for 4 h at Ar atmosphere. +Characterization: An Ultima IV XRD with Cu Kα radiation +(λ  =  0.15418  nm) was used to study the crystalline structure. +Simultaneously, a Rietveld refinement analysis was performed by a +Fullprof software. The nano–micro structures were detected by a TEM +(FEI Talos F200×0) and a SEM (Verios G4, FEI). The chemical structures +were characterized by an XPS (PHI 5000 VersaProbe III) and a Raman +spectrometer (Alpha300R, WITec). The electrochemical performance +was detected by a CHI 660E electrochemical workstation. Beforehand, +all MoS2 solid solutions should be coated on a nickel foam as working +electrodes in the additive of Super P carbon black and polyvinylidene +fluoride in a mass ratio of 8:1:1. Additionally, a saturated calomel +electrode, a Pt foil, and 1 m Na2SO4 aqueous solution was employed as +reference electrode, counter electrode, and electrolyte, respectively. A +vector network analyzer (Anritsu MS46322B) was utilized to characterize +the EM parameters (complex permittivity [εr = ε″ − jε″] and permeability +[μr  =  μ″  −  jμ″]) by a typical coaxial-line method. According to the +transmission line theory, EMW absorption performance was calculated +based on following equation:[65] +20log +0 +0 +RL dB +Z +Z +Z +Z +in +in +( +) = +− ++ + +(1) +tanh +2 +in +0 +Z +Z +j +fd +c +r +r +r +r +µ +ε +π +µ ε += +� +�� +� +�� +(2) +where Zin, Z0, f, d, and c were input impedance of EMAs, impedance of +free space, incident EMW frequency, thickness of absorbers, and velocity +of light. For detection, all EMAs (30.0 wt%) were constructed to 3.0 mm +thick cylinders (Φin = 3.04 mm, Φout = 7.00 mm) in paraffin. +Supporting Information +Supporting Information is available from the Wiley Online Library or +from the author. +Acknowledgements +Z.G. and Z.M. contributed equally to this work. Financial support was +provided by the National Science Foundation of China (Grants nos. +51872238, 21806129, and 52074227), the Fundamental Research Funds +for the Central Universities (Nos. 3102018zy045 and 3102019AX11), +and the Natural Science Basic Research Plan in Shaanxi Province of +China (Nos. 2020JM-118 and 2017JQ5116). The authors acknowledge +the support from The Analytical & Testing Center of Northwestern +Polytechnical University. +Conflict of Interest +The authors declare no conflict of interest. +Data Availability Statement +Research data are not shared. +Keywords +electromagnetic wave absorption, polarization models, solid solution, +synergistic effect +Received: December 2, 2021 +Revised: December 25, 2021 +Published online: +Adv. Funct. Mater. 2022, 2112294 + +www.afm-journal.de +www.advancedsciencenews.com +2112294 (10 of 10) +© 2022 Wiley-VCH GmbH +[1] A. Iqbal, F. Shahzad, K. Hantanasirisakul, M. Kim, J. Kwon, J. Hong, +H. Kim, D. Kim, Y. Gogotsi, C. Koo, Science 2020, 369, 446. +[2] H.  Zhang, J.  Cheng, H.  Wang, Z.  Huang, Q.  Zheng, G.  Zheng, +D.  Zhang, R.  Che, M.  Cao, Adv. Funct. 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Mater. 2022, 2112294 + diff --git a/mdFPT4oBgHgl3EQf4TUa/content/2301.13193v1.pdf b/mdFPT4oBgHgl3EQf4TUa/content/2301.13193v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..dc8451dd54125e83460aaced0bea8f675970da2e --- /dev/null +++ b/mdFPT4oBgHgl3EQf4TUa/content/2301.13193v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:aa3b46f1f2fb60860983003e5eb76c01be3fcd0e25a576009cdd09fca72575a1 +size 3051729 diff --git a/mdFPT4oBgHgl3EQf4TUa/vector_store/index.pkl b/mdFPT4oBgHgl3EQf4TUa/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..41886dfe5d6bc522fa0bcdde75c093604a371709 --- /dev/null +++ b/mdFPT4oBgHgl3EQf4TUa/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0c84f89a24e89c10bcf2c76c5a06166b75c3e2f3f5f3a3dfcd73a350cb5b143f +size 195986 diff --git a/ndFAT4oBgHgl3EQfdB1R/content/tmp_files/2301.08567v1.pdf.txt b/ndFAT4oBgHgl3EQfdB1R/content/tmp_files/2301.08567v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6e1a4852c2aafd181ee2d069a8f6237d39f83e9b --- /dev/null +++ b/ndFAT4oBgHgl3EQfdB1R/content/tmp_files/2301.08567v1.pdf.txt @@ -0,0 +1,3570 @@ +Contributions on complexity bounds for Deterministic Partially +Observed Markov Decision Process +Cyrille Vessaire∗, +Jean-Philippe Chancelier∗, +Michel De Lara∗, +Pierre Carpentier†, +Alejandro Rodr´ıguez-Mart´ınez‡ +January 23, 2023 +Abstract +Markov Decision Processes (Mdps) form a versatile framework used to model a wide range of +optimization problems. The Mdp model consists of sets of states, actions, time steps, rewards, and +probability transitions. +When in a given state and at a given time, the decision maker’s action +generates a reward and determines the state at the next time step according to the probability +transition function. However, Mdps assume that the decision maker knows the state of the controlled +dynamical system. Hence, when one needs to optimize controlled dynamical systems under partial +observation, one often turns toward the formalism of Partially Observed Markov Decision Processes +(Pomdp). Pomdps are often untractable in the general case as Dynamic Programming suffers from +the curse of dimensionality. Instead of focusing on the general Pomdps, we present a subclass where +transitions and observations mappings are deterministic: Deterministic Partially Observed Markov +Decision Processes (Det-Pomdp). That subclass of problems has been studied by (Littman, 1996) +and (Bonet, 2009). It was first considered as a limit case of Pomdps by Littman, mainly used to +illustrate the complexity of Pomdps when considering as few sources of uncertainties as possible. +In this paper, we improve on Littman’s complexity bounds. We then introduce and study an even +simpler class: Separated Det-Pomdps and give some new complexity bounds for this class. This +new class of problems uses a property of the dynamics and observation to push back the curse of +dimensionality. +1 +Introduction +Markov Decision Processes (Mdps) form a versatile framework used to model a wide range of optimization +problems. Indeed, one often uses the formalism of Mdps to optimize controlled dynamical systems. It +is very popular in both optimal control and machine learning community, as it can be used to model +complex real-life problems (see the survey (White, 1993) for common applications). Moreover, it provides +the mathematical foundations for Reinforcement Learning (see (Sutton and Barto, 2018)), and algorithms +such as Policy Iteration and Dynamic Programming can efficiently solve Mdps. +In the Mdp framework, a decision maker can sequentially act upon a controlled dynamical system +and get some rewards. The Mdp model consists of sets of states, actions, time steps, rewards, and +probability transitions. When in a given state and at a given time, the decision maker’s action generates +a reward and determines the state at the next time step according to the probability transition function. +However, Mdps assume that the decision maker knows the state of the controlled dynamical system. +Hence, when one needs to optimize controlled dynamical systems under partial observation, one often +turns toward the formalism of Partially Observed Markov Decision Processes (Pomdp). An extensive +literature exists on Pomdps, most of which focuses on the infinite horizon case. Pomdps can be applied to +numerous fields, from medical models (as in (Steimle et al., 2021)) to robotics (as in (Pajarinen and Kyrki, +2017)) to name a few. Algorithms based on Dynamic Programming (see (Bellman, 1957)) have been +designed to exploit specific structures in Pomdps in order to solve this difficult class of problems. They +∗CERMICS, Ecole des Ponts, Marne-la-Vall´ee, France +†UMA, ENSTA Paris, Institut Polytechnique de Paris, Palaiseau, France +‡IAM, TotalEnergies SE, Pau, France +1 +arXiv:2301.08567v1 [math.OC] 20 Jan 2023 + +do so by first reformulating the problem through the use of beliefs (probability distributions over the state +space), as in (Smallwood and Sondik, 1973). One such algorithm is Sarsop, described in (Kurniawati +et al., 2008). However, Pomdps are often untractable in the general case as Dynamic Programming +suffers from the curse of dimensionality. Indeed, working with beliefs implies working on the space of +distributions over the state space, which is, by nature, an infinite space. +Yet not all Pomdps suffer equally from the curse of dimensionality. Indeed, instead of focusing on the +general Pomdps, we present a subclass where transitions and observations mappings are deterministic: +Deterministic Partially Observed Markov Decision Processes (Det-Pomdp). That subclass of problems +has been studied by (Littman, 1996) and (Bonet, 2009). +It was first considered as a limit case of +Pomdps by Littman, mainly used to illustrate the complexity of Pomdps when considering as few +sources of uncertainties as possible. For Bonet, Det-Pomdps became of interest after some applications +were found. He presented examples in (Bonet, 2009, §2), such as the navigation of a robot in a partially +observed terrain. +In this paper, we improve on Littman’s complexity bounds. We then introduce and study an even +simpler class: Separated Det-Pomdps. This new class of problems uses a property of the dynamics and +observation to push back the curse of dimensionality. +The paper is organized as follows. First, in §2, we present a general formulation of Det-Pomdp. +Second, in §3, we present Dynamic Programming on beliefs for Det-Pomdps with constraints, and +present complexity bounds. +Third, in §4, we introduce a subclass of Det-Pomdp, Separated Det- +Pomdp. +Finally, in §5 we illustrate Separated Det-Pomdp with a toy problem: emptying a tank +containing water when considering partial observation of the level of water in the tank. Meanwhile, in +Appendix A.1, we present technical lemmata and considerations on pushforward measures. Finally, in +Appendix A.2, we present complements on Separated Det-Pomdps. +We now detail our main contributions. In §3, we improve Littman (1996) bound on the cardinality +of the set of reachable beliefs for Det-Pomdps (see Theorem 4). This new bound comes from a new +representation of the belief dynamics in Det-Pomdps using the notion of pushforward measure (see +Lemma 7). In §4, we introduce a subclass of Det-Pomdps, Separated Det-Pomdps. As shown in +Theorem 13, the interest of Separated Det-Pomdps is that they further push back the curse of dimen- +sionality for Dynamic Programming with beliefs (see Theorem 13). Moreover, this last bound is tight +(see Proposition 16). +2 +Formulation of Deterministic Partially Observed Markov De- +cision Processes +A Det-Pomdp is a particular case of Pomdps, itself an extension of Markov Decision Processes (Mdps). +Backgrounds on Mdps can be found in Puterman (1994), whereas backgrounds on Pomdps can be found +in Bertsekas and Shreve (1978). As with Mdps, the model consists of a dynamical system, defined thanks +to states, controls (also called actions), transitions and time steps. At each time-step, the decision maker +(also called the agent) chooses a given action, which generates a random reward depending on the state +of the system and on the time. The state then transits to its next random value. However, in the case of +Det-Pomdp (and Pomdp), the decision maker has only partial knowledge of the state of the dynamical +system. Instead, he has access to functions of the state and controls: the observations. For Det-Pomdps, +the transitions and observations are given by deterministic evolution and observation functions. +First, we present the ingredients of a Det-Pomdp. Second, we present the formulation of a Det- +Pomdp optimization problem. +Ingredients of a Det-Pomdp +A Det-Pomdp is defined by the tuple +D = +� +T , U, O, X, {Lt}t∈T \{T }, {ft}t∈T \{T }, {Uad +t }t∈T \{T }, {ht}t∈T +� +, +(1) +which we now detail1. +The set T = �0, T�2 is the set of time-steps, where the positive integer T ∈ N\{0} is colloquially known +as the horizon. The set U is the set of controls the decision maker can choose from. The set O is the set +1For simplicity, we assume that U, O and X are not indexed by time +2Let t and t′ be two integers, with t′ ≥ t. The set {t, t + 1, . . . , t′} is denoted by �t, t′�. +2 + +of observations available to the decision maker. The set X is the set of states. The collection {Lt}t∈T \{T } +is the collection of instantaneous costs functions: for all time t ∈ T \ {T}, Lt : X × U → R ∪ {+∞}. +Moreover, the final cost function LT is by convention denoted by K : X → R ∪ {+∞}. The collection +{ft}t∈T \{T } is the collection of evolution functions: for all time t ∈ T \{T}, ft : X×U → X. They define +the transitions of the dynamical system. The collection {Uad +t }t∈T \{T } is the collection of admissibility +constraints: for all time t ∈ T \ {T}, Uad +t +: X ⇒ U is a set-valued mapping from X to U, that is, for +all state x ∈ X, the admissible controls at time t are given by the subset of U, Uad +t (x). {ht}t∈T is the +collection of observation functions. The initial observation is given by the mapping h0 : X → O whereas, +for all time t ∈ T \ {0}, the observations are given by the mappings ht : X × U → O. +Let (Ω, F, P) be a probability space, where Ω is the set of possible outcomes, F is a σ-field over Ω +and P is a probability measure on Ω. We denote by E the mathematical expectation operator. +In this paper, we only consider Det-Pomdps which satisfy the following finite sets assumption. +Assumption 1 (Finite sets assumption). The sets of possible outcomes Ω, of states X, of controls U, +and observations O have finite cardinality. Moreover, we consider a finite number of timesteps, i.e. the +horizon is finite: T < +∞. +As we consider finite sets, we introduce a notation for the set of probability distributions on finite +sets. Let Y be a finite set. We denote by ∆(Y) the set of probability distributions on Y. The set ∆(Y) +is in bijection with the simplex ∆|Y| of dimension3 |Y| (hence the notation). +We now present the formulation of the optimization problem which we study in this paper. +Formulation of a Det-Pomdp optimization problem +A finite-horizon Det-Pomdp optimization +problem is formulated as follows +V⋆(b0) = min +X,O,U E +�T −1 +� +t=0 +Lt(Xt, Ut) + K(XT ) +� +(2a) +s.t. PX0 = b0 , +(2b) +Xt+1 = ft(Xt, Ut) , ∀t ∈ T \ {T} , +(2c) +O0 = h0(X0) , +(2d) +Ot+1 = ht+1(Xt+1, Ut) , ∀t ∈ T \ {T} , +(2e) +Ut ∈ Uad +t (Xt) , ∀t ∈ T \ {T} , +(2f) +σ(Ut) ⊂ σ(O0, . . . , Ot, U0, . . . , Ut−1) , ∀t ∈ T \ {T} , +(2g) +where we denote by V⋆(b0) the optimal value of Problem (2), that is, the optimal value of the Det- +Pomdp optimization problem when the initial probability distribution of the state is given by the initial +belief b0 ∈ ∆(X). In Problem (2), there are three processes X = +� +Xt +� +t∈T , U = +� +Ut +� +t∈T \{T } and +O = +� +Ot +� +t∈T . For all time t ∈ T , Xt : Ω → X and Ot : Ω → O are random variables representing +respectively the state and the observation variables of the system at time t, and for all time t ∈ T \ {T}, +Ut : Ω → U are random variables representing the controls at time t. +The optimization criterion of Problem (2) is given by Equation (2a). In this paper, we only consider +the minimization of the expected value in the finite horizon case. +We now detail the constraints of the optimization Problem (2). First, Equation (2b) is the initial- +ization constraint. As the initial state is not fully known, we instead use the probability distribution +b0 ∈ ∆(X) of the initial state of the system for the initialization. Second, Equation (2c) is called the +state evolution equation of the system. It is defined thanks to the dynamics which describe the evolution +of the states of the controlled dynamical system. Third, Equations (2d) and (2e) define the observations +of the system available at each time step. Fourth, Equation (2f) is called the admissibility constraints: +it defines which controls can be applied at each time step. Note that the proper formulation of the +admissibility constraints should contain an added quantification, “∀ω ∈ Ω”, which we omit in this paper +as the set Ω is finite, and we can always assume that P(ω) > 0 for all ω ∈ Ω. Finally, Equation (2g) is the +non-anticipativity constraint: it defines the information available to the decision maker before choosing +3The cardinality of a finite set is the number of its elements and is denoted by | · |. +3 + +a control at each time step. As all sets Ω, X, U and O are assumed to be finite by Assumption 1, all +mappings with domain Ω are random variables and Equation (2a) is well defined because Lt and K takes +their values in R ∪ {+∞}, hence the optimization Problem (2) is well defined. +3 +Complexity analysis of Dynamic Programming for Det-Pomdps +In §3.1, we present Dynamic Programming for Det-Pomdps. Then, in §3.2 we study its complexity, i.e. +the number of “operations” necessary to solve Problem (2). In §3.3, we present a new representation of +beliefs as pushforward measures, that will be used to prove the complexity results. +3.1 +Dynamic Programming for Det-Pomdp +We now present Dynamic Programming Equations with beliefs for Problem (2). +As a Det-Pomdp +is a Pomdp, all the results and numerical methods that apply to Pomdps are carried over to Det- +Pomdps. Notably, it is possible to write Dynamic Programming equations for a finite horizon problem +associated with a Pomdp. +To do so, it is classical to formulate a belief-Mdp where the state is a +probability distribution over the state space, called belief (see (Bertsekas and Shreve, 1978) for details +on the assumptions for general Pomdps). Here, we detail this methodology for the specific Det-Pomdp +case, and extend it to tackle cases with admissibility constraints on the controls. +First, in §3.1.1, we formally define sets and mappings which are necessary for the formulation of +the belief-Mdp. Second, in §3.1.2, we present the Dynamic Programming equations for the resulting +belief-Mdp. +3.1.1 +Beliefs in Det-Pomdp +First, we present the set of beliefs. Second, we present the mappings necessary for the formulation of +the belief-Mdp, notably the beliefs dynamics. +Sets for the beliefs. +The dynamic programming equation for Det-Pomdps is formulated using states +in the set ∆(X), the probability distributions over the “initial” state space X, which are called beliefs. +However, the beliefs dynamics, as described later in Equation (9), may lead to a null measure over the +space X when considering some combination of observations and controls which are in contradiction with +each other. As we want to be able to compose belief dynamics, we combine ∆(X) and the null measure +over X as follows. +We introduce an extended state set X, obtained as the union of the original set X with an extra +element, denoted by ∂ (∂ /∈ X), which is used as the support of the null measure over X. +X = X ∪ {∂} . +(3) +We denote by B the subset of ∆(X) defined by +B = ∆(X) ∪ {δ∂} , +(4) +where we identify the set ∆(X) with {µ ∈ ∆(X) | supp(µ) ⊂ X} and where δ∂ ∈ ∆(X) is the discrete +probability measure on X concentrated on ∂, that is δ∂({∂}) = 1, and where the mapping “supp” is the +support of a nonnegative measure. For any nonnegative measure µ on the finite set Y, we have +supp(µ) = +� +y ∈ Y +�� µ({y}) > 0 +� +. +(5) +We call the probability measure δ∂ the cemetery belief as we will see in Equation (9) that the belief +dynamics, when reaching the belief state δ∂, remains in δ∂ forever. A probability measure ν ∈ ∆(X) is +represented, in some equations, by the ordered pair +� +ν|X, ν(∂) +� +, where ν|X is a nonnegative measure on +the set X and ν(∂) ∈ R+. +Now that the set of beliefs B is defined, we present the beliefs dynamics. +4 + +Beliefs dynamics. +In order to define the beliefs dynamics, we introduce, for each t ∈ T \ {T} two +mappings, Qt+1 : B × U × O → [0, 1] and τt : B × U × O → B. They are defined using partial mappings, +defined as follows. +Let A, D, F and G be sets. Let g : A × D → F, (a, d) �→ g(a, d) be a mapping. We denote by gd the +mapping +gd : A → F , a �→ g(a, d) , +(6) +i.e. the mapping g(·, d) obtained from g by setting its second variable to a fixed value d ∈ D. When +considering mappings with n inputs, we extend this notation to the last n − 1 inputs using a Cartesian +product over the last n − 1 sets. For example, let g : A × D × F → G. We denote by g(d,f) = g(·, d, f) +the mapping g(d,f) : A → G, a �→ g(a, d, f). +Now, the mapping Qt+1 gives the probability of observing o at time t + 1 when applying control u +on the dynamical system when considering belief b at time t, and is given by +∀t ∈ T \ {T} , Qt+1 : (b, u, o) ∋ B × U × O �→ b +� +(hu +t+1 ◦ f u +t )−1(o) +� +, +(7) +where f u +t (·) and hu +t (·) are partial mapping that follow the notation defined in Equation (6): +∀u ∈ U, +f u +t : X → X , x �→ ft(x, u) , +and ∀u ∈ U, +hu +t : X → O , x �→ ht(x, u) , +and where b +� +(hu +t+1 ◦ f u +t )−1(o) +� +is the probability of the set (hu +t+1 ◦ f u +t )−1(o) with respect to the probability +distribution b. Note that, we always have that +Qt+1(δ∂, u, o) = δ∂ +� +(hu +t+1 ◦ f u +t )−1(o) +� += 0 , +(8) +as (hu +t+1 ◦ f u +t )−1(o) is always a subset of X and thus has a null intersection with {∂}. +For all time t ∈ T \ {T}, the mapping τt gives the evolution of the beliefs when applying control u on +the dynamical system when considering belief b at time t and observing o at time t + 1, and is given by +∀y ∈ X , τt(b, u, o)(y) = +� +� +� +b +� +(f u +t )−1(y) +� +Qt+1(b, u, o) +if Qt+1(b, u, o) ̸= 0, and y ∈ +� +hu +t+1 +�−1(o) , +0 +otherwise, +(9a) +τt(b, u, o)(∂) = 1 − τt(b, u, o)(X) . +(9b) +Hence, δ∂ is used as a last resort belief, which appears when it is not possible to observe o after applying +control u to any state of the support of belief b. Indeed, δ∂ is used to ensure that the mappings τt are +well defined for all beliefs, controls and observations. +Using the sequences of mappings {Qt}t∈T \{0} and {τt}t∈T \{T }, we have a properly defined belief- +Mdp, which can be solved by Dynamic Programming. +3.1.2 +Dynamic Programming Equations for Det-Pomdp +In the case of Pomdp (without constraints on the controls), Dynamic Programming equations with beliefs +as new states were first given in (˚Astr¨om, 1965). More general cases (still without explicit constraints on +the controls) are treated in Bertsekas and Shreve (1978, Chapter 10) and in Bertsekas (2000, Chapter +4). Dynamic Programming Equations for Det-Pomdp can be obtained as a special case of Dynamic +Programming for Pomdp. They are given in Equations (10a) and (10b) together with the expression of +the beliefs dynamics {τt}t∈T \{T } (see Equation (9)) in the case where there are no constraints on the +controls in (Littman, 1996). In (Bertsekas and Shreve, 1978) the proof that beliefs are statistics sufficient +for controls was made for Pomdps without any admissibility constraint. We thus cannot directly apply +this result on Problem (2), as it contains Constraint (2f). We extend the classical results by (Bertsekas +and Shreve, 1978) in Proposition 1 in order to tackle such constraints. We identify an admissibility set +for beliefs of the form Ub,ad(b) = � +x∈supp(b) Uad(x). Note that we use an upper index b to distinguish +admissibility sets for beliefs from admissibility sets for states. Also note that, as far as we know, the first +Dynamic Programming equations using such sets Ub,ad(b) were given in (Geffner and Bonet, 1998, §5) +with no explicit proof. +5 + +Proposition 1. Consider a Det-Pomdp optimization problem given by Problem (2) which satisfies the +finite sets Assumption 1. Let B = ∆(X) ∪ {δ∂}, as defined in Equation (4) and consider the sequence +of value functions (Vt : B → R ∪ {+∞})t∈T defined by the following backward induction. First, for all +t ∈ T , we have that Vt(δ∂) = 0. Second, we have that +VT : b ∈ ∆(X) �→ +� +x∈X +b(x)K(x) , +(10a) +Vt : b ∈ ∆(X) �→ +min +u∈Ub,ad +t +(b) +�� +x∈X +b(x)Lt(x, u) + +� +o∈O +Qt+1(b, u, o)Vt+1 +� +τt(b, u, o) +�� +, +(10b) +where Ub,ad +t +(b) = � +x∈supp(b) Uad +t (x). +Then, the optimal value of Problem (2) and the value of the function V0 at the initial belief b0 are +equal, that is, V0(b0) = V⋆(b0). Moreover, a policy π = (π0, . . . , πT −1), defined by a sequence of mappings +πt : B → U, which minimizes the right-hand side of Equation (10b) for each b and t is an optimal policy +of Problem (2): the controls given by Ut = πt(Bt) (where Bt is computed thanks to the recursion +Bt+1 = τt(Bt, Ut, Ot+1), with B0 = b0) are optimal controls of Problem (2). +Proof. We present a sketch of proof of Proposition 1. +First, we rewrite Problem (2) as an equivalent problem, without constraint (2f) by adding charac- +teristic functions of the constraints to the instantaneous costs. The equivalent problem then follows the +framework of (Bertsekas and Shreve, 1978). +Second, we apply the results of (Bertsekas and Shreve, 1978) to the reformulated problem, and obtain +associated Dynamic Programming equations. +Third, the Dynamic Programming equations which solve the equivalent problem are equivalent to +Equations (10) presented in Proposition 1, thus concluding that Equation (10) gives the solution of +Problem (2) as formulated in Proposition 1. This step is a bit technical, but is otherwise straightforward +and does not present any major difficulty. +Now that we have presented Dynamic Programming equations on beliefs, we present the complexity +of Dynamic Programming. +3.2 +Dynamic Programming complexity for Det-Pomdps +According to Proposition 1, we can solve Problem (2) by computing V0(b0) by means of Equations (10). +Solving Dynamic Programming equations (10) implies that we are able to numerically evaluate the value +functions at each reachable belief starting from b0. Thus, we introduce the subsets of reachable beliefs +starting from b0. We start by formally defining the set of reachable beliefs, before we present our first +complexity result on Dynamic Programming for Det-Pomdp. +The set of reachable beliefs BR,D is defined as follows. +Note that we use the upper index D to +recall that we consider the set of reachable beliefs of a Det-Pomdp defined by the data tuple D, in +Equation (1), whereas the upper index R stands for reachable. +Definition 2. Let b0 ∈ ∆(X) be given and consider the sequence {BR,D +t +}t∈T of subsets of the set of +beliefs B = ∆(X) ∪ {δ∂} defined by the induction +BR,D +0 +(b0) = {b0} +and +∀t ∈ T \ {T} , BR,D +t+1 (b0) = τt +� +BR,D +t +(b0), U, O +� +, +(11) +where τt is defined in Equation (9). For any t ∈ T , the subset BR,D +t +(b0) ⊂ B is called the set of reachable +beliefs a time t starting from initial belief b0. +Moreover, we denote by BR,D +�t,t′�(b0) the union, for t′′ in the time interval �t, t′�, t < t′, of the reachable +beliefs at time t′′ starting from the initial belief b0 ∈ ∆(X), that is, +∀(t, t′) ∈ T 2 , t < t′ , BR,D +�t,t′�(b0) = +t′� +t′′=t +BR,D +t′′ (b0) . +(12) +The set BR,D +�1,T �(b0) is called the set of reachable beliefs from the initial belief b0. +6 + +Note that, under Assumption 1, the set BR,D +�1,T �(b0) is finite. +We now present a classical complexity result for Dynamic Programming algorithm (which we call Dp +Algorithm in the rest of this paper). +Proposition 3. Consider a Det-Pomdp optimization problem given by Problem (2) which satisfies +the finite sets Assumption 1. Let b0 ∈ ∆(X). Then, a standard Dp Algorithm (numerically) solves +Problem (2), and its complexity is O(|T ||BR,D +�1,T �(b0)||U||O|), where the set of reachable beliefs BR,D +�1,T �(b0) +is defined in Equation (12). +Proof. First, as we consider that Assumption 1 holds, note that BR,D +�1,T �(b0) is finite and we can apply +Proposition 1 on Problem (2). +We hence solve Problem (2) by computing value functions given by +Equations (10). +For a given time t ∈ T \ {T} and reachable belief b ∈ BR,D +t +(b0), we compute the value function Vt +by evaluating the next value for each control u ∈ U and each resulting observations. We hence need +� +t∈T |BR,D +t +(b0)||U||O| operations to solve Problem (2). Then, since for all time t ∈ T , t > 0, BR,D +t +(b0) ⊂ +BR,D +�1,T �(b0) (see Equation (12)), we have |BR,D +t +(b0)| ≤ |BR,D +�1,T �(b0)|. Moreover, we also have BR,D +�1,T �(b0) ̸= ∅ +(there is always at least one belief in BR,D +1 +(b0), as for a given control u ∈ U and an observation o ∈ O, +τ0(b0, u, o) ∈ BR,D +1 +(b0) ⊂ BR,D +�1,T �(b0)) and BR,D +0 +(b0) = {b0}, hence |BR,D +0 +(b0)| ≤ |BR,D +�1,T �(b0)|. +Hence, � +t∈T |BR,D +t +(b0)||U||O| ≤ |T ||BR,D +�1,T �(b0)||U||O|, and thus we can solve Problem (2) in O(|T ||BR,D +�1,T �(b0)||U||O|) +operations. +In order to apply Proposition 3 on Problem (2) and to get complexity bounds on the Dp Algorithm, +we now study the set of reachable beliefs BR,D +�1,T �(b0), more specifically, we give bounds on its cardinality. +Theorem 4. Consider a Det-Pomdp optimization problem given by Problem (2) which satisfies the +finite sets Assumption 1, and such that |U| > 1. For all initial belief b0 ∈ ∆(X), the cardinality of the +set of reachable beliefs starting from b0, defined in Equation (12), satisfies the following bound +��BR,D +�1,T �(b0) +�� ≤ min +� +(1 + |X|)|supp(b0)| , 1 + |supp(b0)||U||T |� +. +(13) +Proof. A sketch of proof is postponed to §3.3, as it relies on a new representation of the belief dynamics +presented in §3.3. The complete proof can be found in Appendix §A.1.3. +In Theorem 4, we gave a bound on the cardinality of the set BR,D +�1,t�(b0) which improves a previous +result we now recall. +Littman presents in (Littman, 1996, Lemma 6.1) a bound on the set of reachable beliefs starting from +belief b0 ∈ ∆(X): +∀t ∈ T , +��BR,D +�0,t�(b0) +�� ≤ (1 + |X|)|X| . +(14) +Equation (13) is an improvement on the bound given in Equation (14) which takes into account the +support of the initial belief b0: indeed, as b0 ∈ ∆(X) and |supp(b0)| ≤ |X|, Equation (13) is tighter than +Equation (14). +Using Equation (13), we obtain that the number of reachable beliefs of a Det-Pomdp is finite even +when considering the case of an infinite horizon. Indeed, the first inequality in Equation (13) is well +defined even in the infinite horizon case. +A direct consequence of Proposition 3 and Theorem 4 is that the complexity of the Dp Algorithm is +O +� +|BR,D +�1,T �(b0)||T ||U||O| +� +, i.e. in O +� +min +� +(1 + |X|)|supp(b0)| , 1 + |supp(b0)||U||T |� +|T ||U||O| +� +. +Remark 5. As a side note, we can remark that we could also use Theorem 4 to characterize the complex- +ity of a general Pomdp. Indeed, we can reformulate any finite Pomdp with independent noises on the +dynamics {wt}t∈T \{T } and independent noises on the observations {vt}t∈T and admissibility constraints +of the form Uad : X ⇒ U as a finite Det-Pomdp. To do so, we expand the state of the Pomdp with +the realization of all noises. We model the problem as though the realization of the noises are predeter- +mined, but the decision maker does not know the noises in advance. We then obtain a Det-Pomdp, +with states X′, controls U and observations O. However, such reformulation leads to a drastic increase +in the dimension of the state and the cardinality of the support of the initial belief. Indeed, the initial +7 + +belief contains all possible values of the initial state and all the possible realizations of noises, i.e. its +cardinality is multiplied by a factor |V|T +1 × |W|T , with V the set of noises on the observation and W +the set of noises on the dynamics. Hence, we are doubly penalized when considering the bound presented +in Theorem 4: we both increase |X| and |supp(b0)|. This reinforces the point on the difficulty of solving +Pomdps as even the ones with simple structures are far more difficult than similar sized Det-Pomdps. +3.3 +Belief dynamics as pushforward measures +Here, we expose another representation of the beliefs evolution functions {τt}t∈T \{T } defined in Equa- +tion (9), used in the proof of Theorem 4. First, we recall the notion of pushforward measures when +considering finite sets. Second, we introduce the mappings necessary for the new representation. We +then present in Lemma 7 the representation of the belief dynamics as pushforward measures. +Definition 6. Consider two finite sets A and D and a mapping h : A → D. The pushforward measure +(or the image-measure) of a probability measure µ ∈ ∆(A) on the set A by the mapping h is the probability +measure h⋆µ ∈ ∆(D) on the set D defined by +∀d ∈ D , (h⋆µ)(d) = µ +� +h−1(d) +� += +� +a∈A,h(a)=d +µ(a) . +(15) +We also denote by h⋆ the mapping from ∆(A) to ∆(D) such that h⋆(µ) = h⋆µ. +Before presenting Lemma 7, we first introduce some mappings: F u,o +t +, and R. For each pair (u, o) ∈ +U × O, and each t ∈ T \ {T}, we denote by F u,o +t +the self-mapping on the extended state set X = X ∪ {∂} +(defined in Equation (3)), defined by: +F u,o +t +: X → X , x �→ +� +f u +t (x) +if +x ̸= ∂ and f u +t (x) ∈ +� +hu +t+1 +�−1(o) , +∂ +otherwise. +(16) +The mapping F u,o +t +hence applies the dynamics ft, as defined in Problem (2), given control u, and only +keeps the resulting state if it is consistent with observation o. Meanwhile, the renormalization mapping +R : ∆(X) → ∆(X) is defined by +R : ν ∈ ∆(X) �→ +�� +1 +ν(X)ν|X, 0 +� +if ν(X) ̸= 0 , +δ∂ +if ν(X) = 0 . +(17) +We now express the belief dynamics as pushforward measures. +Lemma 7. Let (u, o) ∈ U × O be given, and let t ∈ T \ {T}. We have that +∀b ∈ B , τt(b, u, o) = R ◦ (F u,o +t +)⋆(b) , +(18) +where the pushforward (F u,o +t +)⋆(b) follows Notation (15). +Proof. The proof is detailed in Appendix A.1. +This new representation is of interest as, for all time t ∈ T \ {T}, the composition of belief dynamics +τt is given by the pushforward measure of the composition of mappings F u,o +t +for the relevant pairs (u, o) ∈ +U × O. Indeed, when considering a composition of belief dynamics, we can factorize the renormalization +mapping R. +We thus apply the renormalization mapping R to the composition of the pushforward +measures, which is the pushforward measure of the composition of mappings F u,o +t +. There is therefore +an equivalence between studying the composition for time t ∈ T \ {T} of the belief dynamics τt and +the composition, for the relevant pairs (u, o) ∈ U × O, of the mappings F u,o +t +. Notably, we use this +representation to bound the cardinality of the set of reachable beliefs, and thus study the complexity of +Dynamic Programming for Det-Pomdp. +To do so, we introduce notations for sets and mappings. +8 + +Notation for sets and mappings. +For any given sets Y and V, we denote by L(Y; V) = VY the set +of mappings from Y to V. +• For all G ⊂ L(Y; V), Y ⊂ Y, B ⊂ ∆(Y) and b ∈ ∆(Y) we introduce the notations G(Y ), and +G⋆(B), and G⋆(b) for the sets defined by +G⋆(b) = G⋆({b}) ⊂ ∆(V) . +(19a) +• Given two subsets G′ and G′′ of L(Y; Y) we introduce the subset G′ ◦ G′′ defined by +G′ ◦ G′′ = +� +g′ ◦ g′′ �� g′ ∈ G′ and g′′ ∈ G′′� +⊂ L(Y; Y) . +(19b) +• For any sequence {Gk}k∈N, with Gk ⊂ L(Y; Y) for all k ∈ N, we introduce for any k ∈ N the +subsets G0:k defined by +∀k ∈ N , G0:k = Gk ◦ Gk−1 ◦ · · · ◦ G0 ⊂ L(Y; Y) . +(19c) +For a fixed value of u ∈ U, and o ∈ O, for all t ∈ T \ {T}, we have obtained in Lemma 7 that +τt(·, u, o) = R ◦ (F u,o +t +)⋆. Now, for each t ∈ T , we introduce the sets +TD +t = +� +τt(·, u, o) +�� u ∈ U, o ∈ O +� +⊂ L(B; B) , +(20) +FD +t = +� +F u,o +t +�� u ∈ U, o ∈ O +� +⊂ L(X; X) , +(21) +FD = +� +t∈T \{T } +FD +0:t , +(22) +where the composition of sets of mapping is given by Notation (19b) and (19c). Note that4 FD +0:t ̸= FD +�0,t�. +Moreover, we call FD, defined by Equation (22), the set of pushforwards of the Det-Pomdp defined by +Equation (2). +Lemma 8. Let b0 ∈ ∆(X). We have that +∀t ∈ T \ {0} , BR,D +t +(b0) = TD +0:t−1(b0) = R ◦ (FD)⋆(b0) , +(23) +TD = +� +t∈T \{T } +TD +0:t = R ◦ (FD +0:t)⋆ , +(24) +i.e. +BR,D +�1,T �(b0) = +� +t∈T \{T } +TD +0:t(b0) = R ◦ (FD)⋆(b0) . +(25) +Proof. The proof is detailed in Appendix A.1. +Lemmata 7 and 8 are illustrated in Figures 1 and 2. A direct application of Lemma 8 is that there is +an equivalence between studying the cardinality of BR,D +�1,T �(b0) and studying the cardinality of (FD)⋆(b0). +We now present the postponed sketch of proof of Theorem 4. +A detailed proof can be found in +Appendix §A.1.3. +Sketch of proof of Theorem 4. Let b0 ∈ ∆(X) be given. +By Lemma 8, we have that BR,D +�1,T �(b0) = R ◦ (FD)⋆(b0). +The first inequality |BR,D +�1,T �(b0)| ≤ (1 + |X|)|supp(b0)| comes from the fact that +��(FD)⋆(b0) +�� is bounded +by the number of mappings from supp(b0) to X, as shown in Lemma 21. +Meanwhile, the second inequality +��BR,D +�1,T �(b0) +�� ≤ 1 + |supp(b0)||U||T | comes from the fact that, for all +time and controls (t, u) ∈ T \{T}×U, and for any belief b ∈ ∆(X), we have that � +o∈O +��supp +� +(F u,o +t +)⋆b +��� ≤ +��supp +� +b +��� by Lemma 24. Therefore, for a given sequence of controls u0:t ∈ Ut+1, there can be at most +|supp(b0)| resulting beliefs (see Lemma 23). As there are at most |U||T | such sequences u0:t, t ∈ T \ {T}, +this leads to +��BR,D +�1,T �(b0) +�� ≤ 1 + |supp(b0)||U||T |. +4FD +0:t is the set of compositions of mappings F u,o +t′ +from time t′ = 0 to time t′ = t for all controls u ∈ U and observation +o ∈ O, while the set FD +�0,t� is the set of all mappings F u,o +t +between time 0 and time t. +9 + +∆(X) +b +B = ∆(X) ∪ {δ∂} +τ u,o′ +t +∆(X) +∆(X) +(b, 0) +� +F u,o +t +� +⋆ +R +� +(b′ +|X, b′(∂) +���� +∈R +) +� +Figure 1: Illustration of the beliefs dynamics as +pushforward measures +∆(X) +b +B = ∆(X) ∪ {δ∂} +τ u′,o′ +t+1 ◦ τ u,o +t +∆(X) +∆(X) +∆(X) +� +F u,o +t +� +⋆ +� +F u′,o′ +t+1 +� +⋆ +R += +� +F u′,o′ +t+1 ◦ F u,o +t +� +�� +� +∈X +X +� +⋆ +Figure 2: Illustration of the composition of be- +lief dynamics as pushforward measures. +We now present the subclass of Separated Deterministic Partially Observed Markov Decision Pro- +cesses (Separated Det-Pomdp), which is simpler than Det-Pomdp. +4 +Separated Det-Pomdp and complexity +In this section, we introduce a subclass of Det-Pomdps: Separated Det-Pomdps. First, we define +this subclass in §4.1. Second, in §4.2, we present an improved bound on the cardinality of the set of +reachable beliefs for Separated Det-Pomdps compared to Det-Pomdps. Third, in §4.3, we show that +the improved bound is tight. +4.1 +Definition of (∂)-separated mapping set and Separated Det-Pomdp +Let us first define separated mapping sets. +Definition 9. Let Y1 and Y2 be two given sets. A set G ⊂ L(Y1; Y2) of mappings from Y1 to Y2 is +called a separated mapping set if +∀(g1, g2) ∈ G2 , ∀y ∈ Y1 , +� +g1(y) = g2(y) ⇒ g1 = g2 +� +. +A separated mapping set G ⊂ L(Y1; Y2) is hence a set of mappings where all pairs of mappings are +either different everywhere, or equal everywhere. Otherwise stated, all the evaluation mappings on set +G (i.e. the mappings G → Y2, g �→ g(y), for a fixed y ∈ Y1) are injective for all y ∈ Y1. For example, let +Y1 = �1, n� and Y2 = R. Then, G ⊂ RY1 is identified with G ⊂ Rn, and G is a Separated mapping set if +and only if the projections of G along each axis are injective. +In the special case where Y1 = Y2 = X, with the extended set X = X ∪ {∂} defined in Equation (3), +we want to extend the above notion of separated mapping set to tackle the added point ∂ in a specific +way. We thus introduce the notion of (∂)-separation for a pair of self-mappings on the set X and the +notion of (∂)-separated mapping set. +Definition 10. Let X = X∪{∂}. A pair (g1, g2) ∈ L(X; X) of self-mappings on the set X is (∂)-separated +if the restriction of the pair (g1, g2) to the set g−1 +1 (X) ∩ (g2)−1(X) is separated. Moreover, a set G of +self-mappings on the set X is called a (∂)-separated mapping set if all pairs of mappings (g1, g2) ∈ G2 +are (∂)-separated. +Definition 11. A Separated Det-Pomdp is a Det-Pomdp such that the set of pushforward of the +Det-Pomdp FD, defined in Equation (22), is a (∂)-separated mapping set. +Otherwise stated, for a Separated Det-Pomdp, if two sequences of controls lead to the same state +when starting in state x, then applying the two sequences of controls to another state x′ either leads to +the same state, or at least one sequence of controls leads to the cemetery point ∂. +We now present a link between the notion of separated mapping set and the notion of Separated +Det-Pomdp. This allows us to propose a sufficient condition in order to ensure that a Det-Pomdp is +a Separated Det-Pomdp. +10 + +Proposition 12. If the set � +t∈T \{T } f Ut+1 +0:t += {f u0:t +0:t | ∀t ∈ T \ {T}, ∀u0:t ∈ Ut+1} of the composition +of the evolution functions of Problem (2) is a separated mapping set, as defined if Definition 9, then +Problem (2) is a Separated Det-Pomdp. +Proof. The proof of Proposition 12 is a direct consequence of Corollary 26. The detailed proof is found +in Appendix A.2. +Note that the observation mappings {ht}t∈T \{T } do not play any role in Proposition 12. +Now that we have defined the subclass of Separated Det-Pomdps, we present a bound on the +cardinality of the set of reachable beliefs for this subclass. +4.2 +Complexity analysis of Separated Det-Pomdp +We now present the main interest of Separated Det-Pomdp when compared to Det-Pomdp, namely +that the bound on cardinality of the set of reachable beliefs is lowered from (1 + |X|)|supp(b0)| to 1 + +� +2|supp(b0)| − |supp(b0)| +� +|X|. +Theorem 13. Consider a Separated Det-Pomdp optimization problem given by Problem (2) which +satisfies the finite sets Assumption 1. For all initial belief b0 ∈ ∆(X), the cardinality of the set BR,D +�1,T �(b0) +of reachable beliefs starting from b0 satisfies the following bound +��BR,D +�1,T �(b0) +�� ≤ 1 + +� +2|supp(b0)| − |supp(b0)| +� +|X| . +(26) +Proof. The proof is detailed in Appendix A.2. +We have therefore an improved complexity of the Dp Algorithm for Separated Det-Pomdp compared +with standard Det-Pomdp. +Corollary 14. Consider a Separated Det-Pomdp optimization problem given by Problem (2) which sat- +isfies the finite sets Assumption 1. Then, the Dp Algorithm numerically solves Problem (2) by Dynamic +Programming and its complexity is +O +� +min +� +1 + +� +2|supp(b0)| − |supp(b0)| +� +|X|, 1 + |supp(b0)||U||T |� +|T ||U||O| +� +. +Proof. By Proposition 3, the Dp Algorithm solves Problem (2)and its complexity is O +� +|T ||BR,D +�1,T �(b0)||U||O| +� +. +Then, by Theorem 13, we have +��BR,D +�1,T �(b0) +�� ≤ 1 + +� +2|supp(b0)| − |supp(b0)| +� +|X| and, by Theorem 4, we +have that +��BR,D +�1,T �(b0) +�� ≤ 1 + |supp(b0)||U||T |. +As the bound presented in Theorem 13 depends on the states that can be reached when starting from +states in the support of the initial belief, we can obviously improve the bound when the support of the +belief belongs to a subset of X stable by the dynamics {ft}t∈T . +Remark 15. Assuming that Problem (2) is a Separated Det-Pomdp, that Assumption 1 holds, that +|supp(b0)| > 1, that the evolution functions {ft}t∈T \{T } of Problem (2) satisfy the property that there +exists a subset A ⊂ X such that, for all time t ∈ T \ {T}, ft(A, U) ⊂ A. Assume that supp(b0) ⊂ A. +Then the bound presented in Theorem 13 can be improved as +��BR,D +�1,T �(b0) +�� ≤ 1 + +� +2|supp(b0)| − |supp(b0)| +� +|A| . +(27) +Now that we have a better bound than with non-separated Det-Pomdps, the question is whether it +is tight or not. We now show that it is. +11 + +4.3 +Existence of Separated Det-Pomdps with tight bound +In Theorem 13, we have given an improved bound on the cardinality of the set of reachable beliefs for +Separated Det-Pomdp compared with standard Det-Pomdp. We now prove that the bound is tight. +Proposition 16. There exist a Separated Det-Pomdpsuch that equality is obtained in Equation (26), +that is, +��BR,D +�1,T �(b0) +�� = 1 + +� +2|supp(b0)| − |supp(b0)| +� +|X| . +(28) +Proof. We exhibit a simple Separated Det-Pomdp for which the set of reachable beliefs BR,D +�1,T �(b0) +satisfies Equation (28). Following the framework of §2, let X = {x1, x2, x3} consists of three distinct +states, O = {¯o1, ¯o2} of two distinct observations, and U = {¯u1, ¯u2} of two distinct controls. +The +evolution functions are defined as ∀x ∈ X , f(x, ¯u1) = x, and ∀i ∈ {1, 2, 3}, f(xi, ¯u2) = xmod(i,3)+1, +where mod(i, 3) is the remainder of the euclidean division of i by 3. Finally, the observation mapping is +given by h(x, u) = +� +¯o2 if x = x3 and u = ¯u1 , +¯o1 otherwise . +. +We show in Figure 3 the mappings F (u,o) defined in Equation (16) for this simple case, and we +illustrate the dynamics and observation functions in Figure 4. +F ¯u1,¯o1 +x1 +x2 +x3 +∂ +x1 +x2 +x3 +∂ +F ¯u1,¯o2 +x1 +x2 +x3 +∂ +x1 +x2 +x3 +∂ +F ¯u2,¯o1 +x1 +x2 +x3 +∂ +x1 +x2 +x3 +∂ +F ¯u2,¯o2 +x1 +x2 +x3 +∂ +x1 +x2 +x3 +∂ +Figure 3: Representation of the F (u,o) mappings in the case of +§4.3 +¯u1 +x1 +x2 +x3 +x1 +x2 +x3 +¯o1 +¯o2 +¯u2 +x1 +x2 +x3 +x1 +x2 +x3 +¯o1 +Figure 4: Representation of the dy- +namics and the observations depend- +ing on the control of the case of §4.3 +By adding a cost function L, a horizon T > 0 and admissibility constraints Uad : x ⇒ U, the resulting +problem has all the ingredients of a Det-Pomdp (as presented in §2), where Assumption 1 holds. +We now prove that the resulting Det-Pomdp is a Separated Det-Pomdp. For that purpose, we +enumerate all the possible results of the dynamics before applying Proposition 12. For this purpose, let +us consider a sequence of controls (u1, . . . , ut) ∈ Ut. By denoting f u1:t the compositions of dynamics +(i.e. f u1:t(x) = f ut ◦ · · · ◦ f u1(x)), we have, for all i ∈ �1, 3�, f u1:t(xi) = xmod(i+γ(u1:t)−1,3)+1, where γ +is the function that counts the number of times ¯u2 appears in a sequence of controls. The function γ is +defined as γ : Ut → N, u1:t �→ +��{ui, i ∈ �1, t� | ui = ¯u2} +��. +The set {f u1:t | u1:t ∈ Ut} is thus such that, for all sequences of controls (u1:t, u′ +1:t′) ∈ Ut × Ut′, if +there is a state x ∈ X such that f u1:t(x) = f u′ +1:t′ (x), then for any state x′ ∈ X, f u1:t(x′) = f u′ +1:t′ (x′). +Hence, the set ∪t∈T \{T }f Ut+1 +0:t +is a separated mapping set. By Proposition 12, the optimization problem +is hence a Separated Det-Pomdp. +We now chose an initial belief b0 such that supp(b0) = {x1, x2}, for which we can compute explicitly +the reachable beliefs. We can apply Theorem 13 with such initial belief. Therefore, according to Equa- +tion (26), there can be at most 7 reachable beliefs (including δ∂). In Table 1, we enumerate all possible +supports of the reachable beliefs when starting with belief b0 . +We have therefore 7 different supports for the reachable beliefs, hence at least 7 beliefs in the set of +reachable beliefs starting from b0. As Equation (26) states that there can be at most 7 reachable beliefs, +we obtain that we have exactly 7 reachable beliefs and thus Equation (28) is obtained. +Note that, while the proof of Proposition 16 was made with a Separated Det-Pomdp with |X| = 3, +we can generate a Separated Det-Pomdp such that equality is obtained in Equation (26) for any +|X| = n, n ≥ 3. We need once again that X = {xi}i∈�1,n� consists of n distinct states, O = {¯o1, ¯o2} +12 + +Mapping applied +Support of resulting belief +F ¯u1,¯o1 +{x1, x2} +F ¯u2,¯o1 +{x2, x3} +F ¯u2,¯o1 ◦ F ¯u2,¯o1 +{x3, x1} +F ¯u1,¯o2 ◦ F ¯u2,¯o1 +{x3} +F ¯u2,¯o1 ◦ F ¯u1,¯o2 ◦ F ¯u2,¯o1 +{x1} +F ¯u2,¯o1 ◦ F ¯u2,¯o1 ◦ F ¯u1,¯o2 ◦ F ¯u2,¯o1 +{x2} +F ¯u1,¯o2 +{∂} +Table 1: Resulting support when applying given mappings to the initial belief b0 with supp(b0) = {x1, x2} +of two distinct observations and U = {¯u1, ¯u2} of two distinct controls. Then, the dynamics is given by +∀x ∈ X , f(x, ¯u1) = x, and ∀i ∈ �1, n�, f(xi, ¯u2) = xmod(i,n)+1. Finally, the observation mapping is +given by h(x, u) = +� +¯o2 if x = xn and u = ¯u1 , +¯o1 otherwise . +Now that we have presented the subclass of Separated Det-Pomdps, we give a numerical illustration. +5 +Numerical application on a toy example of Separated Det- +Pomdp +In this section, we present a simple one-dimensional illustration of Separated Det-Pomdp. We consider +that we empty a tank while minimizing an associated cost, as illustrated in Figure 5. The state is one- +dimensional and consists in the volume of water present in the tank. The control is also one-dimensional +and is the amount of water that the decision maker removes during one time step. The decision maker +has access at time t to partial observation, as he only knows that the volume of water in the tank is +between two quantized levels. +5.1 +A partially observed tank as a Separated Det-Pomdp +More precisely, the problem is the following. +• The state x consists of a discrete volume of water in the +tank, with +x ∈ X = {x(1), x(2), . . . , x(n)} ⊂ R+ of finite cardinality +n. +• The observation o consists of a discrete level of water in +the tank, with +o ∈ O = {o(1), o(2), . . . , o(m)} ⊂ R+ of finite cardinality +m. +• The control u consists of a discrete volume of water to be +removed, with +u ∈ U = {u(1), u(2), . . . , u(d)} ⊂ R+ of finite cardinality +d. +• The unitary price of water at each time t ∈ T \ {T} is +given by ct ∈ R. +o(2) +o(3) +o(1) +Figure 5: +Illustration of the wa- +ter tank “quantum” of observation +(m = 3) +13 + +Optimization problem. +We now adapt the Problem (2) to the tank case presented above: +min +X,U,OE +�T −1 +� +t=0 +ctUt +� +(29a) +s.t. PX0 = b0 , +(29b) +Xt+1 = Xt − Ut , ∀t ∈ T \ {T} , +(29c) +Ut ∈ {u(i) ∈ U | u(i) ≤ Xt} , ∀t ∈ T \ {T} , +(29d) +Ot = max{o(j) ∈ O | Xt ≥ o(j)} , ∀t ∈ T , +(29e) +σ(Ut) ⊂ σ (O0, . . . , Ot, U0, . . . , Ut−1) , ∀t ∈ T \ {T} . +(29f) +Equation (29a) represents the objective function of the tank problem, i.e. Equation (2a) of Prob- +lem (2). The instantaneous cost function at time t is defined as Lt(ut) = ctut, and hence only depends on +the controls. The evolution function corresponds to emptying the tank and is given by f : (x, u) �→ x−u, +which gives Equation (29c). The observation function h is given by a piecewise constant function which +does not depend on the controls u: h(x) = max{o(i) | x ≥ o(i)}. +This leads to equation (29e), which is the implementation of (2e). The admissibility set of the tank +problem is given by Uad(Xt) = [0, Xt] (see Equation (29d)). It ensures that we cannot remove more +water than what is in the tank. Note that this could be a problem as we do not observe Xt. +Problem (29) has the same form as Problem (2). It is therefore a Det-Pomdp and all the relevant +results presented in §3.1 hence apply. +Associated beliefs dynamics τ. +Let (b, u, o) ∈ B × U × O, with B = ∆(X) ∪ {δ∂}, as defined in +Equation (4). As the evolution functions and observation functions are stationary, the belief dynamics +are also stationary. +We note I(o) ⊂ X, the set of states compatible with the observation o, i.e. +I(o) = {x ∈ X | h(x) = o} . +(30) +By Equation (29c), we have (f u)−1(y) = y + u. Moreover, we have, by the definition of I(o) in +Equation (30), that +� +hu�−1(o) = I(o). Hence, the function Q in (7) is here +Q : B × U × O → [0, 1] , (b, u, o) �→ +� +x∈I(o)−{u} +b(x) , +and Equation (9) gives +τ(b, u, o)(y) = +� +� +� +� +� +� +� +b(y + u) +� +x′∈I(o)−{u} +b(x′) +if +y ∈ I(o) − {u} , +0 +if +y ̸∈ I(o) − {u} , +where I(o) − {u} is defined in Equation (30). +Bellman equations for the partially observed tank problem. +As Problem (29) is a Det-Pomdp +and the finite sets Assumption 1 holds, we can apply Proposition 1. Equations (10a) and (10b) are here +VT : BR,D +T +(b0) → R , b �→ 0 +(31a) +Vt : BR,D +t +(b0) → R , b �→ +min +u≤minx∈supp(b) x +� +ctu + +� +o∈O +� +x−u∈[o,o] +b(x)Vt+1 +� +τ(b, u, o) +�� +. +(31b) +Indeed, the intersection Ub,ad +t +(b) = � +x∈supp(b) Uad +t (x) is {u(i) ∈ U | u ≤ minx∈supp(b) x}, as the admissi- +bility set is given by Equation (29d), and as +{u(i) ∈ U | u(i) ≤ x(j)} ∩ {u(i) ∈ U | u(i) ≤ x(k)} = {u(i) ∈ U | u(i) ≤ min +� +x(j), x(k)� +} . +14 + +The partially observed tank problem as a Separated Det-Pomdp. +The tank Det-Pomdp is a +Separated Det-Pomdp as a direct consequence of Corollary 29, in Appendix A.2. Indeed, Corollary 29 +states that if the evolution functions ft of a Det-Pomdp are linear, then it is a Separated Det-Pomdp. +As the evolution function f of the partially observed tank is indeed linear, the tank Det-Pomdp is a +Separated Det-Pomdp. +5.2 +Numerical results +We now present some numerical results for the tank problem described by Problem (29). +Presentation of the instances +We made a numerical application with the following parameters: +• X = �0, 300�, +• U = �0, 9�, +• O = {0, 1, 20, 40, 60, 80, 100, 120, 140, 160, 180, 200, 220, 240, 260, 280, 300}, +• T = �0, 100�, +• supp(b0) = �260, 300�, with a randomly generated probability distribution over that support, the +distribution used is detailed in Figure 6. +260 +270 +280 +290 +300 +0 +5 · 10−2 +0.1 +0.15 +x +b0(x) +Figure 6: Probability distribution used as the initial belief b0 for the numerical applications +When considering the initial belief b0 presented in Figure 6 and a “true” (unknown) initial state of +x0 = 290 (used to simulate the observation process depending on the policy), we obtain the tank water +volume represented in Figure 7. +Moreover, we have a set of reachable beliefs BR,D +�0,100� such that |BR,D +�0,100�| = 64, 400. We therefore do +not display value functions, as they are defined on sets with large cardinality. +We also made a second numerical application where the observation O is changed to: +• O = {1, 6, 11, 51, 101, 151, 201, 251} +When considering the new observations set and the same initial belief and initial state, we obtain a +trajectory represented in Figure 8. +Figures 7 and 8 both illustrate some properties of Det-Pomdps. +• First, in both cases, we see that the support of the beliefs decreases with time (the vertical red +slices are non increasing). +• Second, we remark that such a decrease is due to the observations. Indeed, in Problem (29), the +observation function ensures that the support of the beliefs must belong to intervals [ot, ot] when +we observe ot (see Equation (30)). Thus, the supports of the beliefs are reduced along the limit of +15 + +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +0 +50 +100 +150 +200 +250 +300 +Time +Possible States Xt +0 +0.5 +1 +1.5 +2 +2.5 +3 +Prices +Figure 7: Representation of a trajectory of the +volume of water in the tank when applying the +optimal controls when considering the first set of +observations. A vertical slice at time t of the red +area represents the support of the belief held at +time t, the dotted blue curve represents the tra- +jectory of the “true” state, the piecewise constant +green curve is the observation we have access to +at time t, and the dashed orange curve represents +the periodic prices. +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +0 +50 +100 +150 +200 +250 +300 +Time +Possible States Xt +0 +0.5 +1 +1.5 +2 +2.5 +3 +Prices +Figure 8: Representation of a trajectory of the +volume of water in the tank when applying the +optimal controls when considering the second set +of observations. A vertical slice at time t of the +red area represents the support of the belief held +at time t, the dotted blue curve represents the tra- +jectory of the “true” state, the piecewise constant +green curve is the observation we have access to +at time t, and the dashed orange curve represents +the periodic prices. +those intervals, as is more easily seen in Figure 8 between time t = 1 to t = 6 (we apply a control, +i.e. removing some water, and we see that the lower part of the support remains at the observation +value until time t = 7, which is when we change observation and we see that the upper bound of +the support gets just beneath the previous observation, i.e. at x = 249). +• Third, we remark that, as could be expected, the optimal policy consists of removing water when +prices are high, and stopping when prices are low. +• Fourth, we remark that, despite having fewer observations in the second case, the optimal trajectory +in the second case reaches a deterministic belief (i.e. such that |supp(b)| = 1) much sooner in +Figure 8 compared to Figure 7 (at time t = 33 for the second case and time t = 53 for the first +case). Having more observations hence does not guarantee to remove ambiguities at a faster rate. +We now present the computation time of the Dp Algorithm and compare it to another algorithm, Sarsop. +Comparison with Sarsop. +In this paragraph, we focus on the comparison with Sarsop, first in- +troduced in (Kurniawati et al., 2008). We used the Julia implementation of this algorithm, with the +POMDPs package API. The following results were obtained on a computer equipped with a Core i7- +8665U and 32 GB of memory, using Julia v1.7.3, POMDPs v0.9.3 and Sarsop v0.5.5. +However, we must first warn the reader that Sarsop is an algorithm that solves an infinite horizon +Pomdp. We hence reformulate the finite horizon Det-Pomdp as an infinite time Pomdp by extending +the state with the time variable. Such reformulation leads to a much bigger problem in terms of data +and size of the state space, which heavily penalizes Sarsop. Hence, the reformulation prevents any fair +comparison of computation times. We still present some computation time in Table 2. +Note that, for each instance where the computation did not stop (i.e. those without a “>” symbol +in the computation time column) due to hitting the memory limit of the computer, Sarsop and the +Dp Algorithm found the same value, hence Sarsop indeed converged toward the optimal solution of +Problem (29). +16 + +|X| +|U| +|O| +|supp(b0)| +T +Sarsop +Dp Algorithm +computation time (s) +computation time (s) +11 +2 +3 +2 +20 +0.376 +0.002 +21 +2 +5 +2 +25 +0.16 +0.003 +51 +5 +5 +2 +100 +24.9 +0.20 +51 +5 +5 +4 +100 +27.2 +1.20 +51 +5 +5 +6 +100 +29.4 +3.03 +101 +5 +5 +2 +200 +359 +0.96 +101 +5 +5 +10 +200 +1930 +32.2 +101 +10 +5 +10 +200 +1069 +78.2 +201 +5 +5 +10 +200 +3506 +62.1 +201 +10 +5 +10 +200 +15618 +309 +201 +5 +5 +20 +200 +3652 +225 +201 +10 +6 +20 +200 +33562 +497 +301 +5 +6 +10 +200 +4638 +86.8 +301 +10 +6 +10 +300 +> 38000 +762 +(> 19217s of iterations) +Table 2: Computation time of different instances of both Sarsop and the Dp Algorithm +6 +Conclusion +In this paper, we have presented a subclass of Pomdps, Separated Det-Pomdps, which has proper- +ties that contribute to push back the curse of dimensionality for Dynamic Programming. Indeed, we +have shown that the conditions on the dynamics for Separated Det-Pomdp improve the bound on the +cardinality of the set of the reachable beliefs: the bound is reduced from +� +1 + |X| +�|supp(b0)| (in the case +of Det-Pomdp, see Theorem 4) to 2|supp(b0)||X| (Theorem 13), as presented in Table 3. This tighter +bound allows Dynamic Programming algorithms to efficiently solve Separated Det-Pomdp problems, +especially when considering small supports of the initial state distributions. Moreover, the bound is tight +(see Proposition 16). +The Separated Det-Pomdp class is, therefore, an interesting framework for some problems as only +a fraction of the number of beliefs needs to be considered, in comparison with Det-Pomdp or Pomdp. +The Separated Det-Pomdps are therefore tractable with larger instances than regular Pomdps or Det- +Pomdps. +Class +Infinite horizon bound +Finite horizon bound +Det-Pomdp +(1 + |X|)|X| +min +� +(1 + |X|)|X| , +� +|U||O| +�|T |� +(Littman, 1996) +Det-Pomdp +(1 + |X|)|supp(b0)| +min +� +(1 + |X|)|supp(b0)| , 1 + |supp(b0)||U||T |� +improved bounds +Theorem 4 +Theorem 4 +Separated +1 + +� +2|supp(b0)| − |supp(b0)| +� +|X| +min +� +1 + +� +2|supp(b0)| − |supp(b0)| +� +|X|, +Det-Pomdp +1 + |supp(b0)||U||T |� +Theorem 13 +Corollary 14 +Table 3: Summary of the bounds depending on the class of problem +A +Appendix +First, in §A.1, we present technical lemmata used to prove bounds on the cardinality of the sets of +reachable beliefs. Second, in §A.2, we present complementary results on (∂)-separated mappings sets. +17 + +A.1 +Technical lemmata +In this subsection, we present technical lemmata used in the proofs of Theorem 4. We first introduce +in §A.1.1 the notions of forward and backward mappings. Second, in §A.1.2, we present properties on +the composition and pushforward measures by those forward and backward mapping. Third, in §A.1.3, +we present properties on the cardinality of sets of forward and backward mappings used notably in the +proof of Theorem 4. +A.1.1 +Forward and backward mappings +For any subset X ⊂ X, we introduce the notion of X-forward and X-backward mappings. Given a +mapping h : X → X and a subset X ⊂ X, we define a mapping h− +→ +X : from X to X, called a X-forward +mapping, as follows +h− +→ +X : x ∈ X �→ +� +h(x) +if +x ∈ X +and +h(x) ∈ X , +∂ +if +x = ∂ +or +h(x) ̸∈ X . +(32) +We call h− +→ +X : X → X a X-forward mapping as we have h− +→ +X(X) ⊂ X ∪{∂}. X-forward imposes a constraint +on the codomain (set of destinations): we only keep the values that belong to X, whereas the others are +sent to ∂. The set X is thus a subset of the codomain of h. +We also introduce the X-backward mapping h← +− +X : X → X, defined by +h← +− +X : x ∈ X �→ +� +h(x) +if +x ∈ X , +∂ +otherwise. +(33) +We call h← +− +X : X → X a X-backward mapping as we have h← +− +X(X) ⊂ X, and h← +− +X +� +X \ X +� += {∂}. X- +backward imposes a constraint on the domain (set of departures): we only keep the values whose inputs +are in X, whereas the others are sent to ∂. The set X is thus a subset of the domain of h. +It is straightforward to check that we have +∀X ⊂ X , h− +→ +X = h←−−−−− +h−1(X) , +(34a) +∀X ⊂ X , h− +→ +X = h−−−−−−→ +X∩Im(h) , +(34b) +where Im is the image of a mapping, that is Im(h) = h(X). A forward mapping can hence be rewritten +as a backward mapping. The reverse is not true, as we have +h← +− +X = h−−−→ +h(X) ⇔ h−1� +h(X) +� += X . +A.1.2 +Results on pushforward measures by forward and backward mappings sets +We now present properties of the composition of pushforward measures of forward and backward map- +pings. +Definition 17. Let M ⊂ L(X; X) be a subset of self mappings on the set X. We say that G ⊂ L(X; X) +is an +� +M, ←− +X +� +-mappings set (resp. an +� +M, −→ +X +� +-mappings set) if it satisfies the following property +G ⊂ +� +h← +− +X +�� h ∈ M and X ⊂ X +� +, +(35a) +� +resp. +G ⊂ +� +h− +→ +X +�� h ∈ M and X ⊂ X +�� +, +(35b) +where h← +− +X (resp. +h− +→ +X) is defined in Equation (33) (resp. +Equation (32)). +When M = L(X; X), a +� +M, ←− +X +� +-mappings set (resp. +an +� +M, −→ +X +� +-mappings set) is just named a +�←− +X +� +-mappingsset (resp. +an +�−→ +X +� +-mappings set). +We obtain the following properties. +• If G is an +� +M, −→ +X +� +-mappings set, then G is an +� +M, ←− +X +� +-mappings set (using Equality (34a)). +18 + +• +�←− +X +� +-mappings sets are stable by composition, as we easily obtain that +h′←− +X′ ◦ h← +− +X = (h′ ◦ h)←−−−−−−−− +X∩h−1(X′) . +(36) +• Let G be an +�←− +X +� +-mappings set and consider, for any X ⊂ X, the subset G← +− +X of G defined by +G← +− +X = +� +g ∈ G +�� ∃h ∈ L(X; X), g = h← +− +X +� +. +(37) +Then, for any belief b0 ∈ ∆(X), we have +� +R ◦ (G←−−−−−−−− +X∩supp(b0))⋆ +� +(b0) = +� +R ◦ (G← +− +X)⋆ +� +(b0) . +(38) +The Equation (38) is a consequence of the following Lemma 18. Indeed, assuming Lemma 18, +the expression of +� +R ◦ (G← +− +X)⋆ +� +(b0) given by Equation (39b) only depends on the restriction of the +measure b0 to the subset X – which coincides with the restriction of the measure b0 to the subset +X ∩ supp(b0) – as the measure b0 is null outside its support. +Lemma 18. Let X be a subset of X. The mappings R ◦ (h← +− +X)⋆ and R ◦ (h− +→ +X)⋆ in L(∆(X); B), where the +pushforward measure is defined in Equation (15), and the mapping R is defined in Equation (17), have +the following expressions for all ν ∈ ∆(X): +� +R ◦ (h− +→ +X)⋆ +� +(ν) = +� +� +� +� +x ∈ X �→ ν +� +h−1(x) +� +1X(x) +ν +� +h−1(X) +� +� +if +ν +� +h−1(X) +� +̸= 0 , +δ∂ +otherwise, +(39a) +and +� +R ◦ (h← +− +X)⋆ +� +(ν) = +� +� +� +� +x ∈ X �→ ν +� +h−1(x) ∩ X +� +ν +� +h−1(X) ∩ X +� +� +if +ν +� +h−1(X) ∩ X +� +̸= 0 , +δ∂ +otherwise. +(39b) +Proof. For any probability measure ν on the finite set X, it is straightforward, using the definition of +pushforward measure in Equation (15), to obtain that the pushforward of the measure ν through the +mapping h− +→ +X, as defined in Equation (32), is given by +(h− +→ +X)⋆ν : X → R+ +y �→ ν +� +(h− +→ +X)−1(y) +� += +� +� +� +� +� +� +� +ν +� +h−1(y) +� +if +y ∈ X , +� +1 − ν +� +h−1(X) +�� +if +y = ∂ , +0 +if +y ̸= ∂ and y ̸∈ X . +(40) +Thus, we obtain that +∀x ∈ X , +� +(h− +→ +X)⋆ν +� +|X(x) = ν +� +h−1(x) +� +1X(x) , +(41) +and that +� +(h− +→ +X)⋆ν +� +(X) = +� +x∈X +ν +� +h−1(x) +� +1X(x) = ν +� +h−1(X) +� +. +(42) +Hence, using the definition of R in Equation (17), the result follows from Equation (39a). The proof of +Equation (39b) is very similar and left to the reader. +The composition of self-mappings of the form R ◦ (h− +→ +X)⋆ can also be written without resorting to +multiple renormalizations. Instead, we only need to renormalize the composition of the pushforward +measures, as shown below. +19 + +Lemma 19. Assume that h and h′ are self-mappings on the finite set X. Then, for any subsets X and +X′ of X, we have the following composition equalities +R ◦ (h− +→ +X)⋆ ◦ R ◦ (h′−→ +X′)⋆ = R ◦ (h− +→ +X ◦ h′−→ +X′)⋆ , +(43a) +R ◦ (h← +− +X)⋆ ◦ R ◦ (h′←− +X′)⋆ = R ◦ (h← +− +X ◦ h′←− +X′)⋆ . +(43b) +Proof. We just prove Equation (43a) as the proof follows the same lines for Equation (43b). +As a +preliminary, we remark that the mapping R ◦ (h− +→ +X)⋆ is defined on the nonnegative measures on the set +X and not just on probability measures. Now, given µ ∈ ∆(X), we consider the nonnegative measure +µ′ = (µ|X, 0). The two nonnegative measures µ and µ′ coincide on the set X. Thus using the expression of +R ◦ (h− +→ +X)⋆ in Equation (39a) and the fact that X ⊂ X, we obtain that R ◦ (h− +→ +X)⋆(µ) = R ◦ (h− +→ +X)⋆(µ|X, 0). +Now, let ν ∈ ∆(X) be given. We denote by ν′ ∈ ∆(X) the probability measure ν′ = (h′−→ +X′)⋆ν. We +consider two cases: either ν′(X) ̸= 0, or ν′(X) = 0. +First case. We assume that ν′(X) ̸= 0. Then, we successively have +R ◦ (h− +→ +X)⋆ ◦ R ◦ (h′−→ +X′)⋆ν = R ◦ (h− +→ +X)⋆ ◦ R(ν′) +(by replacing (h′−→ +X′)⋆ν by ν′) += R ◦ (h− +→ +X)⋆ +� +1 +ν′(X)ν′ +|X, 0 +� +(using R definition in (17), with ν′(X) ̸= 0) += R ◦ (h− +→ +X)⋆ +� +1 +ν′(X)(ν′ +|X, 0) +� +(factorizing by +1 +ν′(X)) += R +� +1 +ν′(X)(h− +→ +X)⋆ +� +ν′ +|X, 0 +�� +(as (h− +→ +X)⋆ is 1-positively homogeneous) += R +� +(h− +→ +X)⋆ +� +ν′ +|X, 0 +�� +(as R is 0-positively homogeneous) += R +� +(h− +→ +X)⋆(ν′) +� +(using the preliminary part) += R ◦ (h− +→ +X)⋆ ◦ (h′−→ +X′)⋆ν +(as ν′ = (h′−→ +X′)⋆ν) += R ◦ (h− +→ +X ◦ h′−→ +X′)⋆(ν) . +(as f⋆ ◦ h⋆ = (f ◦ h)⋆) +Second case. We assume that ν′(X) = 0. Then, we have that ν′ = δ∂ as ν′ ∈ ∆(X), and we obtain +R ◦ (h− +→ +X)⋆ ◦ R ◦ (h′−→ +X′)⋆ν = R ◦ (h− +→ +X)⋆ ◦ R(δ∂) +(by replacing (h′−→ +X′)⋆ν by ν′ = δ∂) += R ◦ (h− +→ +X)⋆(δ∂) +(as R(δ∂) = δ∂) += R ◦ (h− +→ +X)⋆ ◦ (h′−→ +X′)⋆ν +(by replacing δ∂ = ν′ by (h′−→ +X′)⋆ν) += R ◦ (h− +→ +X ◦ h′−→ +X′)⋆(ν) . +Hence, in both cases, we obtain Equation (43a). +Now that we have exposed technical lemmata on the composition and renormalization of +�−→ +X +� +- +mappings and +�←− +X +� +-mappings, we present lemmata on the cardinality of sets of pushforward measures, +notably the cardinality of pushforward measures by +�−→ +X +� +-mappings and +�←− +X +� +-mappings. +A.1.3 +Results on the cardinality of sets of pushforward measures +We now present results on the cardinality of sets of forward and backward mappings. +Lemma 20. Let {Gk}k∈N be a given sequence where, for each k ∈ N, the set Gk ⊂ L(X; X) is a finite +set of self-mappings on the set X. The sets Gk, for all k ∈ N, are assumed to be either all +�−→ +X +� +-mappings +sets or all +�←− +X +� +-mappings sets. We define the sequence {Φk}k∈N, where, for each k ∈ N, the set Φk ⊂ +L(∆(X); ∆(X)) is a finite set of self-mappings on the set X given by +∀k ∈ N , Φk = R ◦ (Gk)⋆ . +(44) +20 + +Then, for any b0 ∈ ∆(X), we have the following bound +∀n ∈ N , +��� +n +� +k=0 +Φ0:k(b0) +��� ≤ (1 + |X|)|supp(b0)| , +(45) +where Φ0:k = Φk ◦ · · · ◦ Φ0 is defined in Equation (19). +Proof. For all k ∈ N, we have +Φ0:k(b0) = (Φk ◦ Φk−1 ◦ · · · ◦ Φ0)(b0) +(by Equation (19)) += +� +R ◦ (Gk)⋆ ◦ R ◦ (Gk−1)⋆ ◦ · · · ◦ R ◦ (G0)⋆ +� +(b0) +(by Equation (44)) += +� +R ◦ (Gk)⋆ ◦ (Gk−1)⋆ ◦ · · · ◦ (G0)⋆ +� +(b0) +by Lemma (19), as the sets Gk are, by assumption, either all +�−→ +X +� +-mappings sets or all +�←− +X +� +-mappings +sets, += +� +R ◦ (Gk ◦ Gk−1 ◦ · · · ◦ G0)⋆ +� +(b0) +(as f⋆ ◦ h⋆ = (f ◦ h)⋆) += R +� +(G0:k)⋆(b0) +� +. +Thus we have, for all n ∈ N, +��� �n +k=0 Φ0:k(b0) +��� ≤ +��� +��n +k=0 G0:k +� +⋆(b0) +���, and the conclusion follows from the +postponed Lemma 21 with J = �n +k=0 G0:k, Y = V = X, and µ = b0. +Note that we can extend the previous Lemma 20 to cases with sequences {Gk}k∈N of mixes of both +�−→ +X +� +-mappings sets and +�←− +X +� +-mappings sets. Indeed, forward mappings are also backward mappings by +Equation (34a). We can hence write the sequence {Gk}k∈N as a sequence of only +�←− +X +� +-mappings sets. +In the rest of this paper, we consider sequences of only +�−→ +X +� +-mappings sets or only +�←− +X +� +-mappings sets, +and thus only need Lemma 20. +We can bound the cardinality of the set of pushforward of a given nonnegative measure thanks to +the following Lemma 21 (which was previously postponed in the proof of Lemma 20). +Lemma 21. Let J ⊂ L(V; Y) be a set of mappings from the set V to the set Y. Assume that the sets V +and Y are both finite. Then, for any nonnegative measure µ on the set V, we have that +|J⋆µ| ≤ |Y||supp(µ)| , +(46) +where we recall that |J⋆µ| denotes the cardinal of the set +��{j⋆µ | j ∈ J} +�� as exposed in Equation (19a). +Proof. Let µ be a given nonnegative measure on V. For any j ∈ J we denote by j|supp(µ) the restriction +of the mapping j to the subset supp(µ) ⊂ V. For all y ∈ Y, we have that +j⋆µ(y) = µ +� +j−1(y) +� +(by the definition (15) of pushforward measures) += µ +�� +j−1(y) ∩ supp(µ) +� +∪ +� +j−1(y) ∩ (supp(µ))c�� += µ +� +j−1(y) ∩ supp(µ) +� ++ µ +� +j−1(y) ∩ (supp(µ))c� +� +�� +� +=0 += µ +� +j−1 +|supp(µ)(y) +� += +�� +j|supp(µ) +� +⋆µ +� +(y) . +(by (15)) +Thus, defining J|supp(µ) = {j|supp(µ) | j ∈ J}, we get that +|{j⋆µ | j ∈ J}| = |{(j|supp(µ))⋆µ | j ∈ J}| ≤ |J|supp(µ)| ≤ |Ysupp(µ)| = |Y||supp(µ)| . +This ends the proof. +We now present a lemma on the conservation of the cardinality of the support of a measure through +a composition of sets of mappings, if we have conservation of the cardinality for each individual set. +21 + +Lemma 22. Let {Φk}k∈N be a sequence of self-mappings on the set B and assume that, for all k ∈ N, +we have that +∀b ∈ B , +� +h∈Φk +|supp +� +h(b)|X +� +| ≤ |supp(b|X)| . +(47) +Then, for any b0 ∈ ∆(X), we have the following bound +∀k ∈ N , +��Φ0:k(b0) \ {δ∂} +�� ≤ |supp(b0)| , +(48) +where Φ0:k(b0) = Φk ◦ · · · ◦ Φ0(b0) is defined in Equation (19c). +Proof. Let a belief b0 ∈ ∆(X) be given. As a preliminary result we prove, by forward induction on k ∈ N, +that +∀k ∈ N , +� +b∈Φ0:k(b0) +��supp(b|X) +�� ≤ |supp(b0)| . +(49) +First, we consider the case k = 0. As Φ0:0 = Φ0 the result follows from Equation (47) used for k = 0 and +b = b0. Second, we consider 0 < k, and, assuming that Equation (49) is satisfied for k, we prove that it +is also satisfied for k+1 as follows: +� +b∈Φ0:k+1(b0) +��supp(b|X) +�� = +� +h∈Φ0:k+1 +��supp +� +(h(b0))|X +��� +(by (19)) += +� +h′∈Φk+1,h′′∈Φ0:k +���supp +�� +h′(h′′(b0)) +� +|X +���� +(as Φ0:k+1 = Φk+1 ◦ Φ0:k) += +� +h′′∈Φ0:k +� +� +h′∈Φk+1 +���supp +�� +h′(h′′(b0)) +� +|X +���� +� +≤ +� +h′′∈Φ0:k +���supp +�� +h′′(b0) +� +|X +���� +(using Equation (47) for k and b = h′′(b0)) += +� +b∈Φ0:k(b0) +��supp +� +b|X +��� +(by (19)) +≤ |supp(b0)| . +(by induction assumption (49) on k) +We conclude that Equation (49) is satisfied for all k ∈ N. +Now, we turn to the proof of Equation (48). We make the following observation: if b ∈ ∆(X), then +we have that |supp(b|X)| ≥ 1 and if b = δ∂ then |supp(b|X)| = 0. Thus, we have that +|Φ0:k(b0) \ {δ∂}| = +� +b∈Φ0:k(b0)\{δ∂} +1 +(50) +≤ +� +b∈Φ0:k(b0)\{δ∂} +|supp(b|X)| +(as |supp(b|X)| ≥ 1 for b ∈ Φ0:k(b0) \ {δ∂}) += +� +b∈Φ0:k(b0) +|supp(b|X)| +(as |supp(δ∂|X)| = 0) +≤ |supp(b0)| , +(by (49)) +which gives Equation (48). That concludes the proof. +Lemma 23. Let {hk}k∈N be a sequence of self-mappings on the set X and, for all k ∈ N, let {Xk +i }i∈Ik +be a finite family of two by two disjoints subsets of X . Let {Gk}k∈N be the sequence of self-mappings on +the set X, of the following form +∀k ∈ N , Gk = +� +hk−→ +Xk +i +�� i ∈ Ik +� +⊂ X +X , +(51) +where hk−→ +Xk +i : X → X are built following Equation (32). Consider the sequence {Φk}k∈N of self-mappings +on the set B, given, for all k ∈ N, by Φk = R ◦ (Gk)⋆ and the associated sequence (Φ0:k)k∈N as defined +in Equation (19). Then, given b0 ∈ ∆(X), we have +∀k ∈ N , +��Φ0:k(b0) \ {δ∂} +�� ≤ |supp(b0)| . +(52) +22 + +Proof. The proof relies on postponed Lemma 24 from which we obtain that the mappings Φk satisfy +Equation (47) for all k ∈ N, and on Lemma 22. +First, as a preliminary fact, we have that, for all µ ∈ ∆(X), supp +�� +R(µ) +� +|X +� += supp(µ|X). Indeed, +by (17), if µ(X) = 0, then supp +�� +R(µ) +� +|X +� += supp +� +(δ∂)|X +� += ∅ = supp +� +µ|X +� +; whereas if µ(X) ̸= 0, then +we have supp +�� +R(µ) +� +|X +� += supp +� +( +µ|X +µ(X), 0)|X +� += supp +� µ|X +µ(X) +� += supp(µ|X). +Second, we show that the mappings Φk satisfy Equation (47) for all k ∈ N. For that purpose, we fix +k ∈ N, and b ∈ B and we successively have +� +h∈Φk +��supp +� +h(b)|X +��� = +� +i∈Ik +���supp +��� +R ◦ (hk−→ +Xk +i )⋆ +� +(b) +� +|X +���� +(by definition of Φk = R ◦ (Gk)⋆ and Gk in (51)) += +� +i∈Ik +��supp +�� +(hk−→ +Xk +i )⋆(b) +� +|X +��� +(as, by the preliminary fact, ∀µ ∈ ∆(X), supp +�� +R(µ) +� +|X +� += supp(µ|X)) +≤ +��supp +� +b|h−1(⊔i∈Ik Xk +i ) +��� +(by (55) in Lemma 24, applied with Y = V = X and V = X, Vi = Xk +i for i ∈ I = Ik) +≤ +��supp +� +b|X +��� . +(as h−1(⊔i∈IkXk +i ) ⊂ X) +Third, as the assumptions given in Equation (47) are satisfied, the result follows by Lemma 22. +We now present the postponed technical Lemma 24. +Lemma 24. Let h ∈ L(Y; V) be a mapping from the set Y to the set V and assume that the sets Y and +V are both finite. Let V ⊂ V be a subset of V. We define the mapping5 hV : Y → V ∪ {∂V} taking values +in the extended set V = V ∪ {∂V} as follows +hV : y ∈ Y �→ +� +h(y) +if +h(y) ∈ V , +∂V +elsewhere . +(53) +Then, for any nonnegative measure µ on the set Y, we have that +���supp +�� +(hV )⋆µ +� +|V +���� ≤ +��supp +� +µ|h−1(V ) +��� . +(54) +Moreover, for any finite family {Vi}i∈I of pairwise disjoints subsets of V, we have that +� +i∈I +���supp +�� +(hVi)⋆µ +� +|V +���� ≤ +��supp +� +µ|h−1(⊔i∈I Vi) +��� . +(55) +Proof. We prove Equation (54). Let µ ∈ ∆(Y) be given. First, we note that, if the set supp +�� +(hV )⋆µ +� +|V +� +is empty, the result is obvious. +Second, we assume that supp +�� +(hV )⋆µ +� +|V +� +̸= ∅ and consider v ∈ +supp +�� +(hV )⋆µ +� +|V +� +. +Thus, v is restricted to belong to V and, by definition of a pushforward mea- +sure, it must satisfy µ +� +h−1 +V (v) +� +̸= 0. +This implies that h−1 +V (v) ̸= ∅ and, using the definition of +hV (in Equation (53)), we obtain that v must belong to V . +We conclude that there must exist +y ∈ h−1 +V (v) such that µ(y) ̸= 0 which, combined with the fact that the mapping h−1 +V +coincides with +the mapping h−1 on V , gives that y ∈ h−1(v) ∩ supp(µ). +Now, consider the set-valued mapping +Γ : supp +�� +(hV )⋆µ +� +|V +� +⇒ Y , y �→ h−1(v) ∩ supp(µ). +By construction, the set-valued mapping Γ takes values in the subsets of supp(µ|h−1(V )), and we +have just proved that it takes values in the nonempty subsets of µ|h−1(V ). Moreover, the set-valued +5Note that the mapping hV is slightly different from h− +→ +V . Indeed h− +→ +V are defined for self-mappings, whereas hV is +defined for an extended codomain (set of destinations). +23 + +mapping Γ is injective as, for all pairs (v′, v′′) ∈ V 2 of distinct elements, v′ ̸= v′′, we must have that +h−1(v′) ∩ h−1(v′′) = ∅, as otherwise there would exist an element y ∈ Y such that h(y) = v′ and +h(y) = v′′, which is not possible. Thus, the image of Γ is a partition of a subset of supp(µ|h−1(V )) and +we conclude that +��supp +�� +(hV )⋆µ +� +|V +��� = +��Γ +� +supp +�� +(hV )⋆µ +� +|V +���� ≤ |supp(µ|h−1(V ))| , +which gives Equation (54). +Now, we turn to the proof of Inequality (55). We successively have +� +i∈I +���supp +�� +(hVi)⋆µ +� +|V +���� ≤ +� +i∈I +��supp +� +µ|h−1(Vi) +��� +(by (54) for each i ∈ I) += +��supp +� +µ|⊔i∈I h−1(Vi) +��� +(as the family of subsets {h−1(Vi)}i∈I is composed of pairwise disjoints subsets as it was the case for the +family {Vi}i∈I) += +��supp +� +µ|h−1(⊔i∈I Vi) +��� , +(as h−1(⊔i∈IVi) = ⊔i∈Ih−1(Vi)) +which concludes the proof. +This technical Lemma 24 shows that the cardinality of the support of a measure decreases when +the measure is transported by a pushforward measure induced by a mapping of the form given by +Equation (53). A similar result +∀t ∈ T , ∀b ∈ B , ∀u ∈ U , +� +o∈O +��supp +� +τt(b, u, o) +��� ≤ +��supp(b) +�� , +is given in (Littman, 1996, Lemma 6.2) but only for the mappings (τt)t∈T defined in Equation (9), and +with a proof not explicitly connected to pushforward measures. +We now present the postponed proof of Lemma 7, presented in page 8. +Proof of Lemma 7. Fix (u, o) ∈ U × O, t ∈ T \ {T}, and b ∈ B and denote by X ⊂ X the subset +X = +� +hu +t+1 +�−1(o). +We need to prove that we have +τt(b, u, o) = R ◦ (F u,o +t +)⋆(b) . +(56) +Using Equation (7), we have that +Qt+1(b, u, o) = b +� +(hu +t+1 ◦ f u +t )−1(o) +� += b +� +(f u +t )−1(X) +� +. +(57) +Now, using the expression of τt in Equation (9) combined with Equation (57) and the definition of X, +we obtain, for all x ∈ X, that +τt(b, u, o)(x) = +� +� +� +� +� +b +� +(f u +t )−1(x) +� +1X(x) +b +� +(f u +t )−1(X) +� +if b +� +(f u +t )−1(X) +� +̸= 0 , +0 +otherwise . +(58) +Then, Equation (56) follows from Lemma 18 applied with the mapping h = f u +t and with the subset +X = +� +hu +t+1 +�−1(o), as we have +F u,o +t += f u +t −−−−−−−−→ +(hu +t+1)−1(o) , +(59) +where f u +t −−−−−−−−→ +(hu +t+1)−1(o) is defined in Equation (32). +This ends the proof. +We now present the postponed proof of Lemma 8. +24 + +Proof of Lemma 8. We first prove Equation (24). As a preliminary fact, by applying Lemma 7, No- +tation (19a) and the definitions of sets TD +t and FD +t (Equations (20)-(21)), we obtain that, for all time +t ∈ T \ {T}, +TD +t = R ◦ (FD +t )⋆ . +(60) +Second, for all times (t, t′) ∈ +� +T \ {T} +�2 and for all pairs of controls and observations (u, u′) ∈ U2 and +(o, o′) ∈ O2, we can apply Lemma 19 on mappings F u,o +t +and F u′,o′ +t′ +. Indeed, by Equation (59), the +mappings F u,o +t +and F u′,o′ +t′ +are X-forward mappings. We hence have by Equation (43) that R ◦ F u,o +t +◦ R ◦ +F u′,o′ +t′ += R ◦ F u,o +t +◦ F u′,o′ +t′ +. Combined with Equation (60), this leads to +TD +0:t = R ◦ (FD +0:t)⋆ , +(61) +i.e. it leads to Equation (24). +Now, let b0 ∈ ∆(X). We prove Equation (23) by induction on t > 0. By Definition 2 of the set of +reachable beliefs, we have +BR,D +1 +(b0) +(11) += τ0 +� +{b0}, U, O +� (20) += TD +0 (b0) +(60) += R ◦ +� +FD +0 +� +⋆(b0) , +i.e. Equation (23) stands at time 1. Now, assuming Equation (23) is true up to time t ∈ T \ {T}, t > 0, +we have +BR,D +t+1 (b0) +(11) += τt +� +BR,D +t +(b0), U, O +� (20) += TD +t +� +BR,D +t +(b0) +� (23) += TD +t ◦ TD +0:t−1(b0) +(19) += TD +0:t(b0) +(61) += R ◦ +� +FD +0:t +� +⋆(b0) . +By induction on time t, we hence have Equation (23). +Meanwhile, Equation (25) comes from the definition of TD +�1,T � (see Equation (12)), the definition of +FD (Equation (22)) and Equation (23). +We can now present the detailed proof of Theorem 4. +Proof of Theorem 4. Let b0 ∈ ∆(X) be given. +We first prove the inequality |BR,D +�1,T �(b0)| ≤ (1 + |X|)|supp(b0)|, before proving the inequality +��BR,D +�1,T �(b0) +�� ≤ +1 + |supp(b0)||U||T |. +First, by Lemma (8), we have BR,D +�1,T �(b0) = TD(b0). We hence have +|BR,D +�1,t�(b0)| +(23) += |TD(b0)| +(24) += +��� +T −1 +� +i=0 +TD +0:i(b0) +��� +(45) +≤ (1 + |X|)|supp(b0)| . +The last inequality is given by Equation (45), obtained by applying Lemma 20. As all the elements of FD +t +are of the form given in Equation (16), the two sequences {FD +t }t∈�0,T −1� and {TD +t }t∈�0,T −1� satisfy the +assumptions of Lemma 20 where the role of {Φk}k∈N is taken by {TD +t }t∈�0,T −1� and the role of {Gk}k∈N +is taken by {FD +t }t∈�0,T −1� (the proof of Lemma 7 states that set FD +t is an +�−→ +X +� +-mappings set). +We now prove that we have +��BR,D +�1,T �(b0) +�� ≤ 1 + |supp(b0)||U||T | , +(62) +in order to obtain Inequality (13). With the help of the representation of the beliefs evolution mappings +given by Lemma 7, Inequality (62) is obtained as an application of Lemma 23 that we detail now. +For each t ∈ T \ {T} and each ut ∈ U we introduce the sets TD,ut +t += +� +τt(·, ut, o) +�� o ∈ O +� +and FD,ut +t += +� +F ut,o +t +�� o ∈ O +� +. Using set notations described in Equations (19), we obtain that TD,ut +t += R ◦ (FD,ut +t +)⋆. +Then, using the definition of BR,D +t +(b0) in Equation (11), we have that, for all time t ∈ T , t > 0, +BR,D +t +(b0) = +� +u0:t−1∈U0:t−1 +TD,ut−1 +t−1 +◦ TD,ut−2 +t−2 +◦ · · · ◦ TD,u0 +0 +(b0) = +� +u0:t−1∈U0:t−1 +TD,u0:t−1 +0:t−1 +(b0) . +(63) +For a fixed sequence of controls u0:t ∈ U0:t, the associated sequences of mappings {TD,ut +t +}t∈T and +{FD,ut +t +}t∈T satisfy the assumptions of Lemma 23, where the role of {Φk}k∈N is taken by {TD,ut +t +}t∈�−1,T �, +25 + +the role of {Gk}k∈N is taken by {FD,ut +t +}t∈�−1,T � and the role of the family of disjoint sets {Xk +i }i∈Ik is taken +by the family {(hu +t )−1(o)}o∈O,t∈�−1,T � (the proof of Lemma 7 states that the set FD +t is an +�−→ +X +� +-mappings +set). We hence get that +∀t ∈ T \ {T} , +��TD,u0:t0:t(b0) \ {δ∂} +�� ≤ |supp(b0)| . +(64) +Finally, we obtain +��BR,D +�1,T �(b0) +�� = +��� +T� +t=1 +� +BR,D +t +(b0) +���� +(using Equation (12)) +≤ 1 + +��� +T� +t=1 +� +BR,D +t +(b0) \ {δ∂} +���� +(by removing δ∂ from BR,D +t +(b0) for all t) += 1 + +��� +T −1 +� +t=0 +� +u0:t∈U0:t +� +TD,u0:t +0:t +(b0) \ {δ∂} +���� +(using Equation (63)) +≤ 1 + +T −1 +� +t=0 +� +u0:t∈U0:t +��� +TD,u0:t +0:t +(b0) \ {δ∂} +��� +(as |A ∪ B| ≤ |A| + |B|) +≤ 1 + +T −1 +� +t=0 +� +u0:t∈U0:t +|supp(b0)| +(using Equation (64)) +≤ 1 + +T −1 +� +t=0 +|U|t+1|supp(b0)| +(as U0:t = Ut+1) +≤ 1 + |U| +�|U|T − 1 +|U| − 1 +� +|supp(b0)| +(as �N +i=0 xi = xN+1−1 +x−1 +for x ̸= 1) +≤ 1 + |U||T ||supp(b0)| . +(as |T | = T + 1 and |U| > 1) +We have established the Inequality (62) and this concludes the proof. +A.2 +Complementary result on (∂)-separated mapping sets +In this subsection, we present complementary results on (∂)-separated mapping sets by applying the +framework presented in Appendix A.1. We notably apply the notion of forward and backward mappings, +presented in Equations (32) and (33), and the notion of pushforward measures, defined in Equation (15) +in §3.3. +First, in §A.2.1, we present and prove the lemmata used in the proofs of §4. Second, in §A.2.2, we +present a few examples of Separated Det-Pomdps. +A.2.1 +Properties of (∂)-separated mapping sets +Lemma 25. Let G be an +� +M, ←− +X +� +-mappings set as defined in Definition 17. If M is a separated mapping +set, then G is a (∂)-separated mapping set. +Proof. Let g1 and g2 be two mappings in G. In order to prove that G is a (∂)-separated mapping set, +using Definition 10, we need to prove that the restrictions of the two mappings g1 and g2 on the subset +A = g−1 +1 (X)∩g−1 +2 (X) are separated. Using the property of the set G, there exist m1 ∈ M (resp. m2 ∈ M) +and X1 ⊂ X (resp. X2 ⊂ X) such that g1 = m1←− +X1 (resp. g2 = m2←− +X2). Combined with the definition +of m1←− +X1 in Equation (33), this gives that g−1 +1 (X) = (m1)−1(X1) (resp. g−1 +2 (X) = (m2)−1(X2)). We +therefore obtain the equality A = (m1)−1(X1) ∩ (m2)−1(X2). +First, if the set A is empty, it is immediate to prove that g1 and g2 are (∂)-separated. +Second, +assuming that A is not empty and using again the fact that g1 = m1←− +X1, we obtain that g1 coincides with +m1 on the set A, and in the same way we obtain that g2 coincides with m2 on the set A. +26 + +Now, as m1 and m2 belong to a separated mapping set, they are separated mappings, and therefore +their restrictions to A are also separated. We conclude that the restrictions of g1 and g2 on the subset +A = g−1 +1 (X) ∩ g2−1(X) are separated. This ends the proof. +A direct consequence of Lemma 25 is the following Corollary 26. +Corollary 26. Let {Mk}k∈N be a sequence of sets of self-mappings on the set X. Let {Gk}k∈N be a se- +quence of sets of self-mappings on the set X, such that, for all k ∈ N, Gk is an +� +Mk, ←− +X +� +-mappings set. If +the set ∪k∈N +� +Mk ◦ Mk−1 ◦ · · · ◦ M0 +� +of mappings is a separated mapping set, then the set ∪k∈N +� +Gk ◦ Gk−1 ◦ · · · ◦ G0 +� +is a (∂)-separated mapping set. +Proof. Let G1 and G2 be respectively an +� +M1, ←− +X +� +-mappings set and an +� +M2, ←− +X +� +-mappings set. Then, +we have that +G1 ◦ G2 = +� +g1 ◦ g2 +�� g1 ∈ G1 and g2 ∈ G2 +� +(by Notation (19b)) +⊂ +� +m1←− +X1 ◦ m2←− +X2 +�� m1 ∈ M1 , m2 ∈ M2 , X1 ⊂ X , X2 ⊂ X +� +(by (35)) +⊂ +� +(m1 ◦ m2)←−−−−−−−−−−−− +X2∩(m2)−1(X1) +�� m1 ∈ M1 , m2 ∈ M2 , X1 ⊂ X , X2 ⊂ X +� +(by (36)) +⊂ +� +mX +�� m ∈ M1 ◦ M2 and X ⊂ X +� +. +We have obtained that G1◦G2 is a +� +M1 ◦ M2, ←− +X +� +-mappings set. Thus, if M1◦M2 is a separated mapping +set, then the set G1 ◦G2 is a (∂)-separated mapping set by using Lemma 25. The end of the proof follows +by induction on the number of compositions of sets, and by straightforward arguments when considering +unions of +�←− +X +� +-mappings sets. +Before presenting bounds on the cardinality of a (∂)-separated mapping set, we present Lemma 27. +Lemma 27. Let J ⊂ L(X; Y) be a set of mappings from the finite set X to the finite set Y. Assume that +for all pairs of mappings (j, j′) ∈ J2, if there exists x ∈ X such that j(x) = j′(x), then j = j′. Then, we +have that +|J| ≤ |Y| . +(65) +Proof. Fix x ∈ X and consider the evaluation mapping γx : J → Y defined by γx(j) = j(x) for all j ∈ J. +The image γx(J) of the set J by the mapping γx is indeed the subset {j(x) | j ∈ J} of Y. First, the +codomain of the mapping γx being the finite set Y, we immediately have that +��γx(J) +�� ≤ |Y| . +(66) +Second, the mapping γx is injective. Indeed, using the assumption on the set J, two distinct mappings +j and j′ in the set J must satisfy γx(j) = j(x) ̸= j′(x) = γx(j′). Thus, we must have the equality +|J| = +��γx(J) +�� which, combined with Equation (66), gives Inequality (65), and concludes the proof. +We now use the previous Lemma 27 to bound the cardinality of a (∂)-separated mapping set. +Lemma 28. Let X = X ∪ {∂}, and a (∂)-separated mapping set G of self-mappings on the set X. +Moreover, assume that, for all g ∈ G, g(∂) = ∂. For any subsets X and X′ of the set X, we define +GX→X′ as follows +GX→X′ = +� +g ∈ G +�� g−1(X) = X, g(X) ⊂ X′� +. +(67) +Then, we have +��GX→X′�� +� +≤ |X′| +if X ⊂ X , += 0 +if X ∩ {∂} ̸= ∅ . +(68) +Proof. Fix X ⊂ X and X′ ⊂ X. First, we consider the case where X ∩{∂} ̸= ∅. As we have assumed that +g(∂) = ∂, for all g ∈ G, we obtain that g−1(X) ∩ {∂} = ∅. Thus, we conclude that |GX→X′| = |∅| = 0. +Second, we consider the case where X ⊂ X and consider the mapping +Γ : GX→X′ → X′X , g �→ g|X . +(69) +27 + +The mapping Γ is injective. Indeed, if two mappings in GX→X′ have the same restriction on X, they +coincide on X as they are both constant on the set X \ X with value ∂. We therefore obtain that +��GX→X′�� = +��Γ(GX→X′) +�� . +(70) +Now, the set G′ = Γ(GX→X′) is a subset of mappings from X to X′. As G is a (∂)-separated mapping +set, we obtain that G′ is a separated set of mappings from X to X′. Indeed, consider a pair of mappings +(g′ +1, g′ +2) ∈ G′2 and assume that there exists x ∈ X such that g′ +1(x) = g′ +2(x). Using the definition of +G′, we have that g′ +1(x) and g′ +2(x) are both non equal to ∂. Moreover, there exists g1 and g2 in GX→X′ +such that g′ +1 = Γ(g1) and g′ +2 = Γ(g2). Using again the definition of G′ = Γ(GX→X′) we obtain that +g1(x) = g2(x) ̸= ∂. Now, as G is a (∂)-separated mapping set, we obtain that the two mappings g1 and +g2 coincide on X since they both do not take the value ∂ on X. We conclude that their restrictions on +X, the mappings g′ +1 and g′ +2, coincide. Using Lemma 27 in Subsection A.2 we obtain that +��Γ(GX→X′) +�� ≤ |X′| , +(71) +which, combined with Equation (70), gives Equation (68). This concludes the proof. +We now present the postponed proof of Proposition 12, presented in page 11. +Proof of Proposition 12. The proof of Proposition 12 is a direct consequence of Corollary 26. +We assume that the set � +t∈T f Ut+1 +0:t += {f u0:t +0:t | ∀t ∈ T \ {T}, ∀u0:t ∈ Ut+1} of the composition of the +evolution functions of Problem (2) is a separated mapping set. We then prove that Problem (2) is a +Separated Det-Pomdp. +First, for all time t and for all pair (u, o) ∈ U × O, we have F u,o +t += f u +t −−−−−−−−→ +(hu +t+1)−1(o) (see Equation (59)). +Thus, by Equation (34a), there exists X ⊂ X such that F u,o +t += f u +t ← +− +X. Hence, FD +t is of the same form as +in Equation (51), with the role of set Φk taken by {f U +t }. +We hence have that FD = � +t∈T FD +0:t is a (∂)-separated mapping set by Corollary 26, where the role +of {Gk}k∈N is taken by {FD +t }t∈T \{T } and the role of {Φk}k∈N is taken by {f U +t }t∈T \{T }. +Therefore, as FD is a (∂)-separated mapping set, Problem (2) is a Separated Det-Pomdp. +We now present the postponed proof of Theorem 13, presented in page 11. +Proof of Theorem 13. We start by giving preliminary bounds on +��� +� +R ◦ (FD +X→X)⋆ +� +(b0) \ {δ∂} +���, where +FD +X→X is defined by Equation (67), i.e. +FD +X→X = +� +F ∈ FD �� F −1(X) = X, F(X) ⊂ X +� +, +where FD is defined in Equation (21). +We consider three cases depending on the cardinality of the +subset X: +1. When |X| = 0, we have that X = ∅ and +� +R ◦ (FD +∅→X)⋆ +� +(b0) \ {δ∂} = ∅, and thus +��� +� +R ◦ (FD +X→X)⋆ +� +(b0) \ {δ∂} +��� = 0 . +(72a) +2. When |X| = 1, we have that +� +R ◦ (FD +X→X)⋆ +� +(b0) \ {δ∂} ⊂ +� +δx +�� x ∈ X +� +, as the only probability +distributions of ∆(X) with support of cardinality at most 1 are the vertices +� +δx +�� x ∈ X +� +of the +simplex ∆(X), and thus +��� +� +R ◦ (FD +X→X)⋆ +� +(b0) \ {δ∂} +��� ≤ +��� +δx +�� x ∈ X +��� = |X| . +(72b) +3. For |X| ≥ 2, we have by Lemma 28 in Appendix A.1, applied with G = F (as F is a (∂)-separated +mapping set) that +��� +� +R ◦ (FD +X→X)⋆ +� +(b0) \ {δ∂} +��� ≤ +��(FD +X→X)⋆ +�� ≤ |X| . +(72c) +28 + +We have by Equation (25) that +��BR,D +�1,T �(b0) +�� = |TD(b0)|. We now detail the cardinality of TD(b0): +��TD(b0) \ {δ∂} +�� = +��� +R ◦ (FD)⋆ +� +(b0) \ {δ∂} +�� += +��� +� +R ◦ +� � +X⊂X +FD +X→X +� +⋆ +� +(b0) \ {δ∂} +��� +(as � +X⊂X FD +X→X = FD) += +��� +� +X⊂X +� +R ◦ (FD +X→X)⋆ +� +(b0) \ {δ∂} +��� +as ∀(F, F ′) ∈ +� +FD�2, R ◦ +� +F ∪ F ′� += R ◦ F ∪ R ◦ F ′, += +��� +� +X⊂supp(b0) +� +R ◦ (FD +X→X)⋆ +� +(b0) \ {δ∂} +��� +as +� +R ◦ (FD +X∩supp(b0)→X)⋆ +� +(b0) = +� +R ◦ (FD +X→X)⋆ +� +(b0) by Equation (38) in Lemma 18, +≤ +� +X⊂supp(b0) +��� +� +R ◦ (FD +X→X)⋆ +� +(b0) \ {δ∂} +��� += +� +k≥0 +� +X⊂supp(b0) +|X|=k +��� +� +R ◦ (FD +X→X)⋆ +� +(b0) \ {δ∂} +��� +≤ |X| + +� +X⊂supp(b0) +|X|≥2 +|X| +(by Equations (72)) += |X| + +� +2|supp(b0)| − |supp(b0)| − 1 +� +|X| , +(73) +where the last equality comes from the fact that +��{X ⊂ supp(b0) | |X| ≥ 2} +�� is given by +��{X ⊂ supp(b0) | |X| ≥ 2} +�� = +��� +X ⊂ X +�� X ⊂ supp(b0) +��� +� +�� +� +2|supp(b0)| +− +��� +X ⊂ supp(b0) +�� |X| = 1 +��� +� +�� +� +=|supp(b0)| +− +��� +X ⊂ supp(b0) +�� |X| = 0 +��� +� +�� +� +=1 +. +We hence obtain that +��BR,D +�1,T �(b0) +�� (24) += |TD(b0)| +(73) +≤ 1 + +� +2|supp(b0)| − |supp(b0)| +� +|X| . +This ends the proof. +We now present examples of Separated Det-Pomdps. +A.2.2 +Examples of Separated Det-Pomdps +In this subsection, we present examples of Separated Det-Pomdps. Indeed, a direct consequence of +Proposition 12 is that, if the evolution mappings of a Det-Pomdp belong to a separated mapping +set, then the Det-Pomdp is a Separated Det-Pomdp. We now present examples of such evolution +mappings. +Corollary 29. Consider a Det-Pomdp optimization problem given by Problem (2) which satisfies the +finite sets Assumption 1. The notations are those of Problem (2). Assuming that, for all time t ∈ T \{T}, +there exist mappings gt such that, for all states x ∈ X ⊂ Rn, +ft(x, u) = x + gt(u) , +(74) +then Problem (2) is a Separated Det-Pomdp. +29 + +Proof. This corollary is a direct result of Proposition 12. Indeed, we only need to prove that ∪t∈T +� +f Ut+1 +0:t +� +is a separated mapping set. +Let t1 ≤ t′ +1 and t2 ≤ t′ +2 be such that �t1, t′ +1� ⊂ T and �t2, t′ +2� ⊂ T . +Let ut1:t′ +1 ∈ Ut′ +1−t1+1 and +u′ +t2:t′ +2 ∈ Ut′ +2−t2+1 be two sequences of controls. We have, by using Equation (74), that f +ut1:t′ +1 +t1:t′ +1 +: X → +X, x �→ x + � +t∈�t1,t′ +1� gt(ut) , and f +u′ +t2:t′ +2 +t2:t′ +2 +: X → X, x �→ x + � +t∈�t2,t′ +2� gt(u′ +t). +If there exists a state x ∈ X such that f +ut1:t′ +1 +t1:t′ +1 (x) = f +u′ +t2:t′ +2 +t2:t′ +2 (x), we hence have � +t∈�t1,t′ +1� gt(ut) = +� +t∈�t2,t′ +2� gt(u′ +t). +Thus f +ut1:t′ +1 +t1:t′ +1 (x) = f +u′ +t2:t′ +2 +t2:t′ +2 (x) ⇒ f +ut1:t′ +1 +t1:t′ +1 += f +u′ +t2:t′ +2 +t2:t′ +2 . Therefore, the set ∪t∈T +� +f Ut+1 +0:t +� += {f u0:t +0:t | ∀t ∈ T \ {T}, ∀u0:t ∈ Ut+1} +of composition of the evolution mappings is a separated mapping set. We conclude by Proposition 12 +that Problem (2) is a Separated Det-Pomdp. +Corollary 30. Consider a Det-Pomdp optimization problem given by Problem (2) which satisfies the +finite sets Assumption 1. The notations are those of Problem (2). Assuming that, for all time t ∈ T \{T}, +there exist mappings gt such that for all states x ∈ X ⊂ Rn, +ft(x, u) = x × gt(u) , +(75) +and assuming that 0 /∈ X, then Problem (2) is a Separated Det-Pomdp. +Proof. Let t1 ≤ t′ +1 and t2 ≤ t′ +2 such that �t1, t′ +1� ⊂ T and �t2, t′ +2� ⊂ T . Let ut1:t′ +1 ∈ Ut′ +1−t1+1 and +u′ +t2:t′ +2 ∈ Ut′ +2−t2+1 be two sequences of controls . We have, by using Equation (75), f +ut1:t′ +1 +t1:t′ +1 +: X → X, x �→ +x × � +t∈�t1,t′ +1� gt(ut), and f +u′ +t2:t′ +2 +t2:t′ +2 +: X → X, x �→ x × � +t∈�t2,t′ +2� gt(u′ +t). +If there exists a state x ∈ X such that f +ut1:t′ +1 +t1:t′ +1 (x) = f +u′ +t2:t′ +2 +t2:t′ +2 (x), we hence have, as x ̸= 0, � +t∈�t1,t′ +1� gt(ut) = +� +t∈�t2,t′ +2� gt(u′ +t). +Thus f +ut1:t′ +1 +t1:t′ +1 (x) = f +u′ +t2:t′ +2 +t2:t′ +2 (x) ⇒ f +ut1:t′ +1 +t1:t′ +1 += f +u′ +t2:t′ +2 +t2:t′ +2 . Therefore, the set of compositions of the evolution +functions ∪t∈T +� +f Ut+1 +0:t +� += {f u0:t +0:t | ∀t ∈ T \ {T}, ∀u0:t ∈ Ut+1} is a separated mapping set. +References +K. J. ˚Astr¨om. Optimal control of Markov processes with incomplete state information. Journal of Math- +ematical Analysis and Applications, 10(1):174–205, Feb. 1965. doi: 10.1016/0022-247X(65)90154-X. +R. Bellman. Dynamic programming. Princeton Univ. Pr, Princeton, NJ, 1957. +D. P. Bertsekas. Dynamic Programming and Optimal Control. Athena Scientific, Belmont, Massachusetts, +second edition, 2000. Volumes 1 and 2. +D. P. Bertsekas and S. E. Shreve. Stochastic optimal control: the discrete time case. Number v. 139 in +Mathematics in science and engineering. Academic Press, New York, 1978. +B. Bonet. Deterministic pomdps revisited. In Proceedings of the Twenty-Fifth Conference on Uncertainty +in Artificial Intelligence, UAI ’09, page 59–66, Arlington, Virginia, USA, 2009. AUAI Press. +H. Geffner and B. Bonet. Solving Large POMDPs by Real Time Dynamic Programming. In Proc. Fall +AAAI Symposium on POMDPS, Orlando, FL, 1998. +H. Kurniawati, D. Hsu, and W. Sun Lee. SARSOP: Efficient Point-Based POMDP Planning by Approx- +imating Optimally Reachable Belief Spaces. In Robotics: Science and Systems IV. Robotics: Science +and Systems Foundation, June 2008. doi: 10.15607/RSS.2008.IV.009. +M. L. Littman. Algorithms for Sequential Decision Making. PhD thesis, Brown University, 1996. +J. Pajarinen and V. Kyrki. Robotic manipulation of multiple objects as a POMDP. Artificial Intelligence, +247:213–228, June 2017. doi: 10.1016/j.artint.2015.04.001. +30 + +M. L. Puterman. Markov Decision Processes: Discrete Stochastic Dynamic Programming. Wiley Series +in Probability and Statistics. Wiley, 1 edition, Apr. 1994. doi: 10.1002/9780470316887. +R. D. Smallwood and E. J. Sondik. The Optimal Control of Partially Observable Markov Processes over +a Finite Horizon. Operations Research, 21(5):1071–1088, Oct. 1973. doi: 10.1287/opre.21.5.1071. +L. N. Steimle, D. L. Kaufman, and B. T. Denton. Multi-model markov decision processes. IISE Trans- +actions, 53(10):1124–1139, 2021. doi: 10.1080/24725854.2021.1895454. +R. S. Sutton and A. G. Barto. Reinforcement Learning: An Introduction. The MIT Press, second edition, +2018. +D. J. White. A Survey of Applications of Markov Decision Processes. The Journal of the Operational +Research Society, 44(11):1073, Nov. 1993. doi: 10.2307/2583870. +31 + diff --git a/ndFAT4oBgHgl3EQfdB1R/content/tmp_files/load_file.txt b/ndFAT4oBgHgl3EQfdB1R/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9575c8425c8d3b38a8e4727363440edc64cbdcdb --- /dev/null +++ b/ndFAT4oBgHgl3EQfdB1R/content/tmp_files/load_file.txt @@ -0,0 +1,1103 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf,len=1102 +page_content='Contributions on complexity bounds for Deterministic Partially Observed Markov Decision Process Cyrille Vessaire∗, Jean-Philippe Chancelier∗, Michel De Lara∗, Pierre Carpentier†, Alejandro Rodr´ıguez-Mart´ınez‡ January 23, 2023 Abstract Markov Decision Processes (Mdps) form a versatile framework used to model a wide range of optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The Mdp model consists of sets of states, actions, time steps, rewards, and probability transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' When in a given state and at a given time, the decision maker’s action generates a reward and determines the state at the next time step according to the probability transition function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' However, Mdps assume that the decision maker knows the state of the controlled dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Hence, when one needs to optimize controlled dynamical systems under partial observation, one often turns toward the formalism of Partially Observed Markov Decision Processes (Pomdp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Pomdps are often untractable in the general case as Dynamic Programming suffers from the curse of dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Instead of focusing on the general Pomdps, we present a subclass where transitions and observations mappings are deterministic: Deterministic Partially Observed Markov Decision Processes (Det-Pomdp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' That subclass of problems has been studied by (Littman, 1996) and (Bonet, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' It was first considered as a limit case of Pomdps by Littman, mainly used to illustrate the complexity of Pomdps when considering as few sources of uncertainties as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' In this paper, we improve on Littman’s complexity bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We then introduce and study an even simpler class: Separated Det-Pomdps and give some new complexity bounds for this class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' This new class of problems uses a property of the dynamics and observation to push back the curse of dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 1 Introduction Markov Decision Processes (Mdps) form a versatile framework used to model a wide range of optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Indeed, one often uses the formalism of Mdps to optimize controlled dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' It is very popular in both optimal control and machine learning community, as it can be used to model complex real-life problems (see the survey (White, 1993) for common applications).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Moreover, it provides the mathematical foundations for Reinforcement Learning (see (Sutton and Barto, 2018)), and algorithms such as Policy Iteration and Dynamic Programming can efficiently solve Mdps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' In the Mdp framework, a decision maker can sequentially act upon a controlled dynamical system and get some rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The Mdp model consists of sets of states, actions, time steps, rewards, and probability transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' When in a given state and at a given time, the decision maker’s action generates a reward and determines the state at the next time step according to the probability transition function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' However, Mdps assume that the decision maker knows the state of the controlled dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Hence, when one needs to optimize controlled dynamical systems under partial observation, one often turns toward the formalism of Partially Observed Markov Decision Processes (Pomdp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' An extensive literature exists on Pomdps, most of which focuses on the infinite horizon case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Pomdps can be applied to numerous fields, from medical models (as in (Steimle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=', 2021)) to robotics (as in (Pajarinen and Kyrki, 2017)) to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Algorithms based on Dynamic Programming (see (Bellman, 1957)) have been designed to exploit specific structures in Pomdps in order to solve this difficult class of problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' They ∗CERMICS, Ecole des Ponts, Marne-la-Vall´ee, France †UMA, ENSTA Paris, Institut Polytechnique de Paris, Palaiseau, France ‡IAM, TotalEnergies SE, Pau, France 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='08567v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='OC] 20 Jan 2023 do so by first reformulating the problem through the use of beliefs (probability distributions over the state space), as in (Smallwood and Sondik, 1973).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' One such algorithm is Sarsop, described in (Kurniawati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=', 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' However, Pomdps are often untractable in the general case as Dynamic Programming suffers from the curse of dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Indeed, working with beliefs implies working on the space of distributions over the state space, which is, by nature, an infinite space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Yet not all Pomdps suffer equally from the curse of dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Indeed, instead of focusing on the general Pomdps, we present a subclass where transitions and observations mappings are deterministic: Deterministic Partially Observed Markov Decision Processes (Det-Pomdp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' That subclass of problems has been studied by (Littman, 1996) and (Bonet, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' It was first considered as a limit case of Pomdps by Littman, mainly used to illustrate the complexity of Pomdps when considering as few sources of uncertainties as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' For Bonet, Det-Pomdps became of interest after some applications were found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' He presented examples in (Bonet, 2009, §2), such as the navigation of a robot in a partially observed terrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' In this paper, we improve on Littman’s complexity bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We then introduce and study an even simpler class: Separated Det-Pomdps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' This new class of problems uses a property of the dynamics and observation to push back the curse of dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' First, in §2, we present a general formulation of Det-Pomdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Second, in §3, we present Dynamic Programming on beliefs for Det-Pomdps with constraints, and present complexity bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Third, in §4, we introduce a subclass of Det-Pomdp, Separated Det- Pomdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Finally, in §5 we illustrate Separated Det-Pomdp with a toy problem: emptying a tank containing water when considering partial observation of the level of water in the tank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Meanwhile, in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='1, we present technical lemmata and considerations on pushforward measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Finally, in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='2, we present complements on Separated Det-Pomdps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We now detail our main contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' In §3, we improve Littman (1996) bound on the cardinality of the set of reachable beliefs for Det-Pomdps (see Theorem 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' This new bound comes from a new representation of the belief dynamics in Det-Pomdps using the notion of pushforward measure (see Lemma 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' In §4, we introduce a subclass of Det-Pomdps, Separated Det-Pomdps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' As shown in Theorem 13, the interest of Separated Det-Pomdps is that they further push back the curse of dimen- sionality for Dynamic Programming with beliefs (see Theorem 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Moreover, this last bound is tight (see Proposition 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 2 Formulation of Deterministic Partially Observed Markov De- cision Processes A Det-Pomdp is a particular case of Pomdps, itself an extension of Markov Decision Processes (Mdps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Backgrounds on Mdps can be found in Puterman (1994), whereas backgrounds on Pomdps can be found in Bertsekas and Shreve (1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' As with Mdps, the model consists of a dynamical system, defined thanks to states, controls (also called actions), transitions and time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' At each time-step, the decision maker (also called the agent) chooses a given action, which generates a random reward depending on the state of the system and on the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The state then transits to its next random value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' However, in the case of Det-Pomdp (and Pomdp), the decision maker has only partial knowledge of the state of the dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Instead, he has access to functions of the state and controls: the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' For Det-Pomdps, the transitions and observations are given by deterministic evolution and observation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' First, we present the ingredients of a Det-Pomdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Second, we present the formulation of a Det- Pomdp optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Ingredients of a Det-Pomdp A Det-Pomdp is defined by the tuple D = � T , U, O, X, {Lt}t∈T \\{T }, {ft}t∈T \\{T }, {Uad t }t∈T \\{T }, {ht}t∈T � , (1) which we now detail1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The set T = �0, T�2 is the set of time-steps, where the positive integer T ∈ N\\{0} is colloquially known as the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The set U is the set of controls the decision maker can choose from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The set O is the set 1For simplicity, we assume that U, O and X are not indexed by time 2Let t and t′ be two integers, with t′ ≥ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The set {t, t + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' , t′} is denoted by �t, t′�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 2 of observations available to the decision maker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The set X is the set of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The collection {Lt}t∈T \\{T } is the collection of instantaneous costs functions: for all time t ∈ T \\ {T}, Lt : X × U → R ∪ {+∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Moreover, the final cost function LT is by convention denoted by K : X → R ∪ {+∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The collection {ft}t∈T \\{T } is the collection of evolution functions: for all time t ∈ T \\{T}, ft : X×U → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' They define the transitions of the dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The collection {Uad t }t∈T \\{T } is the collection of admissibility constraints: for all time t ∈ T \\ {T}, Uad t : X ⇒ U is a set-valued mapping from X to U, that is, for all state x ∈ X, the admissible controls at time t are given by the subset of U, Uad t (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' {ht}t∈T is the collection of observation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The initial observation is given by the mapping h0 : X → O whereas, for all time t ∈ T \\ {0}, the observations are given by the mappings ht : X × U → O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Let (Ω, F, P) be a probability space, where Ω is the set of possible outcomes, F is a σ-field over Ω and P is a probability measure on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We denote by E the mathematical expectation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' In this paper, we only consider Det-Pomdps which satisfy the following finite sets assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Assumption 1 (Finite sets assumption).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The sets of possible outcomes Ω, of states X, of controls U, and observations O have finite cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Moreover, we consider a finite number of timesteps, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' the horizon is finite: T < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' As we consider finite sets, we introduce a notation for the set of probability distributions on finite sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Let Y be a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We denote by ∆(Y) the set of probability distributions on Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The set ∆(Y) is in bijection with the simplex ∆|Y| of dimension3 |Y| (hence the notation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We now present the formulation of the optimization problem which we study in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Formulation of a Det-Pomdp optimization problem A finite-horizon Det-Pomdp optimization problem is formulated as follows V⋆(b0) = min X,O,U E �T −1 � t=0 Lt(Xt, Ut) + K(XT ) � (2a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' PX0 = b0 , (2b) Xt+1 = ft(Xt, Ut) , ∀t ∈ T \\ {T} , (2c) O0 = h0(X0) , (2d) Ot+1 = ht+1(Xt+1, Ut) , ∀t ∈ T \\ {T} , (2e) Ut ∈ Uad t (Xt) , ∀t ∈ T \\ {T} , (2f) σ(Ut) ⊂ σ(O0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' , Ot, U0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' , Ut−1) , ∀t ∈ T \\ {T} , (2g) where we denote by V⋆(b0) the optimal value of Problem (2), that is, the optimal value of the Det- Pomdp optimization problem when the initial probability distribution of the state is given by the initial belief b0 ∈ ∆(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' In Problem (2), there are three processes X = � Xt � t∈T , U = � Ut � t∈T \\{T } and O = � Ot � t∈T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' For all time t ∈ T , Xt : Ω → X and Ot : Ω → O are random variables representing respectively the state and the observation variables of the system at time t, and for all time t ∈ T \\ {T}, Ut : Ω → U are random variables representing the controls at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The optimization criterion of Problem (2) is given by Equation (2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' In this paper, we only consider the minimization of the expected value in the finite horizon case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We now detail the constraints of the optimization Problem (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' First, Equation (2b) is the initial- ization constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' As the initial state is not fully known, we instead use the probability distribution b0 ∈ ∆(X) of the initial state of the system for the initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Second, Equation (2c) is called the state evolution equation of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' It is defined thanks to the dynamics which describe the evolution of the states of the controlled dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Third, Equations (2d) and (2e) define the observations of the system available at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Fourth, Equation (2f) is called the admissibility constraints: it defines which controls can be applied at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Note that the proper formulation of the admissibility constraints should contain an added quantification, “∀ω ∈ Ω”, which we omit in this paper as the set Ω is finite, and we can always assume that P(ω) > 0 for all ω ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Finally, Equation (2g) is the non-anticipativity constraint: it defines the information available to the decision maker before choosing 3The cardinality of a finite set is the number of its elements and is denoted by | · |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 3 a control at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' As all sets Ω, X, U and O are assumed to be finite by Assumption 1, all mappings with domain Ω are random variables and Equation (2a) is well defined because Lt and K takes their values in R ∪ {+∞}, hence the optimization Problem (2) is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 3 Complexity analysis of Dynamic Programming for Det-Pomdps In §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='1, we present Dynamic Programming for Det-Pomdps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Then, in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='2 we study its complexity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' the number of “operations” necessary to solve Problem (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' In §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='3, we present a new representation of beliefs as pushforward measures, that will be used to prove the complexity results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='1 Dynamic Programming for Det-Pomdp We now present Dynamic Programming Equations with beliefs for Problem (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' As a Det-Pomdp is a Pomdp, all the results and numerical methods that apply to Pomdps are carried over to Det- Pomdps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Notably, it is possible to write Dynamic Programming equations for a finite horizon problem associated with a Pomdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' To do so, it is classical to formulate a belief-Mdp where the state is a probability distribution over the state space, called belief (see (Bertsekas and Shreve, 1978) for details on the assumptions for general Pomdps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Here, we detail this methodology for the specific Det-Pomdp case, and extend it to tackle cases with admissibility constraints on the controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' First, in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='1, we formally define sets and mappings which are necessary for the formulation of the belief-Mdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Second, in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='2, we present the Dynamic Programming equations for the resulting belief-Mdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='1 Beliefs in Det-Pomdp First, we present the set of beliefs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Second, we present the mappings necessary for the formulation of the belief-Mdp, notably the beliefs dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Sets for the beliefs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The dynamic programming equation for Det-Pomdps is formulated using states in the set ∆(X), the probability distributions over the “initial” state space X, which are called beliefs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' However, the beliefs dynamics, as described later in Equation (9), may lead to a null measure over the space X when considering some combination of observations and controls which are in contradiction with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' As we want to be able to compose belief dynamics, we combine ∆(X) and the null measure over X as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We introduce an extended state set X, obtained as the union of the original set X with an extra element, denoted by ∂ (∂ /∈ X), which is used as the support of the null measure over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' X = X ∪ {∂} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (3) We denote by B the subset of ∆(X) defined by B = ∆(X) ∪ {δ∂} , (4) where we identify the set ∆(X) with {µ ∈ ∆(X) | supp(µ) ⊂ X} and where δ∂ ∈ ∆(X) is the discrete probability measure on X concentrated on ∂, that is δ∂({∂}) = 1, and where the mapping “supp” is the support of a nonnegative measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' For any nonnegative measure µ on the finite set Y, we have supp(µ) = � y ∈ Y �� µ({y}) > 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (5) We call the probability measure δ∂ the cemetery belief as we will see in Equation (9) that the belief dynamics, when reaching the belief state δ∂, remains in δ∂ forever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' A probability measure ν ∈ ∆(X) is represented, in some equations, by the ordered pair � ν|X, ν(∂) � , where ν|X is a nonnegative measure on the set X and ν(∂) ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Now that the set of beliefs B is defined, we present the beliefs dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 4 Beliefs dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' In order to define the beliefs dynamics, we introduce, for each t ∈ T \\ {T} two mappings, Qt+1 : B × U × O → [0, 1] and τt : B × U × O → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' They are defined using partial mappings, defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Let A, D, F and G be sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Let g : A × D → F, (a, d) �→ g(a, d) be a mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We denote by gd the mapping gd : A → F , a �→ g(a, d) , (6) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' the mapping g(·, d) obtained from g by setting its second variable to a fixed value d ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' When considering mappings with n inputs, we extend this notation to the last n − 1 inputs using a Cartesian product over the last n − 1 sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' For example, let g : A × D × F → G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We denote by g(d,f) = g(·, d, f) the mapping g(d,f) : A → G, a �→ g(a, d, f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' the mapping Qt+1 gives the probability of observing o at time t + 1 when applying control u on the dynamical system when considering belief b at time t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' and is given by ∀t ∈ T \\ {T} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Qt+1 : (b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' o) ∋ B × U × O �→ b � (hu t+1 ◦ f u t )−1(o) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (7) where f u t (·) and hu t (·) are partial mapping that follow the notation defined in Equation (6): ∀u ∈ U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' f u t : X → X ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' x �→ ft(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' u) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' and ∀u ∈ U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' hu t : X → O ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' x �→ ht(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' u) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' and where b � (hu t+1 ◦ f u t )−1(o) � is the probability of the set (hu t+1 ◦ f u t )−1(o) with respect to the probability distribution b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Note that, we always have that Qt+1(δ∂, u, o) = δ∂ � (hu t+1 ◦ f u t )−1(o) � = 0 , (8) as (hu t+1 ◦ f u t )−1(o) is always a subset of X and thus has a null intersection with {∂}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' For all time t ∈ T \\ {T}, the mapping τt gives the evolution of the beliefs when applying control u on the dynamical system when considering belief b at time t and observing o at time t + 1, and is given by ∀y ∈ X , τt(b, u, o)(y) = � � � b � (f u t )−1(y) � Qt+1(b, u, o) if Qt+1(b, u, o) ̸= 0, and y ∈ � hu t+1 �−1(o) , 0 otherwise, (9a) τt(b, u, o)(∂) = 1 − τt(b, u, o)(X) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (9b) Hence, δ∂ is used as a last resort belief, which appears when it is not possible to observe o after applying control u to any state of the support of belief b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Indeed, δ∂ is used to ensure that the mappings τt are well defined for all beliefs, controls and observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Using the sequences of mappings {Qt}t∈T \\{0} and {τt}t∈T \\{T }, we have a properly defined belief- Mdp, which can be solved by Dynamic Programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='2 Dynamic Programming Equations for Det-Pomdp In the case of Pomdp (without constraints on the controls), Dynamic Programming equations with beliefs as new states were first given in (˚Astr¨om, 1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' More general cases (still without explicit constraints on the controls) are treated in Bertsekas and Shreve (1978, Chapter 10) and in Bertsekas (2000, Chapter 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Dynamic Programming Equations for Det-Pomdp can be obtained as a special case of Dynamic Programming for Pomdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' They are given in Equations (10a) and (10b) together with the expression of the beliefs dynamics {τt}t∈T \\{T } (see Equation (9)) in the case where there are no constraints on the controls in (Littman, 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' In (Bertsekas and Shreve, 1978) the proof that beliefs are statistics sufficient for controls was made for Pomdps without any admissibility constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We thus cannot directly apply this result on Problem (2), as it contains Constraint (2f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We extend the classical results by (Bertsekas and Shreve, 1978) in Proposition 1 in order to tackle such constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We identify an admissibility set for beliefs of the form Ub,ad(b) = � x∈supp(b) Uad(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Note that we use an upper index b to distinguish admissibility sets for beliefs from admissibility sets for states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Also note that, as far as we know, the first Dynamic Programming equations using such sets Ub,ad(b) were given in (Geffner and Bonet, 1998, §5) with no explicit proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 5 Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Consider a Det-Pomdp optimization problem given by Problem (2) which satisfies the finite sets Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Let B = ∆(X) ∪ {δ∂}, as defined in Equation (4) and consider the sequence of value functions (Vt : B → R ∪ {+∞})t∈T defined by the following backward induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' First, for all t ∈ T , we have that Vt(δ∂) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Second, we have that VT : b ∈ ∆(X) �→ � x∈X b(x)K(x) , (10a) Vt : b ∈ ∆(X) �→ min u∈Ub,ad t (b) �� x∈X b(x)Lt(x, u) + � o∈O Qt+1(b, u, o)Vt+1 � τt(b, u, o) �� , (10b) where Ub,ad t (b) = � x∈supp(b) Uad t (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Then, the optimal value of Problem (2) and the value of the function V0 at the initial belief b0 are equal, that is, V0(b0) = V⋆(b0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Moreover, a policy π = (π0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' , πT −1), defined by a sequence of mappings πt : B → U, which minimizes the right-hand side of Equation (10b) for each b and t is an optimal policy of Problem (2): the controls given by Ut = πt(Bt) (where Bt is computed thanks to the recursion Bt+1 = τt(Bt, Ut, Ot+1), with B0 = b0) are optimal controls of Problem (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We present a sketch of proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' First, we rewrite Problem (2) as an equivalent problem, without constraint (2f) by adding charac- teristic functions of the constraints to the instantaneous costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The equivalent problem then follows the framework of (Bertsekas and Shreve, 1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Second, we apply the results of (Bertsekas and Shreve, 1978) to the reformulated problem, and obtain associated Dynamic Programming equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Third, the Dynamic Programming equations which solve the equivalent problem are equivalent to Equations (10) presented in Proposition 1, thus concluding that Equation (10) gives the solution of Problem (2) as formulated in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' This step is a bit technical, but is otherwise straightforward and does not present any major difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Now that we have presented Dynamic Programming equations on beliefs, we present the complexity of Dynamic Programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='2 Dynamic Programming complexity for Det-Pomdps According to Proposition 1, we can solve Problem (2) by computing V0(b0) by means of Equations (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Solving Dynamic Programming equations (10) implies that we are able to numerically evaluate the value functions at each reachable belief starting from b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Thus, we introduce the subsets of reachable beliefs starting from b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We start by formally defining the set of reachable beliefs, before we present our first complexity result on Dynamic Programming for Det-Pomdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The set of reachable beliefs BR,D is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Note that we use the upper index D to recall that we consider the set of reachable beliefs of a Det-Pomdp defined by the data tuple D, in Equation (1), whereas the upper index R stands for reachable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Let b0 ∈ ∆(X) be given and consider the sequence {BR,D t }t∈T of subsets of the set of beliefs B = ∆(X) ∪ {δ∂} defined by the induction BR,D 0 (b0) = {b0} and ∀t ∈ T \\ {T} , BR,D t+1 (b0) = τt � BR,D t (b0), U, O � , (11) where τt is defined in Equation (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' For any t ∈ T , the subset BR,D t (b0) ⊂ B is called the set of reachable beliefs a time t starting from initial belief b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Moreover, we denote by BR,D �t,t′�(b0) the union, for t′′ in the time interval �t, t′�, t < t′, of the reachable beliefs at time t′′ starting from the initial belief b0 ∈ ∆(X), that is, ∀(t, t′) ∈ T 2 , t < t′ , BR,D �t,t′�(b0) = t′� t′′=t BR,D t′′ (b0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (12) The set BR,D �1,T �(b0) is called the set of reachable beliefs from the initial belief b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 6 Note that, under Assumption 1, the set BR,D �1,T �(b0) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We now present a classical complexity result for Dynamic Programming algorithm (which we call Dp Algorithm in the rest of this paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Consider a Det-Pomdp optimization problem given by Problem (2) which satisfies the finite sets Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Let b0 ∈ ∆(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Then, a standard Dp Algorithm (numerically) solves Problem (2), and its complexity is O(|T ||BR,D �1,T �(b0)||U||O|), where the set of reachable beliefs BR,D �1,T �(b0) is defined in Equation (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' First, as we consider that Assumption 1 holds, note that BR,D �1,T �(b0) is finite and we can apply Proposition 1 on Problem (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We hence solve Problem (2) by computing value functions given by Equations (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' For a given time t ∈ T \\ {T} and reachable belief b ∈ BR,D t (b0), we compute the value function Vt by evaluating the next value for each control u ∈ U and each resulting observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We hence need � t∈T |BR,D t (b0)||U||O| operations to solve Problem (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Then, since for all time t ∈ T , t > 0, BR,D t (b0) ⊂ BR,D �1,T �(b0) (see Equation (12)), we have |BR,D t (b0)| ≤ |BR,D �1,T �(b0)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Moreover, we also have BR,D �1,T �(b0) ̸= ∅ (there is always at least one belief in BR,D 1 (b0), as for a given control u ∈ U and an observation o ∈ O, τ0(b0, u, o) ∈ BR,D 1 (b0) ⊂ BR,D �1,T �(b0)) and BR,D 0 (b0) = {b0}, hence |BR,D 0 (b0)| ≤ |BR,D �1,T �(b0)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Hence, � t∈T |BR,D t (b0)||U||O| ≤ |T ||BR,D �1,T �(b0)||U||O|, and thus we can solve Problem (2) in O(|T ||BR,D �1,T �(b0)||U||O|) operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' In order to apply Proposition 3 on Problem (2) and to get complexity bounds on the Dp Algorithm, we now study the set of reachable beliefs BR,D �1,T �(b0), more specifically, we give bounds on its cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Consider a Det-Pomdp optimization problem given by Problem (2) which satisfies the finite sets Assumption 1, and such that |U| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' For all initial belief b0 ∈ ∆(X), the cardinality of the set of reachable beliefs starting from b0, defined in Equation (12), satisfies the following bound ��BR,D �1,T �(b0) �� ≤ min � (1 + |X|)|supp(b0)| , 1 + |supp(b0)||U||T |� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (13) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' A sketch of proof is postponed to §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='3, as it relies on a new representation of the belief dynamics presented in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The complete proof can be found in Appendix §A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' In Theorem 4, we gave a bound on the cardinality of the set BR,D �1,t�(b0) which improves a previous result we now recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Littman presents in (Littman, 1996, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='1) a bound on the set of reachable beliefs starting from belief b0 ∈ ∆(X): ∀t ∈ T , ��BR,D �0,t�(b0) �� ≤ (1 + |X|)|X| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (14) Equation (13) is an improvement on the bound given in Equation (14) which takes into account the support of the initial belief b0: indeed, as b0 ∈ ∆(X) and |supp(b0)| ≤ |X|, Equation (13) is tighter than Equation (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Using Equation (13), we obtain that the number of reachable beliefs of a Det-Pomdp is finite even when considering the case of an infinite horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Indeed, the first inequality in Equation (13) is well defined even in the infinite horizon case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' A direct consequence of Proposition 3 and Theorem 4 is that the complexity of the Dp Algorithm is O � |BR,D �1,T �(b0)||T ||U||O| � , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' in O � min � (1 + |X|)|supp(b0)| , 1 + |supp(b0)||U||T |� |T ||U||O| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' As a side note, we can remark that we could also use Theorem 4 to characterize the complex- ity of a general Pomdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Indeed, we can reformulate any finite Pomdp with independent noises on the dynamics {wt}t∈T \\{T } and independent noises on the observations {vt}t∈T and admissibility constraints of the form Uad : X ⇒ U as a finite Det-Pomdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' To do so, we expand the state of the Pomdp with the realization of all noises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We model the problem as though the realization of the noises are predeter- mined, but the decision maker does not know the noises in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We then obtain a Det-Pomdp, with states X′, controls U and observations O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' However, such reformulation leads to a drastic increase in the dimension of the state and the cardinality of the support of the initial belief.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Indeed, the initial 7 belief contains all possible values of the initial state and all the possible realizations of noises, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' its cardinality is multiplied by a factor |V|T +1 × |W|T , with V the set of noises on the observation and W the set of noises on the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Hence, we are doubly penalized when considering the bound presented in Theorem 4: we both increase |X| and |supp(b0)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' This reinforces the point on the difficulty of solving Pomdps as even the ones with simple structures are far more difficult than similar sized Det-Pomdps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='3 Belief dynamics as pushforward measures Here, we expose another representation of the beliefs evolution functions {τt}t∈T \\{T } defined in Equa- tion (9), used in the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' First, we recall the notion of pushforward measures when considering finite sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Second, we introduce the mappings necessary for the new representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We then present in Lemma 7 the representation of the belief dynamics as pushforward measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Consider two finite sets A and D and a mapping h : A → D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The pushforward measure (or the image-measure) of a probability measure µ ∈ ∆(A) on the set A by the mapping h is the probability measure h⋆µ ∈ ∆(D) on the set D defined by ∀d ∈ D , (h⋆µ)(d) = µ � h−1(d) � = � a∈A,h(a)=d µ(a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (15) We also denote by h⋆ the mapping from ∆(A) to ∆(D) such that h⋆(µ) = h⋆µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Before presenting Lemma 7, we first introduce some mappings: F u,o t , and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' For each pair (u, o) ∈ U × O, and each t ∈ T \\ {T}, we denote by F u,o t the self-mapping on the extended state set X = X ∪ {∂} (defined in Equation (3)), defined by: F u,o t : X → X , x �→ � f u t (x) if x ̸= ∂ and f u t (x) ∈ � hu t+1 �−1(o) , ∂ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (16) The mapping F u,o t hence applies the dynamics ft, as defined in Problem (2), given control u, and only keeps the resulting state if it is consistent with observation o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Meanwhile, the renormalization mapping R : ∆(X) → ∆(X) is defined by R : ν ∈ ∆(X) �→ �� 1 ν(X)ν|X, 0 � if ν(X) ̸= 0 , δ∂ if ν(X) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (17) We now express the belief dynamics as pushforward measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Let (u, o) ∈ U × O be given, and let t ∈ T \\ {T}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We have that ∀b ∈ B , τt(b, u, o) = R ◦ (F u,o t )⋆(b) , (18) where the pushforward (F u,o t )⋆(b) follows Notation (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The proof is detailed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' This new representation is of interest as, for all time t ∈ T \\ {T}, the composition of belief dynamics τt is given by the pushforward measure of the composition of mappings F u,o t for the relevant pairs (u, o) ∈ U × O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Indeed, when considering a composition of belief dynamics, we can factorize the renormalization mapping R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We thus apply the renormalization mapping R to the composition of the pushforward measures, which is the pushforward measure of the composition of mappings F u,o t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' There is therefore an equivalence between studying the composition for time t ∈ T \\ {T} of the belief dynamics τt and the composition, for the relevant pairs (u, o) ∈ U × O, of the mappings F u,o t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Notably, we use this representation to bound the cardinality of the set of reachable beliefs, and thus study the complexity of Dynamic Programming for Det-Pomdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' To do so, we introduce notations for sets and mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 8 Notation for sets and mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' For any given sets Y and V, we denote by L(Y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' V) = VY the set of mappings from Y to V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' For all G ⊂ L(Y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' V), Y ⊂ Y, B ⊂ ∆(Y) and b ∈ ∆(Y) we introduce the notations G(Y ), and G⋆(B), and G⋆(b) for the sets defined by G⋆(b) = G⋆({b}) ⊂ ∆(V) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (19a) Given two subsets G′ and G′′ of L(Y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Y) we introduce the subset G′ ◦ G′′ defined by G′ ◦ G′′ = � g′ ◦ g′′ �� g′ ∈ G′ and g′′ ∈ G′′� ⊂ L(Y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (19b) For any sequence {Gk}k∈N, with Gk ⊂ L(Y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Y) for all k ∈ N, we introduce for any k ∈ N the subsets G0:k defined by ∀k ∈ N , G0:k = Gk ◦ Gk−1 ◦ · · · ◦ G0 ⊂ L(Y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (19c) For a fixed value of u ∈ U, and o ∈ O, for all t ∈ T \\ {T}, we have obtained in Lemma 7 that τt(·, u, o) = R ◦ (F u,o t )⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Now, for each t ∈ T , we introduce the sets TD t = � τt(·, u, o) �� u ∈ U, o ∈ O � ⊂ L(B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' B) , (20) FD t = � F u,o t �� u ∈ U, o ∈ O � ⊂ L(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' X) , (21) FD = � t∈T \\{T } FD 0:t , (22) where the composition of sets of mapping is given by Notation (19b) and (19c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Note that4 FD 0:t ̸= FD �0,t�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Moreover, we call FD, defined by Equation (22), the set of pushforwards of the Det-Pomdp defined by Equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Let b0 ∈ ∆(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We have that ∀t ∈ T \\ {0} , BR,D t (b0) = TD 0:t−1(b0) = R ◦ (FD)⋆(b0) , (23) TD = � t∈T \\{T } TD 0:t = R ◦ (FD 0:t)⋆ , (24) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' BR,D �1,T �(b0) = � t∈T \\{T } TD 0:t(b0) = R ◦ (FD)⋆(b0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (25) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The proof is detailed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Lemmata 7 and 8 are illustrated in Figures 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' A direct application of Lemma 8 is that there is an equivalence between studying the cardinality of BR,D �1,T �(b0) and studying the cardinality of (FD)⋆(b0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We now present the postponed sketch of proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' A detailed proof can be found in Appendix §A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Sketch of proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Let b0 ∈ ∆(X) be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' By Lemma 8, we have that BR,D �1,T �(b0) = R ◦ (FD)⋆(b0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The first inequality |BR,D �1,T �(b0)| ≤ (1 + |X|)|supp(b0)| comes from the fact that ��(FD)⋆(b0) �� is bounded by the number of mappings from supp(b0) to X, as shown in Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Meanwhile, the second inequality ��BR,D �1,T �(b0) �� ≤ 1 + |supp(b0)||U||T | comes from the fact that, for all time and controls (t, u) ∈ T \\{T}×U, and for any belief b ∈ ∆(X), we have that � o∈O ��supp � (F u,o t )⋆b ��� ≤ ��supp � b ��� by Lemma 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Therefore, for a given sequence of controls u0:t ∈ Ut+1, there can be at most |supp(b0)| resulting beliefs (see Lemma 23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' As there are at most |U||T | such sequences u0:t, t ∈ T \\ {T}, this leads to ��BR,D �1,T �(b0) �� ≤ 1 + |supp(b0)||U||T |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 4FD 0:t is the set of compositions of mappings F u,o t′ from time t′ = 0 to time t′ = t for all controls u ∈ U and observation o ∈ O, while the set FD �0,t� is the set of all mappings F u,o t between time 0 and time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 9 ∆(X) b B = ∆(X) ∪ {δ∂} τ u,o′ t ∆(X) ∆(X) (b, 0) � F u,o t � ⋆ R � (b′ |X, b′(∂) ���� ∈R ) � Figure 1: Illustration of the beliefs dynamics as pushforward measures ∆(X) b B = ∆(X) ∪ {δ∂} τ u′,o′ t+1 ◦ τ u,o t ∆(X) ∆(X) ∆(X) � F u,o t � ⋆ � F u′,o′ t+1 � ⋆ R = � F u′,o′ t+1 ◦ F u,o t � �� � ∈X X � ⋆ Figure 2: Illustration of the composition of be- lief dynamics as pushforward measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We now present the subclass of Separated Deterministic Partially Observed Markov Decision Pro- cesses (Separated Det-Pomdp), which is simpler than Det-Pomdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 4 Separated Det-Pomdp and complexity In this section, we introduce a subclass of Det-Pomdps: Separated Det-Pomdps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' First, we define this subclass in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Second, in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='2, we present an improved bound on the cardinality of the set of reachable beliefs for Separated Det-Pomdps compared to Det-Pomdps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Third, in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='3, we show that the improved bound is tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='1 Definition of (∂)-separated mapping set and Separated Det-Pomdp Let us first define separated mapping sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Let Y1 and Y2 be two given sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' A set G ⊂ L(Y1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Y2) of mappings from Y1 to Y2 is called a separated mapping set if ∀(g1, g2) ∈ G2 , ∀y ∈ Y1 , � g1(y) = g2(y) ⇒ g1 = g2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' A separated mapping set G ⊂ L(Y1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Y2) is hence a set of mappings where all pairs of mappings are either different everywhere, or equal everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Otherwise stated, all the evaluation mappings on set G (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' the mappings G → Y2, g �→ g(y), for a fixed y ∈ Y1) are injective for all y ∈ Y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' For example, let Y1 = �1, n� and Y2 = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Then, G ⊂ RY1 is identified with G ⊂ Rn, and G is a Separated mapping set if and only if the projections of G along each axis are injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' In the special case where Y1 = Y2 = X, with the extended set X = X ∪ {∂} defined in Equation (3), we want to extend the above notion of separated mapping set to tackle the added point ∂ in a specific way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We thus introduce the notion of (∂)-separation for a pair of self-mappings on the set X and the notion of (∂)-separated mapping set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Let X = X∪{∂}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' A pair (g1, g2) ∈ L(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' X) of self-mappings on the set X is (∂)-separated if the restriction of the pair (g1, g2) to the set g−1 1 (X) ∩ (g2)−1(X) is separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Moreover, a set G of self-mappings on the set X is called a (∂)-separated mapping set if all pairs of mappings (g1, g2) ∈ G2 are (∂)-separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Definition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' A Separated Det-Pomdp is a Det-Pomdp such that the set of pushforward of the Det-Pomdp FD, defined in Equation (22), is a (∂)-separated mapping set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Otherwise stated, for a Separated Det-Pomdp, if two sequences of controls lead to the same state when starting in state x, then applying the two sequences of controls to another state x′ either leads to the same state, or at least one sequence of controls leads to the cemetery point ∂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We now present a link between the notion of separated mapping set and the notion of Separated Det-Pomdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' This allows us to propose a sufficient condition in order to ensure that a Det-Pomdp is a Separated Det-Pomdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 10 Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' If the set � t∈T \\{T } f Ut+1 0:t = {f u0:t 0:t | ∀t ∈ T \\ {T}, ∀u0:t ∈ Ut+1} of the composition of the evolution functions of Problem (2) is a separated mapping set, as defined if Definition 9, then Problem (2) is a Separated Det-Pomdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The proof of Proposition 12 is a direct consequence of Corollary 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The detailed proof is found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Note that the observation mappings {ht}t∈T \\{T } do not play any role in Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Now that we have defined the subclass of Separated Det-Pomdps, we present a bound on the cardinality of the set of reachable beliefs for this subclass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='2 Complexity analysis of Separated Det-Pomdp We now present the main interest of Separated Det-Pomdp when compared to Det-Pomdp, namely that the bound on cardinality of the set of reachable beliefs is lowered from (1 + |X|)|supp(b0)| to 1 + � 2|supp(b0)| − |supp(b0)| � |X|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Consider a Separated Det-Pomdp optimization problem given by Problem (2) which satisfies the finite sets Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' For all initial belief b0 ∈ ∆(X), the cardinality of the set BR,D �1,T �(b0) of reachable beliefs starting from b0 satisfies the following bound ��BR,D �1,T �(b0) �� ≤ 1 + � 2|supp(b0)| − |supp(b0)| � |X| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (26) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The proof is detailed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We have therefore an improved complexity of the Dp Algorithm for Separated Det-Pomdp compared with standard Det-Pomdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Corollary 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Consider a Separated Det-Pomdp optimization problem given by Problem (2) which sat- isfies the finite sets Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Then, the Dp Algorithm numerically solves Problem (2) by Dynamic Programming and its complexity is O � min � 1 + � 2|supp(b0)| − |supp(b0)| � |X|, 1 + |supp(b0)||U||T |� |T ||U||O| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' By Proposition 3, the Dp Algorithm solves Problem (2)and its complexity is O � |T ||BR,D �1,T �(b0)||U||O| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Then, by Theorem 13, we have ��BR,D �1,T �(b0) �� ≤ 1 + � 2|supp(b0)| − |supp(b0)| � |X| and, by Theorem 4, we have that ��BR,D �1,T �(b0) �� ≤ 1 + |supp(b0)||U||T |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' As the bound presented in Theorem 13 depends on the states that can be reached when starting from states in the support of the initial belief, we can obviously improve the bound when the support of the belief belongs to a subset of X stable by the dynamics {ft}t∈T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Remark 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Assuming that Problem (2) is a Separated Det-Pomdp, that Assumption 1 holds, that |supp(b0)| > 1, that the evolution functions {ft}t∈T \\{T } of Problem (2) satisfy the property that there exists a subset A ⊂ X such that, for all time t ∈ T \\ {T}, ft(A, U) ⊂ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Assume that supp(b0) ⊂ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Then the bound presented in Theorem 13 can be improved as ��BR,D �1,T �(b0) �� ≤ 1 + � 2|supp(b0)| − |supp(b0)| � |A| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (27) Now that we have a better bound than with non-separated Det-Pomdps, the question is whether it is tight or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We now show that it is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='3 Existence of Separated Det-Pomdps with tight bound In Theorem 13, we have given an improved bound on the cardinality of the set of reachable beliefs for Separated Det-Pomdp compared with standard Det-Pomdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We now prove that the bound is tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Proposition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' There exist a Separated Det-Pomdpsuch that equality is obtained in Equation (26), that is, ��BR,D �1,T �(b0) �� = 1 + � 2|supp(b0)| − |supp(b0)| � |X| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (28) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We exhibit a simple Separated Det-Pomdp for which the set of reachable beliefs BR,D �1,T �(b0) satisfies Equation (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Following the framework of §2, let X = {x1, x2, x3} consists of three distinct states, O = {¯o1, ¯o2} of two distinct observations, and U = {¯u1, ¯u2} of two distinct controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The evolution functions are defined as ∀x ∈ X , f(x, ¯u1) = x, and ∀i ∈ {1, 2, 3}, f(xi, ¯u2) = xmod(i,3)+1, where mod(i, 3) is the remainder of the euclidean division of i by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Finally, the observation mapping is given by h(x, u) = � ¯o2 if x = x3 and u = ¯u1 , ¯o1 otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We show in Figure 3 the mappings F (u,o) defined in Equation (16) for this simple case, and we illustrate the dynamics and observation functions in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' F ¯u1,¯o1 x1 x2 x3 ∂ x1 x2 x3 ∂ F ¯u1,¯o2 x1 x2 x3 ∂ x1 x2 x3 ∂ F ¯u2,¯o1 x1 x2 x3 ∂ x1 x2 x3 ∂ F ¯u2,¯o2 x1 x2 x3 ∂ x1 x2 x3 ∂ Figure 3: Representation of the F (u,o) mappings in the case of §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='3 ¯u1 x1 x2 x3 x1 x2 x3 ¯o1 ¯o2 ¯u2 x1 x2 x3 x1 x2 x3 ¯o1 Figure 4: Representation of the dy- namics and the observations depend- ing on the control of the case of §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='3 By adding a cost function L, a horizon T > 0 and admissibility constraints Uad : x ⇒ U, the resulting problem has all the ingredients of a Det-Pomdp (as presented in §2), where Assumption 1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We now prove that the resulting Det-Pomdp is a Separated Det-Pomdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' For that purpose, we enumerate all the possible results of the dynamics before applying Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' For this purpose, let us consider a sequence of controls (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' , ut) ∈ Ut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' By denoting f u1:t the compositions of dynamics (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' f u1:t(x) = f ut ◦ · · · ◦ f u1(x)), we have, for all i ∈ �1, 3�, f u1:t(xi) = xmod(i+γ(u1:t)−1,3)+1, where γ is the function that counts the number of times ¯u2 appears in a sequence of controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The function γ is defined as γ : Ut → N, u1:t �→ ��{ui, i ∈ �1, t� | ui = ¯u2} ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The set {f u1:t | u1:t ∈ Ut} is thus such that, for all sequences of controls (u1:t, u′ 1:t′) ∈ Ut × Ut′, if there is a state x ∈ X such that f u1:t(x) = f u′ 1:t′ (x), then for any state x′ ∈ X, f u1:t(x′) = f u′ 1:t′ (x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Hence, the set ∪t∈T \\{T }f Ut+1 0:t is a separated mapping set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' By Proposition 12, the optimization problem is hence a Separated Det-Pomdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We now chose an initial belief b0 such that supp(b0) = {x1, x2}, for which we can compute explicitly the reachable beliefs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We can apply Theorem 13 with such initial belief.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Therefore, according to Equa- tion (26), there can be at most 7 reachable beliefs (including δ∂).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' In Table 1, we enumerate all possible supports of the reachable beliefs when starting with belief b0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We have therefore 7 different supports for the reachable beliefs, hence at least 7 beliefs in the set of reachable beliefs starting from b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' As Equation (26) states that there can be at most 7 reachable beliefs, we obtain that we have exactly 7 reachable beliefs and thus Equation (28) is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Note that, while the proof of Proposition 16 was made with a Separated Det-Pomdp with |X| = 3, we can generate a Separated Det-Pomdp such that equality is obtained in Equation (26) for any |X| = n, n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We need once again that X = {xi}i∈�1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='n� consists of n distinct states,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' O = {¯o1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' ¯o2} 12 Mapping applied Support of resulting belief F ¯u1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='¯o1 {x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' x2} F ¯u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='¯o1 {x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' x3} F ¯u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='¯o1 ◦ F ¯u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='¯o1 {x3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' x1} F ¯u1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='¯o2 ◦ F ¯u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='¯o1 {x3} F ¯u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='¯o1 ◦ F ¯u1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='¯o2 ◦ F ¯u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='¯o1 {x1} F ¯u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='¯o1 ◦ F ¯u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='¯o1 ◦ F ¯u1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='¯o2 ◦ F ¯u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='¯o1 {x2} F ¯u1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='¯o2 {∂} Table 1: Resulting support when applying given mappings to the initial belief b0 with supp(b0) = {x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' x2} of two distinct observations and U = {¯u1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' ¯u2} of two distinct controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Then, the dynamics is given by ∀x ∈ X , f(x, ¯u1) = x, and ∀i ∈ �1, n�, f(xi, ¯u2) = xmod(i,n)+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Finally, the observation mapping is given by h(x, u) = � ¯o2 if x = xn and u = ¯u1 , ¯o1 otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Now that we have presented the subclass of Separated Det-Pomdps, we give a numerical illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 5 Numerical application on a toy example of Separated Det- Pomdp In this section, we present a simple one-dimensional illustration of Separated Det-Pomdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We consider that we empty a tank while minimizing an associated cost, as illustrated in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The state is one- dimensional and consists in the volume of water present in the tank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The control is also one-dimensional and is the amount of water that the decision maker removes during one time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The decision maker has access at time t to partial observation, as he only knows that the volume of water in the tank is between two quantized levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='1 A partially observed tank as a Separated Det-Pomdp More precisely, the problem is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The state x consists of a discrete volume of water in the tank, with x ∈ X = {x(1), x(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' , x(n)} ⊂ R+ of finite cardinality n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The observation o consists of a discrete level of water in the tank, with o ∈ O = {o(1), o(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' , o(m)} ⊂ R+ of finite cardinality m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The control u consists of a discrete volume of water to be removed, with u ∈ U = {u(1), u(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' , u(d)} ⊂ R+ of finite cardinality d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The unitary price of water at each time t ∈ T \\ {T} is given by ct ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' o(2) o(3) o(1) Figure 5: Illustration of the wa- ter tank “quantum” of observation (m = 3) 13 Optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We now adapt the Problem (2) to the tank case presented above: min X,U,OE �T −1 � t=0 ctUt � (29a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' PX0 = b0 , (29b) Xt+1 = Xt − Ut , ∀t ∈ T \\ {T} , (29c) Ut ∈ {u(i) ∈ U | u(i) ≤ Xt} , ∀t ∈ T \\ {T} , (29d) Ot = max{o(j) ∈ O | Xt ≥ o(j)} , ∀t ∈ T , (29e) σ(Ut) ⊂ σ (O0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' , Ot, U0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' , Ut−1) , ∀t ∈ T \\ {T} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (29f) Equation (29a) represents the objective function of the tank problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Equation (2a) of Prob- lem (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The instantaneous cost function at time t is defined as Lt(ut) = ctut, and hence only depends on the controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The evolution function corresponds to emptying the tank and is given by f : (x, u) �→ x−u, which gives Equation (29c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The observation function h is given by a piecewise constant function which does not depend on the controls u: h(x) = max{o(i) | x ≥ o(i)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' This leads to equation (29e), which is the implementation of (2e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The admissibility set of the tank problem is given by Uad(Xt) = [0, Xt] (see Equation (29d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' It ensures that we cannot remove more water than what is in the tank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Note that this could be a problem as we do not observe Xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Problem (29) has the same form as Problem (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' It is therefore a Det-Pomdp and all the relevant results presented in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='1 hence apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Associated beliefs dynamics τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Let (b, u, o) ∈ B × U × O, with B = ∆(X) ∪ {δ∂}, as defined in Equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' As the evolution functions and observation functions are stationary, the belief dynamics are also stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We note I(o) ⊂ X, the set of states compatible with the observation o, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' I(o) = {x ∈ X | h(x) = o} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (30) By Equation (29c), we have (f u)−1(y) = y + u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Moreover, we have, by the definition of I(o) in Equation (30), that � hu�−1(o) = I(o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Hence, the function Q in (7) is here Q : B × U × O → [0, 1] , (b, u, o) �→ � x∈I(o)−{u} b(x) , and Equation (9) gives τ(b, u, o)(y) = � � � � � � � b(y + u) � x′∈I(o)−{u} b(x′) if y ∈ I(o) − {u} , 0 if y ̸∈ I(o) − {u} , where I(o) − {u} is defined in Equation (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Bellman equations for the partially observed tank problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' As Problem (29) is a Det-Pomdp and the finite sets Assumption 1 holds, we can apply Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Equations (10a) and (10b) are here VT : BR,D T (b0) → R , b �→ 0 (31a) Vt : BR,D t (b0) → R , b �→ min u≤minx∈supp(b) x � ctu + � o∈O � x−u∈[o,o] b(x)Vt+1 � τ(b, u, o) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (31b) Indeed, the intersection Ub,ad t (b) = � x∈supp(b) Uad t (x) is {u(i) ∈ U | u ≤ minx∈supp(b) x}, as the admissi- bility set is given by Equation (29d), and as {u(i) ∈ U | u(i) ≤ x(j)} ∩ {u(i) ∈ U | u(i) ≤ x(k)} = {u(i) ∈ U | u(i) ≤ min � x(j), x(k)� } .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 14 The partially observed tank problem as a Separated Det-Pomdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The tank Det-Pomdp is a Separated Det-Pomdp as a direct consequence of Corollary 29, in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Indeed, Corollary 29 states that if the evolution functions ft of a Det-Pomdp are linear, then it is a Separated Det-Pomdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' As the evolution function f of the partially observed tank is indeed linear, the tank Det-Pomdp is a Separated Det-Pomdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='2 Numerical results We now present some numerical results for the tank problem described by Problem (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Presentation of the instances We made a numerical application with the following parameters: X = �0, 300�, U = �0, 9�, O = {0, 1, 20, 40, 60, 80, 100, 120, 140, 160, 180, 200, 220, 240, 260, 280, 300}, T = �0, 100�, supp(b0) = �260, 300�, with a randomly generated probability distribution over that support, the distribution used is detailed in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 260 270 280 290 300 0 5 · 10−2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='15 x b0(x) Figure 6: Probability distribution used as the initial belief b0 for the numerical applications When considering the initial belief b0 presented in Figure 6 and a “true” (unknown) initial state of x0 = 290 (used to simulate the observation process depending on the policy), we obtain the tank water volume represented in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Moreover, we have a set of reachable beliefs BR,D �0,100� such that |BR,D �0,100�| = 64, 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We therefore do not display value functions, as they are defined on sets with large cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We also made a second numerical application where the observation O is changed to: O = {1, 6, 11, 51, 101, 151, 201, 251} When considering the new observations set and the same initial belief and initial state, we obtain a trajectory represented in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Figures 7 and 8 both illustrate some properties of Det-Pomdps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' First, in both cases, we see that the support of the beliefs decreases with time (the vertical red slices are non increasing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Second, we remark that such a decrease is due to the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Indeed, in Problem (29), the observation function ensures that the support of the beliefs must belong to intervals [ot, ot] when we observe ot (see Equation (30)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Thus, the supports of the beliefs are reduced along the limit of 15 10 20 30 40 50 60 70 80 90 100 0 50 100 150 200 250 300 Time Possible States Xt 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='5 3 Prices Figure 7: Representation of a trajectory of the volume of water in the tank when applying the optimal controls when considering the first set of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' A vertical slice at time t of the red area represents the support of the belief held at time t, the dotted blue curve represents the tra- jectory of the “true” state, the piecewise constant green curve is the observation we have access to at time t, and the dashed orange curve represents the periodic prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 10 20 30 40 50 60 70 80 90 100 0 50 100 150 200 250 300 Time Possible States Xt 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='5 3 Prices Figure 8: Representation of a trajectory of the volume of water in the tank when applying the optimal controls when considering the second set of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' A vertical slice at time t of the red area represents the support of the belief held at time t, the dotted blue curve represents the tra- jectory of the “true” state, the piecewise constant green curve is the observation we have access to at time t, and the dashed orange curve represents the periodic prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' those intervals, as is more easily seen in Figure 8 between time t = 1 to t = 6 (we apply a control, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' removing some water, and we see that the lower part of the support remains at the observation value until time t = 7, which is when we change observation and we see that the upper bound of the support gets just beneath the previous observation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' at x = 249).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Third, we remark that, as could be expected, the optimal policy consists of removing water when prices are high, and stopping when prices are low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Fourth, we remark that, despite having fewer observations in the second case, the optimal trajectory in the second case reaches a deterministic belief (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' such that |supp(b)| = 1) much sooner in Figure 8 compared to Figure 7 (at time t = 33 for the second case and time t = 53 for the first case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Having more observations hence does not guarantee to remove ambiguities at a faster rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We now present the computation time of the Dp Algorithm and compare it to another algorithm, Sarsop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Comparison with Sarsop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' In this paragraph, we focus on the comparison with Sarsop, first in- troduced in (Kurniawati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=', 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We used the Julia implementation of this algorithm, with the POMDPs package API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The following results were obtained on a computer equipped with a Core i7- 8665U and 32 GB of memory, using Julia v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='3, POMDPs v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='3 and Sarsop v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' However, we must first warn the reader that Sarsop is an algorithm that solves an infinite horizon Pomdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We hence reformulate the finite horizon Det-Pomdp as an infinite time Pomdp by extending the state with the time variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Such reformulation leads to a much bigger problem in terms of data and size of the state space, which heavily penalizes Sarsop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Hence, the reformulation prevents any fair comparison of computation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We still present some computation time in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Note that, for each instance where the computation did not stop (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' those without a “>” symbol in the computation time column) due to hitting the memory limit of the computer, Sarsop and the Dp Algorithm found the same value, hence Sarsop indeed converged toward the optimal solution of Problem (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 16 |X| |U| |O| |supp(b0)| T Sarsop Dp Algorithm computation time (s) computation time (s) 11 2 3 2 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='376 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='002 21 2 5 2 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='003 51 5 5 2 100 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='20 51 5 5 4 100 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='20 51 5 5 6 100 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='03 101 5 5 2 200 359 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='96 101 5 5 10 200 1930 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='2 101 10 5 10 200 1069 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='2 201 5 5 10 200 3506 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='1 201 10 5 10 200 15618 309 201 5 5 20 200 3652 225 201 10 6 20 200 33562 497 301 5 6 10 200 4638 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='8 301 10 6 10 300 > 38000 762 (> 19217s of iterations) Table 2: Computation time of different instances of both Sarsop and the Dp Algorithm 6 Conclusion In this paper, we have presented a subclass of Pomdps, Separated Det-Pomdps, which has proper- ties that contribute to push back the curse of dimensionality for Dynamic Programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Indeed, we have shown that the conditions on the dynamics for Separated Det-Pomdp improve the bound on the cardinality of the set of the reachable beliefs: the bound is reduced from � 1 + |X| �|supp(b0)| (in the case of Det-Pomdp, see Theorem 4) to 2|supp(b0)||X| (Theorem 13), as presented in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' This tighter bound allows Dynamic Programming algorithms to efficiently solve Separated Det-Pomdp problems, especially when considering small supports of the initial state distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Moreover, the bound is tight (see Proposition 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The Separated Det-Pomdp class is, therefore, an interesting framework for some problems as only a fraction of the number of beliefs needs to be considered, in comparison with Det-Pomdp or Pomdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The Separated Det-Pomdps are therefore tractable with larger instances than regular Pomdps or Det- Pomdps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Class Infinite horizon bound Finite horizon bound Det-Pomdp (1 + |X|)|X| min � (1 + |X|)|X| ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' � |U||O| �|T |� (Littman,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 1996) Det-Pomdp (1 + |X|)|supp(b0)| min � (1 + |X|)|supp(b0)| ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 1 + |supp(b0)||U||T |� improved bounds Theorem 4 Theorem 4 Separated 1 + � 2|supp(b0)| − |supp(b0)| � |X| min � 1 + � 2|supp(b0)| − |supp(b0)| � |X|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Det-Pomdp 1 + |supp(b0)||U||T |� Theorem 13 Corollary 14 Table 3: Summary of the bounds depending on the class of problem A Appendix First,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' in §A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='1, we present technical lemmata used to prove bounds on the cardinality of the sets of reachable beliefs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Second, in §A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='2, we present complementary results on (∂)-separated mappings sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 17 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='1 Technical lemmata In this subsection, we present technical lemmata used in the proofs of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We first introduce in §A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='1 the notions of forward and backward mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Second, in §A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='2, we present properties on the composition and pushforward measures by those forward and backward mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Third, in §A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='3, we present properties on the cardinality of sets of forward and backward mappings used notably in the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='1 Forward and backward mappings For any subset X ⊂ X, we introduce the notion of X-forward and X-backward mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Given a mapping h : X → X and a subset X ⊂ X, we define a mapping h− → X : from X to X, called a X-forward mapping, as follows h− → X : x ∈ X �→ � h(x) if x ∈ X and h(x) ∈ X , ∂ if x = ∂ or h(x) ̸∈ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (32) We call h− → X : X → X a X-forward mapping as we have h− → X(X) ⊂ X ∪{∂}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' X-forward imposes a constraint on the codomain (set of destinations): we only keep the values that belong to X, whereas the others are sent to ∂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The set X is thus a subset of the codomain of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We also introduce the X-backward mapping h← − X : X → X, defined by h← − X : x ∈ X �→ � h(x) if x ∈ X , ∂ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (33) We call h← − X : X → X a X-backward mapping as we have h← − X(X) ⊂ X, and h← − X � X \\ X � = {∂}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' X- backward imposes a constraint on the domain (set of departures): we only keep the values whose inputs are in X, whereas the others are sent to ∂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The set X is thus a subset of the domain of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' It is straightforward to check that we have ∀X ⊂ X , h− → X = h←−−−−− h−1(X) , (34a) ∀X ⊂ X , h− → X = h−−−−−−→ X∩Im(h) , (34b) where Im is the image of a mapping, that is Im(h) = h(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' A forward mapping can hence be rewritten as a backward mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The reverse is not true, as we have h← − X = h−−−→ h(X) ⇔ h−1� h(X) � = X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='2 Results on pushforward measures by forward and backward mappings sets We now present properties of the composition of pushforward measures of forward and backward map- pings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Definition 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Let M ⊂ L(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' X) be a subset of self mappings on the set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We say that G ⊂ L(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' X) is an � M, ←− X � mappings set (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' an � M, −→ X � mappings set) if it satisfies the following property G ⊂ � h← − X �� h ∈ M and X ⊂ X � , (35a) � resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' G ⊂ � h− → X �� h ∈ M and X ⊂ X �� , (35b) where h← − X (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' h− → X) is defined in Equation (33) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Equation (32)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' When M = L(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' X), a � M, ←− X � mappings set (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' an � M, −→ X � mappings set) is just named a �←− X � mappingsset (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' an �−→ X � mappings set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We obtain the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' If G is an � M, −→ X � mappings set, then G is an � M, ←− X � mappings set (using Equality (34a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 18 �←− X � mappings sets are stable by composition, as we easily obtain that h′←− X′ ◦ h← − X = (h′ ◦ h)←−−−−−−−− X∩h−1(X′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (36) Let G be an �←− X � mappings set and consider, for any X ⊂ X, the subset G← − X of G defined by G← − X = � g ∈ G �� ∃h ∈ L(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' X), g = h← − X � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (37) Then, for any belief b0 ∈ ∆(X), we have � R ◦ (G←−−−−−−−− X∩supp(b0))⋆ � (b0) = � R ◦ (G← − X)⋆ � (b0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (38) The Equation (38) is a consequence of the following Lemma 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Indeed, assuming Lemma 18, the expression of � R ◦ (G← − X)⋆ � (b0) given by Equation (39b) only depends on the restriction of the measure b0 to the subset X – which coincides with the restriction of the measure b0 to the subset X ∩ supp(b0) – as the measure b0 is null outside its support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Lemma 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Let X be a subset of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The mappings R ◦ (h← − X)⋆ and R ◦ (h− → X)⋆ in L(∆(X);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' B), where the pushforward measure is defined in Equation (15), and the mapping R is defined in Equation (17), have the following expressions for all ν ∈ ∆(X): � R ◦ (h− → X)⋆ � (ν) = � � � � x ∈ X �→ ν � h−1(x) � 1X(x) ν � h−1(X) � � if ν � h−1(X) � ̸= 0 , δ∂ otherwise, (39a) and � R ◦ (h← − X)⋆ � (ν) = � � � � x ∈ X �→ ν � h−1(x) ∩ X � ν � h−1(X) ∩ X � � if ν � h−1(X) ∩ X � ̸= 0 , δ∂ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (39b) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' For any probability measure ν on the finite set X, it is straightforward, using the definition of pushforward measure in Equation (15), to obtain that the pushforward of the measure ν through the mapping h− → X, as defined in Equation (32), is given by (h− → X)⋆ν : X → R+ y �→ ν � (h− → X)−1(y) � = � � � � � � � ν � h−1(y) � if y ∈ X , � 1 − ν � h−1(X) �� if y = ∂ , 0 if y ̸= ∂ and y ̸∈ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (40) Thus, we obtain that ∀x ∈ X , � (h− → X)⋆ν � |X(x) = ν � h−1(x) � 1X(x) , (41) and that � (h− → X)⋆ν � (X) = � x∈X ν � h−1(x) � 1X(x) = ν � h−1(X) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (42) Hence, using the definition of R in Equation (17), the result follows from Equation (39a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The proof of Equation (39b) is very similar and left to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The composition of self-mappings of the form R ◦ (h− → X)⋆ can also be written without resorting to multiple renormalizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Instead, we only need to renormalize the composition of the pushforward measures, as shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 19 Lemma 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Assume that h and h′ are self-mappings on the finite set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Then, for any subsets X and X′ of X, we have the following composition equalities R ◦ (h− → X)⋆ ◦ R ◦ (h′−→ X′)⋆ = R ◦ (h− → X ◦ h′−→ X′)⋆ , (43a) R ◦ (h← − X)⋆ ◦ R ◦ (h′←− X′)⋆ = R ◦ (h← − X ◦ h′←− X′)⋆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (43b) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We just prove Equation (43a) as the proof follows the same lines for Equation (43b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' As a preliminary, we remark that the mapping R ◦ (h− → X)⋆ is defined on the nonnegative measures on the set X and not just on probability measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Now, given µ ∈ ∆(X), we consider the nonnegative measure µ′ = (µ|X, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The two nonnegative measures µ and µ′ coincide on the set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Thus using the expression of R ◦ (h− → X)⋆ in Equation (39a) and the fact that X ⊂ X, we obtain that R ◦ (h− → X)⋆(µ) = R ◦ (h− → X)⋆(µ|X, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Now, let ν ∈ ∆(X) be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We denote by ν′ ∈ ∆(X) the probability measure ν′ = (h′−→ X′)⋆ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We consider two cases: either ν′(X) ̸= 0, or ν′(X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' First case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We assume that ν′(X) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' we successively have R ◦ (h− → X)⋆ ◦ R ◦ (h′−→ X′)⋆ν = R ◦ (h− → X)⋆ ◦ R(ν′) (by replacing (h′−→ X′)⋆ν by ν′) = R ◦ (h− → X)⋆ � 1 ν′(X)ν′ |X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 0 � (using R definition in (17),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' with ν′(X) ̸= 0) = R ◦ (h− → X)⋆ � 1 ν′(X)(ν′ |X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 0) � (factorizing by 1 ν′(X)) = R � 1 ν′(X)(h− → X)⋆ � ν′ |X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 0 �� (as (h− → X)⋆ is 1-positively homogeneous) = R � (h− → X)⋆ � ν′ |X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 0 �� (as R is 0-positively homogeneous) = R � (h− → X)⋆(ν′) � (using the preliminary part) = R ◦ (h− → X)⋆ ◦ (h′−→ X′)⋆ν (as ν′ = (h′−→ X′)⋆ν) = R ◦ (h− → X ◦ h′−→ X′)⋆(ν) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (as f⋆ ◦ h⋆ = (f ◦ h)⋆) Second case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We assume that ν′(X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Then, we have that ν′ = δ∂ as ν′ ∈ ∆(X), and we obtain R ◦ (h− → X)⋆ ◦ R ◦ (h′−→ X′)⋆ν = R ◦ (h− → X)⋆ ◦ R(δ∂) (by replacing (h′−→ X′)⋆ν by ν′ = δ∂) = R ◦ (h− → X)⋆(δ∂) (as R(δ∂) = δ∂) = R ◦ (h− → X)⋆ ◦ (h′−→ X′)⋆ν (by replacing δ∂ = ν′ by (h′−→ X′)⋆ν) = R ◦ (h− → X ◦ h′−→ X′)⋆(ν) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Hence, in both cases, we obtain Equation (43a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Now that we have exposed technical lemmata on the composition and renormalization of �−→ X � mappings and �←− X � mappings, we present lemmata on the cardinality of sets of pushforward measures, notably the cardinality of pushforward measures by �−→ X � mappings and �←− X � mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='3 Results on the cardinality of sets of pushforward measures We now present results on the cardinality of sets of forward and backward mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Lemma 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Let {Gk}k∈N be a given sequence where, for each k ∈ N, the set Gk ⊂ L(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' X) is a finite set of self-mappings on the set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The sets Gk, for all k ∈ N, are assumed to be either all �−→ X � mappings sets or all �←− X � mappings sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We define the sequence {Φk}k∈N, where, for each k ∈ N, the set Φk ⊂ L(∆(X);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' ∆(X)) is a finite set of self-mappings on the set X given by ∀k ∈ N , Φk = R ◦ (Gk)⋆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (44) 20 Then, for any b0 ∈ ∆(X), we have the following bound ∀n ∈ N , ��� n � k=0 Φ0:k(b0) ��� ≤ (1 + |X|)|supp(b0)| , (45) where Φ0:k = Φk ◦ · · · ◦ Φ0 is defined in Equation (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' For all k ∈ N, we have Φ0:k(b0) = (Φk ◦ Φk−1 ◦ · · · ◦ Φ0)(b0) (by Equation (19)) = � R ◦ (Gk)⋆ ◦ R ◦ (Gk−1)⋆ ◦ · · · ◦ R ◦ (G0)⋆ � (b0) (by Equation (44)) = � R ◦ (Gk)⋆ ◦ (Gk−1)⋆ ◦ · · · ◦ (G0)⋆ � (b0) by Lemma (19), as the sets Gk are, by assumption, either all �−→ X � mappings sets or all �←− X � mappings sets, = � R ◦ (Gk ◦ Gk−1 ◦ · · · ◦ G0)⋆ � (b0) (as f⋆ ◦ h⋆ = (f ◦ h)⋆) = R � (G0:k)⋆(b0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Thus we have, for all n ∈ N, ��� �n k=0 Φ0:k(b0) ��� ≤ ��� ��n k=0 G0:k � ⋆(b0) ���, and the conclusion follows from the postponed Lemma 21 with J = �n k=0 G0:k, Y = V = X, and µ = b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Note that we can extend the previous Lemma 20 to cases with sequences {Gk}k∈N of mixes of both �−→ X � mappings sets and �←− X � mappings sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Indeed, forward mappings are also backward mappings by Equation (34a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We can hence write the sequence {Gk}k∈N as a sequence of only �←− X � mappings sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' In the rest of this paper, we consider sequences of only �−→ X � mappings sets or only �←− X � mappings sets, and thus only need Lemma 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We can bound the cardinality of the set of pushforward of a given nonnegative measure thanks to the following Lemma 21 (which was previously postponed in the proof of Lemma 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Let J ⊂ L(V;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Y) be a set of mappings from the set V to the set Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Assume that the sets V and Y are both finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Then, for any nonnegative measure µ on the set V, we have that |J⋆µ| ≤ |Y||supp(µ)| , (46) where we recall that |J⋆µ| denotes the cardinal of the set ��{j⋆µ | j ∈ J} �� as exposed in Equation (19a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Let µ be a given nonnegative measure on V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' For any j ∈ J we denote by j|supp(µ) the restriction of the mapping j to the subset supp(µ) ⊂ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' For all y ∈ Y, we have that j⋆µ(y) = µ � j−1(y) � (by the definition (15) of pushforward measures) = µ �� j−1(y) ∩ supp(µ) � ∪ � j−1(y) ∩ (supp(µ))c�� = µ � j−1(y) ∩ supp(µ) � + µ � j−1(y) ∩ (supp(µ))c� � �� � =0 = µ � j−1 |supp(µ)(y) � = �� j|supp(µ) � ⋆µ � (y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (by (15)) Thus, defining J|supp(µ) = {j|supp(µ) | j ∈ J}, we get that |{j⋆µ | j ∈ J}| = |{(j|supp(µ))⋆µ | j ∈ J}| ≤ |J|supp(µ)| ≤ |Ysupp(µ)| = |Y||supp(µ)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' This ends the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We now present a lemma on the conservation of the cardinality of the support of a measure through a composition of sets of mappings, if we have conservation of the cardinality for each individual set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 21 Lemma 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Let {Φk}k∈N be a sequence of self-mappings on the set B and assume that, for all k ∈ N, we have that ∀b ∈ B , � h∈Φk |supp � h(b)|X � | ≤ |supp(b|X)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (47) Then, for any b0 ∈ ∆(X), we have the following bound ∀k ∈ N , ��Φ0:k(b0) \\ {δ∂} �� ≤ |supp(b0)| , (48) where Φ0:k(b0) = Φk ◦ · · · ◦ Φ0(b0) is defined in Equation (19c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Let a belief b0 ∈ ∆(X) be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' As a preliminary result we prove, by forward induction on k ∈ N, that ∀k ∈ N , � b∈Φ0:k(b0) ��supp(b|X) �� ≤ |supp(b0)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (49) First, we consider the case k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' As Φ0:0 = Φ0 the result follows from Equation (47) used for k = 0 and b = b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Second,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' we consider 0 < k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' assuming that Equation (49) is satisfied for k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' we prove that it is also satisfied for k+1 as follows: � b∈Φ0:k+1(b0) ��supp(b|X) �� = � h∈Φ0:k+1 ��supp � (h(b0))|X ��� (by (19)) = � h′∈Φk+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='h′′∈Φ0:k ���supp �� h′(h′′(b0)) � |X ���� (as Φ0:k+1 = Φk+1 ◦ Φ0:k) = � h′′∈Φ0:k � � h′∈Φk+1 ���supp �� h′(h′′(b0)) � |X ���� � ≤ � h′′∈Φ0:k ���supp �� h′′(b0) � |X ���� (using Equation (47) for k and b = h′′(b0)) = � b∈Φ0:k(b0) ��supp � b|X ��� (by (19)) ≤ |supp(b0)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (by induction assumption (49) on k) We conclude that Equation (49) is satisfied for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Now, we turn to the proof of Equation (48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We make the following observation: if b ∈ ∆(X), then we have that |supp(b|X)| ≥ 1 and if b = δ∂ then |supp(b|X)| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Thus, we have that |Φ0:k(b0) \\ {δ∂}| = � b∈Φ0:k(b0)\\{δ∂} 1 (50) ≤ � b∈Φ0:k(b0)\\{δ∂} |supp(b|X)| (as |supp(b|X)| ≥ 1 for b ∈ Φ0:k(b0) \\ {δ∂}) = � b∈Φ0:k(b0) |supp(b|X)| (as |supp(δ∂|X)| = 0) ≤ |supp(b0)| , (by (49)) which gives Equation (48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' That concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Lemma 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Let {hk}k∈N be a sequence of self-mappings on the set X and, for all k ∈ N, let {Xk i }i∈Ik be a finite family of two by two disjoints subsets of X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Let {Gk}k∈N be the sequence of self-mappings on the set X, of the following form ∀k ∈ N , Gk = � hk−→ Xk i �� i ∈ Ik � ⊂ X X , (51) where hk−→ Xk i : X → X are built following Equation (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Consider the sequence {Φk}k∈N of self-mappings on the set B, given, for all k ∈ N, by Φk = R ◦ (Gk)⋆ and the associated sequence (Φ0:k)k∈N as defined in Equation (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Then, given b0 ∈ ∆(X), we have ∀k ∈ N , ��Φ0:k(b0) \\ {δ∂} �� ≤ |supp(b0)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (52) 22 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The proof relies on postponed Lemma 24 from which we obtain that the mappings Φk satisfy Equation (47) for all k ∈ N, and on Lemma 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' First, as a preliminary fact, we have that, for all µ ∈ ∆(X), supp �� R(µ) � |X � = supp(µ|X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Indeed, by (17), if µ(X) = 0, then supp �� R(µ) � |X � = supp � (δ∂)|X � = ∅ = supp � µ|X � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' whereas if µ(X) ̸= 0, then we have supp �� R(µ) � |X � = supp � ( µ|X µ(X), 0)|X � = supp � µ|X µ(X) � = supp(µ|X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Second, we show that the mappings Φk satisfy Equation (47) for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' For that purpose,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' we fix k ∈ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' and b ∈ B and we successively have � h∈Φk ��supp � h(b)|X ��� = � i∈Ik ���supp ��� R ◦ (hk−→ Xk i )⋆ � (b) � |X ���� (by definition of Φk = R ◦ (Gk)⋆ and Gk in (51)) = � i∈Ik ��supp �� (hk−→ Xk i )⋆(b) � |X ��� (as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' by the preliminary fact,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' ∀µ ∈ ∆(X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' supp �� R(µ) � |X � = supp(µ|X)) ≤ ��supp � b|h−1(⊔i∈Ik Xk i ) ��� (by (55) in Lemma 24,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' applied with Y = V = X and V = X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Vi = Xk i for i ∈ I = Ik) ≤ ��supp � b|X ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (as h−1(⊔i∈IkXk i ) ⊂ X) Third, as the assumptions given in Equation (47) are satisfied, the result follows by Lemma 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We now present the postponed technical Lemma 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Lemma 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Let h ∈ L(Y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' V) be a mapping from the set Y to the set V and assume that the sets Y and V are both finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Let V ⊂ V be a subset of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We define the mapping5 hV : Y → V ∪ {∂V} taking values in the extended set V = V ∪ {∂V} as follows hV : y ∈ Y �→ � h(y) if h(y) ∈ V , ∂V elsewhere .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (53) Then, for any nonnegative measure µ on the set Y, we have that ���supp �� (hV )⋆µ � |V ���� ≤ ��supp � µ|h−1(V ) ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (54) Moreover, for any finite family {Vi}i∈I of pairwise disjoints subsets of V, we have that � i∈I ���supp �� (hVi)⋆µ � |V ���� ≤ ��supp � µ|h−1(⊔i∈I Vi) ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (55) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We prove Equation (54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Let µ ∈ ∆(Y) be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' First, we note that, if the set supp �� (hV )⋆µ � |V � is empty, the result is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Second, we assume that supp �� (hV )⋆µ � |V � ̸= ∅ and consider v ∈ supp �� (hV )⋆µ � |V � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Thus, v is restricted to belong to V and, by definition of a pushforward mea- sure, it must satisfy µ � h−1 V (v) � ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' This implies that h−1 V (v) ̸= ∅ and, using the definition of hV (in Equation (53)), we obtain that v must belong to V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We conclude that there must exist y ∈ h−1 V (v) such that µ(y) ̸= 0 which, combined with the fact that the mapping h−1 V coincides with the mapping h−1 on V , gives that y ∈ h−1(v) ∩ supp(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Now, consider the set-valued mapping Γ : supp �� (hV )⋆µ � |V � ⇒ Y , y �→ h−1(v) ∩ supp(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' By construction, the set-valued mapping Γ takes values in the subsets of supp(µ|h−1(V )), and we have just proved that it takes values in the nonempty subsets of µ|h−1(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Moreover, the set-valued 5Note that the mapping hV is slightly different from h− → V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Indeed h− → V are defined for self-mappings, whereas hV is defined for an extended codomain (set of destinations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 23 mapping Γ is injective as, for all pairs (v′, v′′) ∈ V 2 of distinct elements, v′ ̸= v′′, we must have that h−1(v′) ∩ h−1(v′′) = ∅, as otherwise there would exist an element y ∈ Y such that h(y) = v′ and h(y) = v′′, which is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Thus, the image of Γ is a partition of a subset of supp(µ|h−1(V )) and we conclude that ��supp �� (hV )⋆µ � |V ��� = ��Γ � supp �� (hV )⋆µ � |V ���� ≤ |supp(µ|h−1(V ))| , which gives Equation (54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Now, we turn to the proof of Inequality (55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We successively have � i∈I ���supp �� (hVi)⋆µ � |V ���� ≤ � i∈I ��supp � µ|h−1(Vi) ��� (by (54) for each i ∈ I) = ��supp � µ|⊔i∈I h−1(Vi) ��� (as the family of subsets {h−1(Vi)}i∈I is composed of pairwise disjoints subsets as it was the case for the family {Vi}i∈I) = ��supp � µ|h−1(⊔i∈I Vi) ��� , (as h−1(⊔i∈IVi) = ⊔i∈Ih−1(Vi)) which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' This technical Lemma 24 shows that the cardinality of the support of a measure decreases when the measure is transported by a pushforward measure induced by a mapping of the form given by Equation (53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' A similar result ∀t ∈ T , ∀b ∈ B , ∀u ∈ U , � o∈O ��supp � τt(b, u, o) ��� ≤ ��supp(b) �� , is given in (Littman, 1996, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='2) but only for the mappings (τt)t∈T defined in Equation (9), and with a proof not explicitly connected to pushforward measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We now present the postponed proof of Lemma 7, presented in page 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Proof of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Fix (u, o) ∈ U × O, t ∈ T \\ {T}, and b ∈ B and denote by X ⊂ X the subset X = � hu t+1 �−1(o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We need to prove that we have τt(b, u, o) = R ◦ (F u,o t )⋆(b) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (56) Using Equation (7), we have that Qt+1(b, u, o) = b � (hu t+1 ◦ f u t )−1(o) � = b � (f u t )−1(X) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (57) Now, using the expression of τt in Equation (9) combined with Equation (57) and the definition of X, we obtain, for all x ∈ X, that τt(b, u, o)(x) = � � � � � b � (f u t )−1(x) � 1X(x) b � (f u t )−1(X) � if b � (f u t )−1(X) � ̸= 0 , 0 otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (58) Then, Equation (56) follows from Lemma 18 applied with the mapping h = f u t and with the subset X = � hu t+1 �−1(o), as we have F u,o t = f u t −−−−−−−−→ (hu t+1)−1(o) , (59) where f u t −−−−−−−−→ (hu t+1)−1(o) is defined in Equation (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' This ends the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We now present the postponed proof of Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 24 Proof of Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We first prove Equation (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' As a preliminary fact, by applying Lemma 7, No- tation (19a) and the definitions of sets TD t and FD t (Equations (20)-(21)), we obtain that, for all time t ∈ T \\ {T}, TD t = R ◦ (FD t )⋆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (60) Second, for all times (t, t′) ∈ � T \\ {T} �2 and for all pairs of controls and observations (u, u′) ∈ U2 and (o, o′) ∈ O2, we can apply Lemma 19 on mappings F u,o t and F u′,o′ t′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Indeed, by Equation (59), the mappings F u,o t and F u′,o′ t′ are X-forward mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We hence have by Equation (43) that R ◦ F u,o t R ◦ F u′,o′ t′ = R ◦ F u,o t F u′,o′ t′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Combined with Equation (60), this leads to TD 0:t = R ◦ (FD 0:t)⋆ , (61) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' it leads to Equation (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Now, let b0 ∈ ∆(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We prove Equation (23) by induction on t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' By Definition 2 of the set of reachable beliefs, we have BR,D 1 (b0) (11) = τ0 � {b0}, U, O � (20) = TD 0 (b0) (60) = R ◦ � FD 0 � ⋆(b0) , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Equation (23) stands at time 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Now, assuming Equation (23) is true up to time t ∈ T \\ {T}, t > 0, we have BR,D t+1 (b0) (11) = τt � BR,D t (b0), U, O � (20) = TD t � BR,D t (b0) � (23) = TD t ◦ TD 0:t−1(b0) (19) = TD 0:t(b0) (61) = R ◦ � FD 0:t � ⋆(b0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' By induction on time t, we hence have Equation (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Meanwhile, Equation (25) comes from the definition of TD �1,T � (see Equation (12)), the definition of FD (Equation (22)) and Equation (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We can now present the detailed proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Let b0 ∈ ∆(X) be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We first prove the inequality |BR,D �1,T �(b0)| ≤ (1 + |X|)|supp(b0)|, before proving the inequality ��BR,D �1,T �(b0) �� ≤ 1 + |supp(b0)||U||T |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' First, by Lemma (8), we have BR,D �1,T �(b0) = TD(b0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We hence have |BR,D �1,t�(b0)| (23) = |TD(b0)| (24) = ��� T −1 � i=0 TD 0:i(b0) ��� (45) ≤ (1 + |X|)|supp(b0)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The last inequality is given by Equation (45), obtained by applying Lemma 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' As all the elements of FD t are of the form given in Equation (16), the two sequences {FD t }t∈�0,T −1� and {TD t }t∈�0,T −1� satisfy the assumptions of Lemma 20 where the role of {Φk}k∈N is taken by {TD t }t∈�0,T −1� and the role of {Gk}k∈N is taken by {FD t }t∈�0,T −1� (the proof of Lemma 7 states that set FD t is an �−→ X � mappings set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We now prove that we have ��BR,D �1,T �(b0) �� ≤ 1 + |supp(b0)||U||T | , (62) in order to obtain Inequality (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' With the help of the representation of the beliefs evolution mappings given by Lemma 7, Inequality (62) is obtained as an application of Lemma 23 that we detail now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' For each t ∈ T \\ {T} and each ut ∈ U we introduce the sets TD,ut t = � τt(·, ut, o) �� o ∈ O � and FD,ut t = � F ut,o t �� o ∈ O � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Using set notations described in Equations (19), we obtain that TD,ut t = R ◦ (FD,ut t )⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Then, using the definition of BR,D t (b0) in Equation (11), we have that, for all time t ∈ T , t > 0, BR,D t (b0) = � u0:t−1∈U0:t−1 TD,ut−1 t−1 TD,ut−2 t−2 · · · ◦ TD,u0 0 (b0) = � u0:t−1∈U0:t−1 TD,u0:t−1 0:t−1 (b0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (63) For a fixed sequence of controls u0:t ∈ U0:t, the associated sequences of mappings {TD,ut t }t∈T and {FD,ut t }t∈T satisfy the assumptions of Lemma 23, where the role of {Φk}k∈N is taken by {TD,ut t }t∈�−1,T �, 25 the role of {Gk}k∈N is taken by {FD,ut t }t∈�−1,T � and the role of the family of disjoint sets {Xk i }i∈Ik is taken by the family {(hu t )−1(o)}o∈O,t∈�−1,T � (the proof of Lemma 7 states that the set FD t is an �−→ X � mappings set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We hence get that ∀t ∈ T \\ {T} , ��TD,u0:t0:t(b0) \\ {δ∂} �� ≤ |supp(b0)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (64) Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' we obtain ��BR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='D �1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='T �(b0) �� = ��� T� t=1 � BR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='D t (b0) ���� (using Equation (12)) ≤ 1 + ��� T� t=1 � BR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='D t (b0) \\ {δ∂} ���� (by removing δ∂ from BR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='D t (b0) for all t) = 1 + ��� T −1 � t=0 � u0:t∈U0:t � TD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='u0:t 0:t (b0) \\ {δ∂} ���� (using Equation (63)) ≤ 1 + T −1 � t=0 � u0:t∈U0:t ��� TD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='u0:t 0:t (b0) \\ {δ∂} ��� (as |A ∪ B| ≤ |A| + |B|) ≤ 1 + T −1 � t=0 � u0:t∈U0:t |supp(b0)| (using Equation (64)) ≤ 1 + T −1 � t=0 |U|t+1|supp(b0)| (as U0:t = Ut+1) ≤ 1 + |U| �|U|T − 1 |U| − 1 � |supp(b0)| (as �N i=0 xi = xN+1−1 x−1 for x ̸= 1) ≤ 1 + |U||T ||supp(b0)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (as |T | = T + 1 and |U| > 1) We have established the Inequality (62) and this concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='2 Complementary result on (∂)-separated mapping sets In this subsection, we present complementary results on (∂)-separated mapping sets by applying the framework presented in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We notably apply the notion of forward and backward mappings, presented in Equations (32) and (33), and the notion of pushforward measures, defined in Equation (15) in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' First, in §A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='1, we present and prove the lemmata used in the proofs of §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Second, in §A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='2, we present a few examples of Separated Det-Pomdps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='1 Properties of (∂)-separated mapping sets Lemma 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Let G be an � M, ←− X � mappings set as defined in Definition 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' If M is a separated mapping set, then G is a (∂)-separated mapping set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Let g1 and g2 be two mappings in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' In order to prove that G is a (∂)-separated mapping set, using Definition 10, we need to prove that the restrictions of the two mappings g1 and g2 on the subset A = g−1 1 (X)∩g−1 2 (X) are separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Using the property of the set G, there exist m1 ∈ M (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' m2 ∈ M) and X1 ⊂ X (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' X2 ⊂ X) such that g1 = m1←− X1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' g2 = m2←− X2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Combined with the definition of m1←− X1 in Equation (33), this gives that g−1 1 (X) = (m1)−1(X1) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' g−1 2 (X) = (m2)−1(X2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We therefore obtain the equality A = (m1)−1(X1) ∩ (m2)−1(X2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' First, if the set A is empty, it is immediate to prove that g1 and g2 are (∂)-separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Second, assuming that A is not empty and using again the fact that g1 = m1←− X1, we obtain that g1 coincides with m1 on the set A, and in the same way we obtain that g2 coincides with m2 on the set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 26 Now, as m1 and m2 belong to a separated mapping set, they are separated mappings, and therefore their restrictions to A are also separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We conclude that the restrictions of g1 and g2 on the subset A = g−1 1 (X) ∩ g2−1(X) are separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' This ends the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' A direct consequence of Lemma 25 is the following Corollary 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Corollary 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Let {Mk}k∈N be a sequence of sets of self-mappings on the set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Let {Gk}k∈N be a se- quence of sets of self-mappings on the set X, such that, for all k ∈ N, Gk is an � Mk, ←− X � mappings set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' If the set ∪k∈N � Mk ◦ Mk−1 ◦ · · · ◦ M0 � of mappings is a separated mapping set, then the set ∪k∈N � Gk ◦ Gk−1 ◦ · · · ◦ G0 � is a (∂)-separated mapping set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Let G1 and G2 be respectively an � M1, ←− X � mappings set and an � M2, ←− X � mappings set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Then, we have that G1 ◦ G2 = � g1 ◦ g2 �� g1 ∈ G1 and g2 ∈ G2 � (by Notation (19b)) ⊂ � m1←− X1 ◦ m2←− X2 �� m1 ∈ M1 , m2 ∈ M2 , X1 ⊂ X , X2 ⊂ X � (by (35)) ⊂ � (m1 ◦ m2)←−−−−−−−−−−−− X2∩(m2)−1(X1) �� m1 ∈ M1 , m2 ∈ M2 , X1 ⊂ X , X2 ⊂ X � (by (36)) ⊂ � mX �� m ∈ M1 ◦ M2 and X ⊂ X � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We have obtained that G1◦G2 is a � M1 ◦ M2, ←− X � mappings set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Thus, if M1◦M2 is a separated mapping set, then the set G1 ◦G2 is a (∂)-separated mapping set by using Lemma 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The end of the proof follows by induction on the number of compositions of sets, and by straightforward arguments when considering unions of �←− X � mappings sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Before presenting bounds on the cardinality of a (∂)-separated mapping set, we present Lemma 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Lemma 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Let J ⊂ L(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Y) be a set of mappings from the finite set X to the finite set Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Assume that for all pairs of mappings (j, j′) ∈ J2, if there exists x ∈ X such that j(x) = j′(x), then j = j′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Then, we have that |J| ≤ |Y| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (65) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Fix x ∈ X and consider the evaluation mapping γx : J → Y defined by γx(j) = j(x) for all j ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The image γx(J) of the set J by the mapping γx is indeed the subset {j(x) | j ∈ J} of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' First, the codomain of the mapping γx being the finite set Y, we immediately have that ��γx(J) �� ≤ |Y| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (66) Second, the mapping γx is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Indeed, using the assumption on the set J, two distinct mappings j and j′ in the set J must satisfy γx(j) = j(x) ̸= j′(x) = γx(j′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Thus, we must have the equality |J| = ��γx(J) �� which, combined with Equation (66), gives Inequality (65), and concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We now use the previous Lemma 27 to bound the cardinality of a (∂)-separated mapping set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Lemma 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Let X = X ∪ {∂}, and a (∂)-separated mapping set G of self-mappings on the set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Moreover, assume that, for all g ∈ G, g(∂) = ∂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' For any subsets X and X′ of the set X, we define GX→X′ as follows GX→X′ = � g ∈ G �� g−1(X) = X, g(X) ⊂ X′� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (67) Then, we have ��GX→X′�� � ≤ |X′| if X ⊂ X , = 0 if X ∩ {∂} ̸= ∅ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (68) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Fix X ⊂ X and X′ ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' First, we consider the case where X ∩{∂} ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' As we have assumed that g(∂) = ∂, for all g ∈ G, we obtain that g−1(X) ∩ {∂} = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Thus, we conclude that |GX→X′| = |∅| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Second, we consider the case where X ⊂ X and consider the mapping Γ : GX→X′ → X′X , g �→ g|X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (69) 27 The mapping Γ is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Indeed, if two mappings in GX→X′ have the same restriction on X, they coincide on X as they are both constant on the set X \\ X with value ∂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We therefore obtain that ��GX→X′�� = ��Γ(GX→X′) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (70) Now, the set G′ = Γ(GX→X′) is a subset of mappings from X to X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' As G is a (∂)-separated mapping set, we obtain that G′ is a separated set of mappings from X to X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Indeed, consider a pair of mappings (g′ 1, g′ 2) ∈ G′2 and assume that there exists x ∈ X such that g′ 1(x) = g′ 2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Using the definition of G′, we have that g′ 1(x) and g′ 2(x) are both non equal to ∂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Moreover, there exists g1 and g2 in GX→X′ such that g′ 1 = Γ(g1) and g′ 2 = Γ(g2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Using again the definition of G′ = Γ(GX→X′) we obtain that g1(x) = g2(x) ̸= ∂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Now, as G is a (∂)-separated mapping set, we obtain that the two mappings g1 and g2 coincide on X since they both do not take the value ∂ on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We conclude that their restrictions on X, the mappings g′ 1 and g′ 2, coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Using Lemma 27 in Subsection A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='2 we obtain that ��Γ(GX→X′) �� ≤ |X′| , (71) which, combined with Equation (70), gives Equation (68).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We now present the postponed proof of Proposition 12, presented in page 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Proof of Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The proof of Proposition 12 is a direct consequence of Corollary 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We assume that the set � t∈T f Ut+1 0:t = {f u0:t 0:t | ∀t ∈ T \\ {T}, ∀u0:t ∈ Ut+1} of the composition of the evolution functions of Problem (2) is a separated mapping set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We then prove that Problem (2) is a Separated Det-Pomdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' First, for all time t and for all pair (u, o) ∈ U × O, we have F u,o t = f u t −−−−−−−−→ (hu t+1)−1(o) (see Equation (59)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Thus, by Equation (34a), there exists X ⊂ X such that F u,o t = f u t ← − X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Hence, FD t is of the same form as in Equation (51), with the role of set Φk taken by {f U t }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We hence have that FD = � t∈T FD 0:t is a (∂)-separated mapping set by Corollary 26, where the role of {Gk}k∈N is taken by {FD t }t∈T \\{T } and the role of {Φk}k∈N is taken by {f U t }t∈T \\{T }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Therefore, as FD is a (∂)-separated mapping set, Problem (2) is a Separated Det-Pomdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We now present the postponed proof of Theorem 13, presented in page 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Proof of Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We start by giving preliminary bounds on ��� � R ◦ (FD X→X)⋆ � (b0) \\ {δ∂} ���, where FD X→X is defined by Equation (67), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' FD X→X = � F ∈ FD �� F −1(X) = X, F(X) ⊂ X � , where FD is defined in Equation (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We consider three cases depending on the cardinality of the subset X: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' When |X| = 0, we have that X = ∅ and � R ◦ (FD ∅→X)⋆ � (b0) \\ {δ∂} = ∅, and thus ��� � R ◦ (FD X→X)⋆ � (b0) \\ {δ∂} ��� = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (72a) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' When |X| = 1, we have that � R ◦ (FD X→X)⋆ � (b0) \\ {δ∂} ⊂ � δx �� x ∈ X � , as the only probability distributions of ∆(X) with support of cardinality at most 1 are the vertices � δx �� x ∈ X � of the simplex ∆(X), and thus ��� � R ◦ (FD X→X)⋆ � (b0) \\ {δ∂} ��� ≤ ��� δx �� x ∈ X ��� = |X| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (72b) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' For |X| ≥ 2, we have by Lemma 28 in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='1, applied with G = F (as F is a (∂)-separated mapping set) that ��� � R ◦ (FD X→X)⋆ � (b0) \\ {δ∂} ��� ≤ ��(FD X→X)⋆ �� ≤ |X| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (72c) 28 We have by Equation (25) that ��BR,D �1,T �(b0) �� = |TD(b0)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We now detail the cardinality of TD(b0): ��TD(b0) \\ {δ∂} �� = ��� R ◦ (FD)⋆ � (b0) \\ {δ∂} �� = ��� � R ◦ � � X⊂X FD X→X � ⋆ � (b0) \\ {δ∂} ��� (as � X⊂X FD X→X = FD) = ��� � X⊂X � R ◦ (FD X→X)⋆ � (b0) \\ {δ∂} ��� as ∀(F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' F ′) ∈ � FD�2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' R ◦ � F ∪ F ′� = R ◦ F ∪ R ◦ F ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' = ��� � X⊂supp(b0) � R ◦ (FD X→X)⋆ � (b0) \\ {δ∂} ��� as � R ◦ (FD X∩supp(b0)→X)⋆ � (b0) = � R ◦ (FD X→X)⋆ � (b0) by Equation (38) in Lemma 18,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' ≤ � X⊂supp(b0) ��� � R ◦ (FD X→X)⋆ � (b0) \\ {δ∂} ��� = � k≥0 � X⊂supp(b0) |X|=k ��� � R ◦ (FD X→X)⋆ � (b0) \\ {δ∂} ��� ≤ |X| + � X⊂supp(b0) |X|≥2 |X| (by Equations (72)) = |X| + � 2|supp(b0)| − |supp(b0)| − 1 � |X| ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' (73) where the last equality comes from the fact that ��{X ⊂ supp(b0) | |X| ≥ 2} �� is given by ��{X ⊂ supp(b0) | |X| ≥ 2} �� = ��� X ⊂ X �� X ⊂ supp(b0) ��� � �� � 2|supp(b0)| − ��� X ⊂ supp(b0) �� |X| = 1 ��� � �� � =|supp(b0)| − ��� X ⊂ supp(b0) �� |X| = 0 ��� � �� � =1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We hence obtain that ��BR,D �1,T �(b0) �� (24) = |TD(b0)| (73) ≤ 1 + � 2|supp(b0)| − |supp(b0)| � |X| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' This ends the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We now present examples of Separated Det-Pomdps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='2 Examples of Separated Det-Pomdps In this subsection, we present examples of Separated Det-Pomdps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Indeed, a direct consequence of Proposition 12 is that, if the evolution mappings of a Det-Pomdp belong to a separated mapping set, then the Det-Pomdp is a Separated Det-Pomdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We now present examples of such evolution mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Corollary 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Consider a Det-Pomdp optimization problem given by Problem (2) which satisfies the finite sets Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The notations are those of Problem (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Assuming that, for all time t ∈ T \\{T}, there exist mappings gt such that, for all states x ∈ X ⊂ Rn, ft(x, u) = x + gt(u) , (74) then Problem (2) is a Separated Det-Pomdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 29 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' This corollary is a direct result of Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Indeed, we only need to prove that ∪t∈T � f Ut+1 0:t � is a separated mapping set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Let t1 ≤ t′ 1 and t2 ≤ t′ 2 be such that �t1, t′ 1� ⊂ T and �t2, t′ 2� ⊂ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Let ut1:t′ 1 ∈ Ut′ 1−t1+1 and u′ t2:t′ 2 ∈ Ut′ 2−t2+1 be two sequences of controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We have, by using Equation (74), that f ut1:t′ 1 t1:t′ 1 : X → X, x �→ x + � t∈�t1,t′ 1� gt(ut) , and f u′ t2:t′ 2 t2:t′ 2 : X → X, x �→ x + � t∈�t2,t′ 2� gt(u′ t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' If there exists a state x ∈ X such that f ut1:t′ 1 t1:t′ 1 (x) = f u′ t2:t′ 2 t2:t′ 2 (x), we hence have � t∈�t1,t′ 1� gt(ut) = � t∈�t2,t′ 2� gt(u′ t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Thus f ut1:t′ 1 t1:t′ 1 (x) = f u′ t2:t′ 2 t2:t′ 2 (x) ⇒ f ut1:t′ 1 t1:t′ 1 = f u′ t2:t′ 2 t2:t′ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Therefore, the set ∪t∈T � f Ut+1 0:t � = {f u0:t 0:t | ∀t ∈ T \\ {T}, ∀u0:t ∈ Ut+1} of composition of the evolution mappings is a separated mapping set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We conclude by Proposition 12 that Problem (2) is a Separated Det-Pomdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Corollary 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Consider a Det-Pomdp optimization problem given by Problem (2) which satisfies the finite sets Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' The notations are those of Problem (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Assuming that, for all time t ∈ T \\{T}, there exist mappings gt such that for all states x ∈ X ⊂ Rn, ft(x, u) = x × gt(u) , (75) and assuming that 0 /∈ X, then Problem (2) is a Separated Det-Pomdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Let t1 ≤ t′ 1 and t2 ≤ t′ 2 such that �t1, t′ 1� ⊂ T and �t2, t′ 2� ⊂ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Let ut1:t′ 1 ∈ Ut′ 1−t1+1 and u′ t2:t′ 2 ∈ Ut′ 2−t2+1 be two sequences of controls .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' We have, by using Equation (75), f ut1:t′ 1 t1:t′ 1 : X → X, x �→ x × � t∈�t1,t′ 1� gt(ut), and f u′ t2:t′ 2 t2:t′ 2 : X → X, x �→ x × � t∈�t2,t′ 2� gt(u′ t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' If there exists a state x ∈ X such that f ut1:t′ 1 t1:t′ 1 (x) = f u′ t2:t′ 2 t2:t′ 2 (x), we hence have, as x ̸= 0, � t∈�t1,t′ 1� gt(ut) = � t∈�t2,t′ 2� gt(u′ t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Thus f ut1:t′ 1 t1:t′ 1 (x) = f u′ t2:t′ 2 t2:t′ 2 (x) ⇒ f ut1:t′ 1 t1:t′ 1 = f u′ t2:t′ 2 t2:t′ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Therefore, the set of compositions of the evolution functions ∪t∈T � f Ut+1 0:t � = {f u0:t 0:t | ∀t ∈ T \\ {T}, ∀u0:t ∈ Ut+1} is a separated mapping set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' References K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' ˚Astr¨om.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Optimal control of Markov processes with incomplete state information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Journal of Math- ematical Analysis and Applications, 10(1):174–205, Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 1965.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content='1016/0022-247X(65)90154-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Bellman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Dynamic programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Princeton Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Pr, Princeton, NJ, 1957.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Bertsekas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Dynamic Programming and Optimal Control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Athena Scientific, Belmont, Massachusetts, second edition, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Volumes 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Bertsekas and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Shreve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Stochastic optimal control: the discrete time case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Number v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' 139 in Mathematics in science and engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Academic Press, New York, 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Bonet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' Deterministic pomdps revisited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' In Proceedings of the Twenty-Fifth Conference on Uncertainty in Artificial Intelligence, UAI ’09, page 59–66, Arlington, Virginia, USA, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' AUAI Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFAT4oBgHgl3EQfdB1R/content/2301.08567v1.pdf'} +page_content=' H.' 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[physics.flu-dyn] 12 Jan 2023 +Long-distance migration with minimal energy consumption in a +thermal turbulent environment +Ao Xu,1, 2, ∗ Hua-Lin Wu,1 and Heng-Dong Xi1, 2 +1School of Aeronautics, Northwestern Polytechnical University, Xi’an 710072, China +2Institute of Extreme Mechanics, Northwestern +Polytechnical University, Xi’an 710072, China +(Dated: January 13, 2023) +1 + +Abstract +We adopt the reinforcement learning algorithm to train the self-propelling agent migrating long- +distance in a thermal turbulent environment. We choose the Rayleigh–B´enard turbulent convection +cell with an aspect ratio (Γ, which is defined as the ratio between cell length and cell height) of 2 as +the training environment. Our results showed that, compared to a naive agent that moves straight +from the origin to the destination, the smart agent can learn to utilize the carrier flow currents to +save propelling energy. We then apply the optimal policy obtained from the Γ = 2 cell and test +the smart agent migrating in convection cells with Γ up to 32. In a larger Γ cell, the dominant flow +modes of horizontally stacked rolls are less stable, and the energy contained in higher-order flow +modes increases. We found that the optimized policy can be successfully extended to convection +cells with a larger Γ. In addition, the ratio of propelling energy consumed by the smart agent to +that of the naive agent decreases with the increase of Γ, indicating more propelling energy can +be saved by the smart agent in a larger Γ cell. We also evaluate the optimized policy when the +agents are being released from the randomly chosen origin, which aims to test the robustness of the +learning framework, and possible solutions to improve the success rate are suggested. This work has +implications for long-distance migration problems, such as unmanned aerial vehicles patrolling in +a turbulent convective environment, where planning energy-efficient trajectories can be beneficial +to increase their endurance. +I. +INTRODUCTION +Humans have long been fascinated with flight, and we can learn how to fly efficiently +from birds. Some soaring birds can fly long distances during their trips without flapping +their wings, and they spend the greatest effort only during the take-off or landing stage. +For example, Weimerskirch et al. [1] showed that frigate birds can stay aloft for up to 48 +days during transoceanic flight. Williams et al. [2] recorded that an Andean condor flew for +over 5 hours without flapping, which covers 172 kilometers. Croxall et al. [3] revealed that +the fastest gray-headed albatrosses can make global circumnavigations in just 46 days. It +was not until 1885, when Lancaster published his pioneer observations and deductions [4], +that the mystery of flying birds not flapping their wings is gradually solved. The secret of +birds is that they can utilize warm rising atmospheric currents (also known as thermals) to +∗ Author to whom correspondence should be addressed: axu@nwpu.edu.cn +2 + +reduce the expenditure of energy. Thermals are part of the convection flows that develop +in the convective layer of the atmosphere (i.e., the troposphere). During sunny days, heat +from the sun warms the earth and the earth warms the air above it. Warm air expands and +lighter air rises, and the resulting column of rising air is called thermals. +Not only birds but also gliders and unmanned aerial vehicles (UAVs) can utilize the +updrafts of thermals to increase endurance and save energy. For example, MacCready [5] +determined the optimal gliding speed to fly between thermals to maximize speed and energy +gain. Allen et al. [6] estimated a UAV with a nominal endurance of 2 hours can achieve a +12-hours increase in the summer and a 6-hours increase in the winter. To investigate the use +of the convective lift, various thermal models have been developed, such as chimney models +and bubble models [7]. Chimney thermals are continuous columns of rising air, which extend +from the ground surface to the highest level of the troposphere [8]. Bubble thermals are +closed updraft masses that form a rising vortex ring near the ground, and the updraft at +the core of the vortex ring is provided by the buoyancy of the air [9]. When the air leaves +the bubble core, it cools down and loses buoyancy, thus moving downward on the outside of +the vortex ring to complete a cycle. Both the chimney and bubble thermal models describe +simplified situations, where there is no turbulent motion or the fluctuations are modeled as +Gaussian white noise. However, in the troposphere, the wind field exhibits strong turbulent +fluctuations. Akos et al. [10] found that turbulent fluctuations of the environment bring +challenges in identifying effective thermal soaring strategies. Laurent et al. [11] further +pointed out that turbulence leaves an imprint on all modes of flight, and they revealed the +analogy between the flight trajectories of a golden eagle and the trajectories of particles +carried by turbulent flows. They also reinforced the need to fully incorporate turbulence +into understanding the movement and behaviors of the flying object. +To model the flow patterns of the wind in strong convective weather, a paradigmatic +turbulent convection system, known as Rayleigh-B´enard (RB) convection, can describe tur- +bulent flows driven by buoyancy forces [12–14]. The control parameters of the RB system +mainly include the Prandtl number (Pr, defined later in the paper), the Rayleigh number +(Ra), and the cell aspect ratio (Γ). The Pr describes the thermophysical properties of the +fluid and Pr = 0.71 for the air. The Ra describes the ratio of buoyancy forces relative to +the viscous forces due to temperature differences, and 1018 ≤ Ra ≤ 1022 in the atmosphere +[13]. The Γ characterizes the geometric information of the convection system, and Γ ≈ 100 +for mesoscale convective system [15]. In the RB turbulent convection, important coherent +3 + +structures include small-scale thermal plumes, large-scale circulation rolls, and the very- +large-scale superstructure. The thermal plumes are detached from the hot or cold boundary +layers, it then collides and merges, further self-organize into large-scale circulation rolls. If +the convection system extends several times the distance in the horizontal direction than +that in the vertical direction, thermal plumes form a web of connected ridgelike structures +of cold downwelling and hot upwelling fluids, also known as the superstructure of thermal +turbulence [16, 17]. +Adopting the RB turbulent environment, Reddy et al. [18] numerically trained a glider +to rise on thermals using reinforcement learning algorithms. The trained glider can ascend +from low altitude to high altitude in a spiral form, which has a similar pattern to soaring +birds in nature [19]. They analyzed the changes in the glider’s flight strategy when the +turbulent intensity varied. Thereafter, they equipped a glider with a two-meter wingspan +and trained the glider in the field to navigate atmospheric thermals autonomously [20]. In +Reddy et al.’s works [18, 20], the main goal is to train the glider to ascend higher; whilst +for practical application of UAVs, a more frequently encountered scenario is to fly from one +position to another. To minimize energy consumption during the point-to-point migration in +a thermal turbulent environment, Xu et al. [21] optimized the trajectory for a self-propelling +agent in the RB turbulent convection, such that the agent can utilize the kinetic energy of +the thermal turbulence as much as possible. Compared with the straight-line propelling +trajectory, the optimized trajectory allows the agent to save around two-thirds of its energy +consumption. +In the previous work on soaring within the RB turbulent environment [18, 21], the simu- +lated RB convection cells have an aspect ratio of Γ ≤ 2. However, migration often occurs in +a large-aspect-ratio convection system and covers a long distance. In this work, our motiva- +tion is to train the self-propelling agent to migrate in a large-aspect-ratio RB cell that has +multiple circulation rolls. The rest of this paper is organized as follows. In Section II, we +introduce numerical details for the simulation of the turbulent environment, including the +mathematical model and the in-house numerical solver for the RB convection. In Section +III, we present details of the dynamics of the self-propelling agent and the reinforcement +learning algorithm to train the agent to find an energy-efficient trajectory. In Section IV, +we first present general flow patterns in the RB convection, followed by training results for +the agent migrating in a Γ = 2 cell, and then test the agent migrating in larger Γ cells. In +Section V, the main findings of this work are summarized. +4 + +II. +SIMULATION OF THE TURBULENT ENVIRONMENT +II.1. +Mathematical model for the RB turbulent thermal convection +We simulate the turbulent environment in the RB convection cells based on the Boussinesq +approximation. We assume the fluid flow is incompressible, and we treat the temperature +as an active scalar that influences the velocity field through the buoyancy. +The viscous +heat dissipation and compression work are neglected, and all the transport coefficients are +assumed to be constants. Then, the governing equations for the RB thermal convection can +be written as +∇ · u = 0 +(1) +∂u +∂t + u · ∇u = − 1 +ρ0 +∇P + ν∇2u + gβT(T − T0)ˆy +(2) +∂T +∂t + u · ∇T = αT∇2T +(3) +where u = (u, v), P and T are the velocity, pressure, and temperature of the fluid, respec- +tively. ρ0 and T0 are reference density and temperature, respectively. ˆy is the unit parallel +to gravity. With the scaling +x∗ = x/H, +t∗ = t/ +� +H/(βTg∆T), +u∗ = u/ +� +βT gH∆T, +P ∗ = P/(ρ0gβT∆TH), +T ∗ = (T − T0)/∆T +(4) +Then, Eqs. 1, 2, 3 can be rewritten in dimensionless from as +∇ · u∗ = 0 +(5) +∂u∗ +∂t∗ + u∗ · ∇u∗ = −∇P ∗ + +� +Pr +Ra∇2u∗ + T ∗˜y +(6) +∂T ∗ +∂t∗ + u∗ · ∇T ∗ = +� +1 +PrRa∇2T ∗ +(7) +Here, H is the cell height and it is chosen as the characteristics length. tf = +� +H/(βTg∆T) is +the free-fall time and it is chosen as the characteristic time. ∆T is the temperature difference +between heating and cooling walls. The two dimensionless parameters are the Ra and the +Pr, which are defined as +Ra = gβT∆T H3 +ναT +, +Pr = ν +αT +(8) +5 + +II.2. +The lattice Boltzmann method for thermal convection +We adopt the lattice Boltzmann (LB) method to simulate thermal convection. +The +advantages of the LB method include easy implementation and parallelization as well as +high computing efficiency [22]. Specifically, we chose a D2Q9 model for the Navier–Stokes +equations to simulate fluid flows and a D2Q5 model for the energy equation to simulate heat +transfer. To enhance the numerical stability, the multi-relaxation-time collision operator is +adopted in the evolution equations of both density and temperature distribution functions. +The evolution equation of the density distribution function is written as +fi(x + eiδt, t + δt) − fi(x, t) = − +� +M−1S +� +ij +� +mj(x, t) − m(eq) +j +(x, t) +� ++ δtF ′ +i +(9) +where fi is the density distribution function. x is the fluid parcel position, t is the time, +δt is the time step. ei is the discrete velocity along the ith direction. The forcing term F ′ +i +on the right-hand side of Eq. 9 is given by F′ = M−1 (I − S/2) M˜F, and the term M˜F +is given as [23] M˜F = [0, 6u · F, −6u · F, Fx, −Fx, Fy, −Fy, 2uFx − 2vFy, uFx + vFy]T +where F = ρgβT(T − T0)ˆy. The macroscopic density ρ and velocity u are obtained from +ρ = �8 +i=0 fi, u = +��8 +i=0 eifi + F/2 +� +/ρ. +The evolution equation of the temperature distribution function is written as +gi(x + eiδt, t + δt) − gi(x, t) = − +� +N−1Q +� +ij +� +nj(x, t) − n(eq) +j +(x, t) +� +(10) +where gi is the temperature distribution function. The macroscopic temperature T is ob- +tained from T = �4 +i=0 gi. More numerical details of the LB method and validation of the +in-house solver can be found in our previous work [24–26]. +II.3. +Simulation settings for the turbulent thermal convection +We consider a two-dimensional RB cell with length L and height H. The top and bot- +tom walls of the cell are kept at constant cold and hot temperatures, respectively; while +the other two vertical walls are adiabatic; all four walls impose no-slip velocity boundary +conditions. We set the cell aspect ratio (Γ = L/H) as 2 ≤ Γ ≤ 32, and we fix the Prandtl +number as Pr = 0.71 (corresponds to the working fluids of air) and the Rayleigh number +as Ra = 108. Although the Ra is far less than that in the atmosphere because of limited +computing resources to simulate ultra-high Ra convection, we note the RB convection at +6 + +Ra = 108 already exhibits strong turbulent fluctuations and the flows fall in the ’hard turbu- +lence’ regime [27]. We also checked the turbulent database and confirmed that statistically +stationary states have been reached and the initial transient effects of the simulations are +washed out. +III. +DYNAMICS AND CONTROL OF THE SELF-PROPELLING AGENT +III.1. +Kinematic model of the self-propelling agent +The dynamics of the self-propelling agent can be described as +uagent(t) = ufluid(t) + upropel(t) +(11) +xagent(t + dt) = xagent(t) + uagent(t) · dt +(12) +Here, dt is the time step, uagent and xagent denote the velocity and position of the agent, +respectively. ufluid denotes the velocity of the carrier fluids, and upropel denotes the velocity +generated by the agent. We assume that, without control, the velocity of the agent equals +that of the carrier flows; whilst, with control, the velocity of the agent is the superposition of +the carrier flow and agent’s propulsion. Similar dynamics of the agent have been previously +adopted by Krishna et al. [28]. To mimic the limited propelling ability of the agent in +real-world scenarios, we restricted the maximum propelling velocity of the agent ∥uagent∥ +to be less than one-third of the largest carrier flow velocity. On the other hand, a more +complex kinematic model for the self-propelling agent, such as the one that includes inertial +and rotational dynamics [29–33], fluttering and tumbling [34], multimodal locomotion [35] +of the propelling agent can be considered in the future work. +III.2. +Optimal control via the reinforcement learning +We adopt the RL algorithm to optimize the control of the agent to migrate in an energy- +efficient trajectory. The advantages of the RL algorithm include agnostic for control and +optimization tasks, easy to be re-used to speed-up optimization in a similar system configu- +ration, and robust to the disturbances in the chaotic system [36, 37]. In the RL algorithm, +the agent observes the state of the environment and decides to take an action interacting +with the environment. If the agent then receives a reward (or a penalty), it is more likely +7 + +to repeat (or forego) that action in the future. Overall, the agent learns by trial and er- +ror, with the long-term goal to maximize the cumulative expected return, and eventually +improve its performance. Applying the RL algorithm, we can obtain the optimal policy, +which advises the favorable action to take for the agent [38–41]. The model-free RL algo- +rithm can generally be classified into policy-based methods and value-based methods. In +the policy-based method, such as the policy gradient method, the parameter of the policy +network θ is optimized to maximize the performance objective J(πθ). +Here, πθ denotes +the parameterized stochastic policy. The policy-based methods are inefficient in sampling, +thus leading to slow learning, and are not suitable for complex flow problems [42]. In the +value-based methods, such as the Q-learning method, the agent takes action a that tried to +maximize the optimal action-value function, i.e., a(s) = arg maxa Qθ(s, a). Here, s denotes +the state of the environment and Qθ(s, a) approximates the optimal action-value function +Q∗(s, a). Using the Q-learning method, Colabrese et al. [29] showed that gravitactic swim- +mers can reach high altitudes in steady Taylor-Green vortex flow; Mui˜nos-Landin et al. +[43] demonstrated the artificial self-thermophoretic micro-swimmers can navigate under the +influence of Brownian motion; Monderkamp et al. [44] trained active Brownian particles +through complex motility landscapes; Gazzola et al. [45] and Verma et al. [46] found op- +timal swimming strategies that minimize drag and energy consumption in the school of +fish. The above five examples adopt the off-policy learning techniques, which means that +each update stochastic samples the data collected at any point during training, namely, +Q(st, at) = Q(st, at) + α [rt+1 + γ maxa Q(st+1, a) − Q(st, at)]. +Earlier, Reddy et al. +[18] +adopted the state-action-reward-state-action (SARSA) method, which can be regarded as a +variation of the Q-learning method. The main difference is that the SARSA method adopts +the on-policy learning technique, which uses the action performed by the current policy to +learn the Q-value, namely, Q(st, at) = Q(st, at) + α [rt+1 + γQ(st+1, at+1) − Q(st, at)]. How- +ever, the value-based method can only work in discrete state and action spaces; whilst in +most real-world scenarios, such as training a vehicle to navigate, the continuous state and +action spaces are preferred to develop more versatile motion for complex navigation tasks. +In this work, we adopt the soft actor-critic (SAC) algorithm, which is an interpolation +between policy-based methods and value-based methods. In the SAC algorithm, the agent +decides the next action via the actor network, whilst that action is further evaluated by +the critic network to guide the training process. In addition, the actor aims to maximize +the expected reward (i.e., succeed at the task) while also maximizing entropy (i.e., acting +8 + +as randomly as possible). In entropy regularized reinforcement learning, the optimization +problem can be described as +π∗(θ) = arg max +π +Eτ∼π +� ∞ +� +t=0 +(R(st, at, st+1) + αH(π(·|st))) +� +(13) +In the above equation, π∗ is the optimal policy. The reward function r depends on the current +state of the environment st, the current action at, and the next state of the environment +st+1. α is the trade-off coefficient. The entropy H of τ is computed from its distribution +π as H(π(·|st)) = Eτ∼π[− log π(τ)]. More details on the SAC algorithm can be found by +Haarnoja et al. [47]. +Key ingredients in the RL framework include the environmental cues that the agent can +observe (i.e., the current state st of the environment), the actions the agent takes (i.e., at), +and the response of the agent to its behavior (i.e., the reward rt). In this work, to migrate +in a large-aspect-ratio convection cell, the observation variables for the agent include the +carrier flow velocity ufluid, the agent’s spatial coordinate in the vertical direction yagent, and +the fluid temperature T. +In Section IV.2, we provide a detailed discussion on selecting +observation variables. The action variable is the propelling velocity upropel generated by the +agent. Following the previous work of Xu et al. [21], we assume the rewards received by +the agent are simultaneously affected by the current state, energy consumption, and time +consumption of the agent +r(t) = rs(t) + re(t) + rh(t) +(14) +Here, the rs denotes the reward affected by the current state of the agent. If the agent +migrates out of the flow domain through the top or the bottom boundaries, it will receive +a penalty of -φ; if the agent moves rightward and gets closer to the right-side boundary, it +will receive a basic reward ebasic with an empirical pre-factor ε. Thus, rs is written as +rs(t) = + + + + + + + + + +− φ, +agent is out of the flow domain +εebasic, +xt +agent > xt−1 +agent +0, +otherwise +(15) +Here, we adopt φ = 10 and ε = 10. A detailed discussion on the sensitivity of the hyper- +parameters in the reward function can be found in Appendix A. In Eq. 14, the re denotes +the reward affected by the energy consumption of the agent. If the propelling velocity of +the agent upropel is in alignment with that of the background flow ufluid, namely, the angle +between these two vectors (denoted by θ) is less than 90◦, the agent will receive a reward +9 + +of ε[ebasic + (emax − e)], where e = 0.5||upropel||2 and ebasic = emax = 0.5(||upropel||)2 +max; if +the angle between upropel and ufluid is greater than 90◦, the agent will receive a penalty of +−2ε(ebasic + e). The above designs also imply that, when the agent migrates in alignment +with the carrier flow direction, it will receive a lower reward if the propelling velocity is +higher; when the agent migrates against the carrier flow direction, it will receive a higher +penalty if the propelling velocity is higher. Thus, re is written as +re(t) = + + + +ε[ebasic + (emax − e)], +0◦ ≤ θ ≤ 90◦ +− 2ε(ebasic + e), +90◦ ≤ θ ≤ 180◦ +(16) +In Eq. 14, the rh denotes the reward affected by the time consumption of the agent. If +the agent migrates out of the domain via the right-side boundary, we assume the agent +completes the task and it will receive a reward that is inversely proportional to the time +taken. This design implies that the sooner the agent reaches the destination, the higher the +reward it receives. Thus, rh is written as +rh(t) = + + + +(tmax − t)/ε, +xt +agent > L +0, +otherwise +(17) +IV. +RESULTS AND DISCUSSION +IV.1. +General flow patterns in the RB convection with a large aspect ratio +Typical snapshots of the temperature field at Ra = 108, Pr = 0.71, and 2 ≤ Γ ≤ 32 are +shown in Fig. 1. We can distinguish small-scale thermal plumes and large-scale circulation +rolls. Thin thermal boundary layers appear near the bottom heating wall and top cooling +wall. Plumes that are released from the boundary layers penetrate upwards (or downwards) +towards the opposite wall of lower (or higher temperature), and intense mixing occurs in +the central region. Thermals are almost periodically released from relatively fixed locations, +and neighboring thermals that move in opposite directions entrain the surrounding fluid and +self-organize into circulation rolls. Previously, Wang et al. [48] found that depending on the +initial conditions, the flow system at a given aspect ratio evolves to different final turbulent +states with different roll numbers. In our work, the convection roll number n is 2, 4, 9, 18 +and 34, in the Γ = 2, 4, 8, 16 and 32 convection cells, respectively; their corresponding mean +aspect ratios are Γr = Γ/n = 1, 1, 0.889, 0.889 and 0.941, which is consistent with that +predicted by the elliptical instability theory [48]. +10 + +(a) +(b) +(c) +(d) +(e) +0.2 0.3 0.4 0.5 0.6 0.7 0.8 +Temperature +FIG. 1. Typical instantaneous temperature field at Ra = 108, Pr = 0.71, (a) Γ = 2, (b) Γ = 4, (c) +Γ = 8, (d) Γ = 16, (e) Γ = 32. +We then examine the global response parameter of Reynolds number (Re) on the control +parameter Γ. Here, the global flow strength is calculated as Re = +� +⟨(u2 + v2)⟩V,tH/ν and +⟨· · · ⟩V,t denotes the spatial and temporal average. The measured Re as a function of Γ is +shown in Fig. 2(a), and we can observe enhanced global flow strength with the increase of +cell aspect ratio. In addition, with the increase of Γ, the Re gradually reaches an asymptotic +value, similar to that in the 3D convection cell [16]. Previously, Xu et al. [21] found the +optimized energy-efficient strategy obtained from the reinforcement learning algorithm is +not sensitive to small perturbation of the global flow strength, but it changes when the +global flow strength increases (or decreases) more than one magnitude of order. Because +the self-propelling agent was trained in the Γ = 2 cell, we further checked that even in the +Γ = 32 cell, the flow strength increases around 22% compared to that in the Γ = 2 cell, +indicating the flow strength in a larger aspect ratio cell does not increase significantly. To +11 + +.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.� +Re +10 +0 +10 +1 +10 +2 +3500 +4000 +4500 +5000 +||u|| / ||u||rms +PDF +0 +2 +4 +6 +10 +-8 +10 +-7 +10 +-6 +10 +-5 +10 +-4 +10 +-3 +10 +-2 +10 +-1 +10 +0 +10 +1 +� = 2 +� = 4 +� = 8 +� = 16 +� = 32 +(a) +(b) +fluid +fluid rms + / +u +u +100 +10-2 +10-4 +10-6 +10-8 +FIG. 2. (a) Reynolds number, and (b) probability density functions (PDFs) of normalized velocity +magnitude ∥ufluid∥/∥ufluid∥rms, for various Γ at Ra = 108 and Pr = 0.71. +quantify the fluctuations of velocity magnitude, we plot the probability density functions +(PDFs) of normalized velocity magnitude ∥ufluid∥/∥ufluid∥rms for various Γ, as shown in Fig. +2(b). +We can see the PDF heads collapse for different Γ, whilst the PDF tails become +slightly extended with the increase of Γ, which implies an increased degree of fluctuations +for the velocity magnitude ∥ufluid∥. +To extract the coherent flow structure from the turbulent database, we adopt the proper +orthogonal decomposition (POD) analysis. +In the POD, the spatiotemporal vector field +X(r, t) is decomposed as a superposition of orthogonal eigenfunctions φi(r) and their am- +plitudes ai(t) as +X(r, t) = +∞ +� +i=1 +ai(t)φi(r) +(18) +Here, the vector field X(r, t) is chosen as the flow velocity field X = (u, v). Practically, we +can use the singular value decomposition (SVD) on the dataset X to obtain the flow mode +φi(r) and the corresponding mode amplitude ai(t) [49]. Because the database for turbulent +convection in a large-aspect-ratio cell is huge, which consists of fine spatial resolution and +long temporal evolution data, the memory consumption to perform standard SVD is intense. +Here, we adopt the randomized SVD to reduce the computational load [50]. The shape of the +first POD mode φ1(r) at various Γ is shown in Fig. 3. The most energetic POD mode consists +of horizontally stacked circulation primary rolls rotating in either the clockwise direction or +the anti-clockwise direction, and these primary rolls exhibit a periodical pattern. Large +values of velocity magnitude appear near the vortex edge, indicating a strong energy barrier +for the agent to move across the vortex edge. It is noteworthy that at much higher Ra +(i.e., Ra > 1010), when the large-scale-circulation is weaker and the flow consists of multiple +12 + +(a) +(b) +(c) +(d) +(e) +0.2 0.3 0.4 0.5 0.6 0.7 0.8 +U / Umax +FIG. 3. Contour of the first proper orthogonal decomposition (POD) mode φ1(r) at Ra = 108, +Pr = 0.71, (a) Γ = 2, (b) Γ = 4, (c) Γ = 8, (d) Γ = 16, (e) Γ = 32. Here, U = +√ +u2 + v2 is the +velocity magnitude, and Umax denotes the maximum value of U for normalization. +mobile and orbital small vortices [51–53], whether the current learning framework still works +deserves further study. +We further calculate the stability of the first POD mode as S1 = +� +⟨a1(t)⟩t/σa1, such that +a larger value of S1 indicates a more stable pattern of circulation rolls. Similar estimations +of the roll stability have been previously used in the Fourier mode decomposition of the +turbulent thermal convection [49, 54]. +From Fig. +4(a), we can see the stability of the +first POD mode decreases with the increase of Γ. We also analyze the energy contained +in the first POD mode and calculate the energy percentage as λ1/ �∞ +i=1 λi. Here, we have +λiδij = ⟨ai(t)aj(t)⟩t and λi denotes the energy of the ith POD mode, δij is the Kronecker +symbol, and ⟨· · ·⟩t denotes the temporal average. From Fig. 4(b), we can see the energy +percentage in the first POD mode also decreases with the increase of Γ. Although the first +13 + +.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.� +Stability +10 +0 +10 +1 +10 +2 +10 +0 +10 +1 +10 +2 +� +Energy pct. (%) +10 +0 +10 +1 +10 +2 +80 +90 +100 +(a) +(b) +FIG. 4. (a) The stability of the first POD mode, (b) the energy contained in the first POD mode +at various Γ. +mode is still the dominant flow structure (e.g., in the Γ = 32 cell, the energy contained in the +first POD mode accounts for more than 83.8% of the total energy), higher-order flow modes +become stronger with the increase of Γ. Thus, in a large-aspect-ratio cell, the horizontally +stacked circulation rolls that form a periodical pattern are less stable, and those higher-order +modes lead to a more irregular flow pattern. Because the agent was trained in the Γ = 2 +cell, the above-mentioned complex flow features bring challenges for the agent to identify an +energy-efficient trajectory in a larger Γ cell. +IV.2. +Training the agent to migrate in the RB cell with an aspect ratio of 2 +We first train the agent to migrate across the turbulent RB cell with Γ = 2. The agent +starts from the point of (0, 0) at the bottom-left corner of the cell, and its goal is to reach the +right-side boundary of the cell (i.e., the vertical line of x = 2). Here, the simple Γ = 2 cell +consists of the characteristic large-scale coherent structure (i.e., a clockwise rotating primary +roll and an anti-clockwise rotating primary roll), thus it serves as a paradigm environment +for the learning agent. In Fig. 5, we show the instantaneous trajectories of the smart agent +after training, and the corresponding video can be viewed in the supplementary movie. +Initially, the agent moves upward driven by the clockwise rotating corner roll at the bottom- +left corner [see Fig. 5(a)]. When the agent reaches the edge of the primary roll, which is +rotating in the anti-clockwise direction, it moves along with the horizontal currents [see Fig. +5(b)], until it meets the rising thermals. The agent then rises on the thermals and ascends +higher [see Fig. 5(c)]. After reaching the top layer of the cell, due to the right-directed +propelling velocity, the agent moves rightward and utilizes the horizontal currents [see Fig. +14 + +(a) +(b) +(c) +(d) +0.2 0.3 0.4 0.5 0.6 0.7 0.8 +Temperature +FIG. 5. Trajectory (black dotted line) of the smart agent in the RB convection at (a) t = 18, (b) +t = 36, (c) t = 54, and (d) t = 72. The contour shows the typical instantaneous temperature field, +and the vectors denote the velocity field of the convection. +5(d)]. The migration task is completed when the agent reaches the right-side boundary of +the cell. Overall, the smart agent tries to follow the carrier currents as much as possible, +and it discovers an effective policy of moving along the edge of the rolls. +A comparison between the smart agent and the naive agent is performed to highlight +the differences in propelling behaviors and the savings in energy expenditure. Here, the +naive agent refers to the agent that moves straight from the origin to the destination. We +set the naive agent to spend the same amount of total time ttotal as that of the smart +agent to complete the migration task, and its destination point is also the same as the +point where the smart agent left the right-side boundary. Thus, the velocity of the naive +agent keeps a constant direction pointing from the origin to the destination, and it keeps +a constant magnitude of ∥uagent∥ = ∥xgoal − xstart∥/ttotal. In Figs. 6(a) and 6(b), we show +trajectories of the smart agent and the naive agent, respectively. From the instantaneous +velocity magnitude shown on the color-coded trajectories, we can see that the naive agent +generally migrates slower than the smart agent, because the naive agent travels a shorter +15 + +.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.distance during the same ttotal. Although the smart agent migrates faster, it does not indicate +that the smart agent will consume more energy, since the smart agent can utilize the carrier +flow currents to save energy. Along with the trajectories, we also plot the propelling velocity +vector (denoted by the red arrows) and the fluid velocity vector (denoted by the blue arrows). +We calculate the correlation coefficient between the orientation of the propelling velocity +vector (i.e., θpropel) and the orientation of the fluid velocity vector (i.e., θfluid) as +C = ⟨[θpropel(t) − ⟨θpropel⟩] [θfluid(t) − ⟨θfluid⟩]⟩/ +� +σθpropelσθfluid +� +(19) +Here, the orientation is in the range from -180◦ to 180◦. The positive orientation is defined +as anti-clockwise rotating the x-axis, and the negative orientation is defined as clockwise ro- +tating the x-axis. For the smart agent, the resulting correlation coefficient of 0.56 suggests +positive statistical relevance between them, revealing that the smart agent adjusts its migra- +tion direction in response to the changing carrier flow, enabling energy-efficient migration; +for the naive agent, the resulting correlation coefficient of -0.74 implies negative statistical +relevance. In addition, to quantitatively describe the angles between the propelling velocity +vector and the fluid velocity vector, in Figs. 6(c) and 6(d), we plot the histogram of those +angles for the smart agent and the naive agent, respectively. We can see for the smart agent, +the angles are generally less than 90◦, and the frequency exhibits a peak around 20◦, which +is another evidence that the agent tries to follow the carrier currents. For the naive agent, +the frequency of those angles exhibits a peak around 120◦, indicating that the naive agent +has to generate propelling velocity against the carrier flow currents to keep the shortest +migrating path. +We assume that only the propelling velocity upropel affects the propelling energy con- +sumption for the agents. Thus, the accumulative energy consumption is calculated as +Epropel(t) = +� t +0 +1 +2∥upropel(τ)∥2dτ +(20) +In Fig. 7(a), we plot the time series of accumulative energy consumed by the agents. After +completing the migration task, the smart agent consumed around 38% of the propelling +energy compared to that of the naive agent, meaning migrating in the shortest path does +not always save energy. We also calculate the instantaneous energy consumption as +epropel(t) = 1 +2∥upropel(t)∥2 +(21) +From Fig. 7(b), we can see the epropel for the smart agent is generally lower than that of +the naive agent, and the epropel keeps smaller values for the smart agent during the whole +16 + +Angle +Frequency +0 +60 +120 +180 +0 +0.1 +0.2 +Angle +Frequency +0 +60 +120 +180 +0 +0.1 +0.2 +(a) +(b) +(c) +(d) +Velocity +Angle (�) +Angle (�) +FIG. 6. (a, b) Trajectories for the smart agent enabling energy-efficient migration and a naive +agent moving straightly, respectively. The red arrows denote the propelling velocity and the blue +arrows denote the fluid velocity. The trajectories are color-coded by the instantaneous velocity +magnitude of the agent. (c, d) Histogram of angles between the propelling velocity vector and the +fluid velocity vector for the smart agent and the naive agent, respectively. +migration process. For the naive agent, epropel exhibits a first peak around t ≈ 3, when it +moves across the edges of the left-bottom corner roll that requires a high energy barrier. At +around t ≈ 23, epropel drops to the minimum, because the naive agent reaches the location +where flow currents are in alignment with the migration direction. The second peak of epropel +appears at around t ≈ 38, when the naive agent crosses the edge of the primary roll and +drifts to the clockwise rotating primary roll. The third peak of epropel appears at around +t ≈ 67, when the naive agent approaches the right-side boundary. We also compare the +accumulative total kinetic energy of the agents [see Fig. 7(c)], which is calculated as +Etotal(t) = +� t +0 +1 +2∥uagent∥2dτ +(22) +After completing the migration task, the total kinetic energy of the smart agent is almost +twice that of the naive agent, mostly contributed by the kinetic energy of the carrier flow. +17 + +t +Epropel +0 +20 +40 +60 +0.00 +0.01 +0.02 +0.03 +smart +naive +t +epropel +0 +20 +40 +60 +0.0000 +0.0004 +0.0008 +smart +naive +t +Etotal +0 +20 +40 +60 +0.00 +0.02 +0.04 +0.06 +smart +naive +t +etotal +0 +20 +40 +60 +0.0000 +0.0006 +0.0012 +0.0018 +smart +naive +(a) +(b) +(c) +(d) +FIG. 7. Comparison of the (a) accumulative energy consumption Epropel, (b) instantaneous energy +consumption epropel, (c) accumulative total kinetic energy Etotal, and (d) instantaneous total kinetic +energy etotal for the smart agent and the naive agent. +Similarly, we calculate the instantaneous total kinetic energy etotal as +etotal(t) = 1 +2∥uagent(t)∥2 +(23) +From Fig. 7(d), we can see the etotal of the smart agent is generally higher than that of +the naive agent and keeps larger values during the whole migration process. Three peaks +appear when the smart agents migrate in alignment with the flow direction, thus utilizing +more kinetic energy of the carrier flow. On the other hand, the etotal of the naive agent keeps +constant due to the constant value of ∥uagent(t)∥ in the simulation settings. +Choosing appropriate environment cues that the agent can observe is crucial in the RL +training. Here, we numerically determine the set of observation variables by comparing the +evolution of cumulative reward during the training process. We train five different instances +of each observation variable set with different random seeds, and each set performs one +evaluation rollout every 1000 environment steps. The solid curves correspond to the mean +and the shaded region to the minimum and maximum returns over the five trials, and it +18 + +represents a moving average with a window of 20 timesteps. We first consider the agent is +flow-blinded and cannot sense the surrounding flow and temperature information. The agent +can have access to its position information, but only the vertical component, i.e., s = {y}. +Here, we do not consider the horizontal position information (i.e., x /∈ {s} ); otherwise, the +agent trained in the Γ = 2 cell would fail to find optimal trajectory in a larger Γ cell once +its horizontal position is x > 2. As shown in Fig. 8(a), the agent performs poorly in the +case of flow-blinded, which shows similar behavior as that of navigating through unsteady +cylinder flow [55]. We then consider the agent can also sense the carrier flow velocity, i.e., +s = {y, u, v}, and plot the results in Fig. 8(b). We can see that the agent performs much +better, and the cumulative reward is higher than that of the flow-blinded agent. In addition, +we consider the agent can sense extra vorticity information (i.e., s = {y, u, v, ω}) [29, 56], +and plot the results in Fig. 8(c). With the consideration of additional flow field information, +the cumulative reward converges at an earlier time (i.e., t ≈ 0.5 × 105 for s = {y, u, v, ω}) +compared to that of velocity information (i.e., t ≈ 1.0 × 105 for s = {y, u, v}). On the +other hand, in the turbulent RB convection, the temperature acts as an active scalar that +influences the velocity, we next consider the agent can sense extra temperature information +(i.e., s = {y, u, v, T}) [21]. As shown in Fig. 8(d), the agent outperforms previous ones and +the cumulative reward converges at an earlier time and almost remains steady, suggesting +temperature is an important environment cue for the agent to migrate in thermal convection. +In the Appendix B, we evaluate more combinations and then choose s = {y, u, v, T} as +observation variables. On the other hand, Kubo and Shimizu [57] proposed a framework +that can perform fluid flow control with partial observables. We expect the extension of +that framework to turbulent flows will simplify the selection of observables. +IV.3. +Testing the agent to migrate in the RB cell with a larger aspect ratio +We now apply the obtained policy to test whether the smart agent can migrate in an +energy-efficient way in convection cells with larger Γ. The flow mode analysis presented +in Section IV.1 indicates that in a larger Γ cell, the dominant flow modes of horizontally +stacked rolls are less stable, and the energy contained in higher-order flow modes increases. +Despite these challenges brought by the complex flow features, we can still obtain optimized +trajectories, as shown in Fig. 9. Starting from the origin of (0, 0) point, the smart agent +first escapes the corner roll at the left-bottom cell corner, it then ascends higher and rises on +19 + +(a) +(b) +(c) +(d) +�105 +�105 +Timesteps +Timesteps +�105 +Timesteps +Rewards +�105 +Timesteps +Rewards +Rewards +Rewards +FIG. 8. Evolution of the cumulative reward during training for different combinations of observation +variables: (a) s = {y}; (b) s = {y, u, v}; (c) s = {y, u, v, ω}; (d) s = {y, u, v, T}. +the thermal (if the first primary roll is clockwise rotating), or follows the horizontal currents +(if the first primary roll is anti-clockwise rotating). Afterward, the smart agent always tries +to migrate along the edges of the primary rolls, where the carrier fluid flows fast and plenty +of kinetic energy from the flow is available. +We then compare the propelling energy consumed by the smart agent and the naive agent +in the convection cell with various Γ. In Fig. 10(a), we plot the accumulative propelling +energy for the agents when they complete the migration task. Generally, for both agents, +the energy consumption increases with the increase of Γ, due to longer migration distance. +For the smart agent, it enables an energy-efficient migration strategy via migrating along +the edges of horizontally stacked multiple primary rolls, thus we have Epropel ∝ Γ. +To +quantitatively describe how much propelling energy can be saved, we plot the ratio of energy +consumed by the smart agent to that of the naive agent, as shown in Fig. 10(b). We can see +that in a larger Γ cell, the ratio of Esmart/Enaive is smaller, meaning more propelling energy +20 + +(a) +(b) +(c) +(d) +0.2 0.3 0.4 0.5 0.6 0.7 0.8 +Temperature +FIG. 9. Trajectory (black dotted line) of the smart agent in the convection cell with (a) Γ = 4, +(b) Γ = 8, (c) Γ = 16, and (d) Γ = 32. The contour shows the typical instantaneous temperature +field. +can be saved by the smart agent. The reason is that in a larger Γ cell, the naive agent has +to cross more edges of the circulation rolls and overcome higher energy barriers, whilst the +smart agent follows the carrier currents in an energy-efficient way. +The above results are obtained with the prescribed and fixed origin position, namely, the +location at the (0, 0) point. We further test the robustness of the energy-efficient policy +concerning random origin position. We first release the agent in the position where the +local flow velocity is weak, i.e., ∥ufluid∥ < 0.03, and 100 example trajectories are plotted +in Fig. 11. We can see regardless of the random origin position, the successful attempts +gradually converge to a similar path line, and the agent migrates along the edges of the +primary rolls. It should be noted that we restrict these ’random’ positions to be x < Γ/2, +which prevents the agent from being released too close to the outlet. The average success +21 + +.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.� +Epropel +10 +0 +10 +1 +10 +2 +10 +-2 +10 +-1 +10 +0 +smart +naive +� +Esmart / Enaive +10 +0 +10 +1 +10 +2 +0.2 +0.4 +(a) +(b) +FIG. 10. (a) The propelling energy consumed by the smart agent and the naive agent, (b) the +ratio of energy consumed by the smart agent to that of the naive agent as a function of Γ. +rate to complete the migration task in the Γ = 4, 8, 16, and 32 cell is 74%, 87%, 92%, and +97%, respectively. We then release the agent in the position where the local flow velocity is +strong, i.e., ∥ufluid∥ > 0.03. The average success rate is much higher, and it is 100%, 100%, +100%, and 99% in the Γ = 4, 8, 16, and 32 cell, respectively. For the sake of clarity, we do +not repeat plotting the example trajectories. +The above results indicate that the success rate to complete the migration task increases +with the increase of Γ. On the other hand, in a larger Γ cell, the flow structure is more +complex, and we expect it would be more challenging for the agent to complete the migration +task and earn a higher success rate. To understand such behaviors, we further consider the +following scenarios: (i) the agents being released where carrier flow velocity is ∥ufluid∥ < 0.01; +(ii) the agents being released where carrier flow velocity is 0.01 < ∥ufluid∥ < 0.02; (iii) the +agents being released where carrier flow velocity is 0.02 < ∥ufluid∥ < 0.03. We can see from +Fig. 12 that in convection cells with the same Γ, the average success rate is higher if the +carrier flow velocity at the origin is stronger. Because the global flow strength is stronger +in a larger Γ cell (see discussion in Section IV.1), the agents are more likely to be released +where carrier flow is strong. Utilizing stronger carrier flow currents, the agents can complete +the migration task within a shorter time and earn a higher reward, thus the agent is more +likely to repeat that action in the future. The higher sampling frequency for the agent in +areas with stronger flow strength leads to higher success rate in larger Γ cell. It should be +noted that such a trend is only obvious when the origin of agents possesses weak carrier flow +velocity; with strong carrier flow velocity, the success rates are always near 100%. +22 + +(a) +(b) +(c) +(d) +0.2 0.3 0.4 0.5 0.6 0.7 0.8 +Temperature +FIG. 11. Example trajectories in the (a) Γ = 4, (b) Γ = 8, (c) Γ = 16, and (d) Γ = 32 cell (the +agents are released where ∥ufluid∥ < 0.03). Green lines represent successful attempts to complete +the migration, while red lines represent unsuccessful attempts. +The contour shows the typical +instantaneous temperature field. +V. +CONCLUSIONS +In this work, using the reinforcement learning algorithm, we performed numerical train- +ing of the self-propelling agent migrating long-distance in a thermal turbulent environment. +We choose the paradigmatic turbulent RB convection cell as the flow environment, which +can incorporate strong fluctuations of velocity and temperature. To build up the reinforce- +ment learning framework, we designed a reward function that simultaneously considers the +current state, energy consumption, and time consumption of the agent. We also compare +the evolution of cumulative reward for different combinations of observation variables. We +select the position of the agent, as well as the velocity and temperature of the carrier flow +23 + +.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.� +Success rate (%) +10 +0 +10 +1 +10 +2 +60 +80 +100 +fluid +0.01 +� +u +fluid +0.01 +0.02 +� +� +u +fluid +0.02 +0.03 +� +� +u +fluid +0.03 +� +u +FIG. 12. Average success rate as functions of Γ when the agents are randomly released at the +positions with the different local carrier flow velocities. +as appropriate environmental cues. The simulation results in a Γ = 2 RB cell showed that, +compared to a naive agent that moves straight from the origin to the destination, the smart +agent can learn to utilize the carrier flow currents to save propelling energy. +We then apply the optimal policy obtained in the Γ = 2 cell and test the smart agent +migrating in convection cells with larger Γ. From flow mode analysis, we found the dominant +flow modes in a larger Γ RB cell consist of less stable horizontally stacked rolls, and the +energy contained in higher-order flow modes increases with the increase of Γ. Although these +complex flow features bring challenges to optimizing the trajectories for the smart agent, we +can still obtain energy-efficient migrating trajectories using the policy trained in the Γ = 2 +RB cell. In addition, we found the ratio of propelling energy consumed by the smart agent to +that of the naive agent decreases with the increase of Γ, meaning more propelling energy can +be saved by the smart agent in a larger Γ cell. The reason is that in a larger Γ cell, the naive +agent has to cross more edges of the circulation rolls and overcome higher energy barriers, +whilst the smart agent always tries to follow the carrier currents as much as possible. +We also evaluate the optimized policy when the agents are being released from the ran- +domly chosen origin, which aims to test the robustness of the learning framework. We found +the success rate increases with the increase of Γ, despite the flow structures being more com- +plex in a larger Γ cell. The main reason is that, in a larger Γ cell, the global flow strength +is stronger (evident by the relationship between Re and Γ), and the agent is more likely +24 + +to be released in positions where the carrier flow velocity is stronger. Utilizing stronger +carrier flow velocity, the agent can complete the migration task within a shorter time and +receive a higher reward, thus leading to a higher success rate. Our work has implications +for long-distance migration problems, for example, the UAVs patrolling in the convective +layer of the atmosphere. Migrating in energy-efficient trajectories, the UAVs can increase +endurance and cover a wider range. +ACKNOWLEDGMENTS +This work was supported by the National Natural Science Foundation of China (NSFC) +through Grant Nos. 12272311 and 12125204, the National Key Project via No. GJXM92579, +and the 111 project of China (No. B17037). The authors acknowledge the Beijing Beilong +Super Cloud Computing Co., Ltd for providing HPC resources that have contributed to the +research results reported within this paper (URL: http://www.blsc.cn/). +Appendix A: Sensitivity of the hyperparameters in the reward function +In our designed reward function (see Eqs. 14-17), we have two hyperparameters: one is φ, +which represents the penalty when the agent is out of the flow domain; the other is ε, which +is the reward scale coefficient. We tuned these two parameters separately to determine the +optimal hyperparameters. It should be noted that we compared the value of the normalized +reward (i.e., varied between 0 and 1) rather than the absolute value of the reward. We +can see from Fig. 13 (a) that, small values of the penalty φ (e.g., φ = 1 and 5) results in +substantial degradation of performance; large values of the penalty (e.g., φ ≥ 10) almost lead +to the same performance. As for the reward scale coefficient ε, it is almost insensitive for the +investigated value and they can all give optimal policy, as shown in Fig. 13 (b). However, +a large value (e.g., ε = 100, not shown here for clarity) would result in �(rs + re) > � rh +during training, and the agent’s failure to explore the successful trajectory within the given +time of tmax. +25 + +Timesteps +Normalized reward +0 +400000 +800000 +0.4 +0.6 +0.8 +1 +� = 1 +� = 5 +� = 10 +� = 20 +� = 100 +Timesteps +Normalized reward +0 +400000 +800000 +0.4 +0.6 +0.8 +1 +� = 1 +� = 5 +� = 10 +� = 20 +(a) +(b) +FIG. 13. Sensitivity of the hyperparameters in the reward function: (a) the penalty φ when the +agent is out of the flow domain, (b) the reward scale coefficient ε. +Appendix B: Evaluation of different combinations of observation variables +In addition to the observation variables described in Section IV.2, we also consider the +following different combinations: (i) the agent has access to position and velocity informa- +tion, and it can sense strain rate [58, 59], i.e., s = {y, u, v, sxx} and s = {y, u, v, sxy}. Here, +sxx = ∂xu and sxy = (∂yu + ∂xv)/2. We did not consider the syy = ∂yv component of +the strain rate tensor, because flow continuity equation gives sxx + syy = 0 in 2D flows, +which means sxx and syy are negatively correlated. (ii) the agent has access to position and +velocity information, and it can sense temperature gradient, i.e., s = {y, u, v, (∇T)x} and +s = {y, u, v, (∇T)y}. Here, (∇T)x essentially represents the vorticity produced by buoyancy +in the 2D convection flow [60]. (iii) the agent has access to position, velocity and tempera- +ture information, and it can sense additional vorticity, strain rate, or temperature gradient, +i.e., s = {y, u, v, T, sxx}, s = {y, u, v, T, sxy}, s = {y, u, v, T, ω}, s = {y, u, v, T, (∇T)x}, +and s = {y, u, v, T, (∇T)y}. In Fig. 14, we plot the evolution of the cumulative reward +during training for the above nine combinations of observation variables. We can see these +combinations only slightly changes the converging speed of the training, not the asymptotic +accumulative reward value. Among them, the s = {y, u, v, T, (∇T)x} shown in Fig. 14(h) +outperforms other combinations. In practical applications, velocity or temperature sensing +could be implemented via a variety of methods, such as pitot tubes, hot wire, and so on; +while vorticity, shear strain component, and temperature gradient should be computed from +several velocities or temperature sensors, which increases the complexity that the agent has +to sense. Thus, as described in Section IV.2, we deliberately keep simple the environmental +26 + +(a) +(b) +Rewards +(c) +(d) +(e) +(f ) +(g) +�105 +�105 +�105 +�105 +�105 +�105 +�105 +Rewards +Rewards +Timesteps +Timesteps +Timesteps +(h) +(i) +�105 +�105 +FIG. 14. +Evolution of the cumulative reward during training for different combinations of ob- +servation variables: (a) s = {y, u, v, sxx}; (b) s = {y, u, v, sxy}; (c) s = {y, u, v, (∇T)x}; (d) +s = {y, u, v, (∇T)y}; (e) s = {y, u, v, T, sxx}; (f ) s = {y, u, v, T, sxy}; (g) s = {y, u, v, T, ω}; (h) +s = {y, u, v, T, (∇T)x}; (i) s = {y, u, v, T, (∇T)y}. +cues of local information s = {y, u, v, T} that the agent can see to guide its migration, such +that the amount of data storage by the agent can be reduced in practical applications. +[1] H. Weimerskirch, C. Bishop, T. Jeanniard-du Dot, A. Prudor, and G. Sachs, Frigate birds +track atmospheric conditions over months-long transoceanic flights, Science 353, 74 (2016). +[2] H. J. Williams, E. 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Fluids 34, 013609 (2022). +31 + diff --git a/rdE3T4oBgHgl3EQf8wtj/content/tmp_files/load_file.txt b/rdE3T4oBgHgl3EQf8wtj/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cf0bbde929c1effadd1fcef1639d700de75d996f --- /dev/null +++ b/rdE3T4oBgHgl3EQf8wtj/content/tmp_files/load_file.txt @@ -0,0 +1,1120 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf,len=1119 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='04810v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='flu-dyn] 12 Jan 2023 Long-distance migration with minimal energy consumption in a thermal turbulent environment Ao Xu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' ∗ Hua-Lin Wu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='1 and Heng-Dong Xi1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 2 1School of Aeronautics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Northwestern Polytechnical University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Xi’an 710072,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' China 2Institute of Extreme Mechanics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Northwestern Polytechnical University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Xi’an 710072,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' China (Dated: January 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 2023) 1 Abstract We adopt the reinforcement learning algorithm to train the self-propelling agent migrating long- distance in a thermal turbulent environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We choose the Rayleigh–B´enard turbulent convection cell with an aspect ratio (Γ, which is defined as the ratio between cell length and cell height) of 2 as the training environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Our results showed that, compared to a naive agent that moves straight from the origin to the destination, the smart agent can learn to utilize the carrier flow currents to save propelling energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We then apply the optimal policy obtained from the Γ = 2 cell and test the smart agent migrating in convection cells with Γ up to 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' In a larger Γ cell, the dominant flow modes of horizontally stacked rolls are less stable, and the energy contained in higher-order flow modes increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We found that the optimized policy can be successfully extended to convection cells with a larger Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' In addition, the ratio of propelling energy consumed by the smart agent to that of the naive agent decreases with the increase of Γ, indicating more propelling energy can be saved by the smart agent in a larger Γ cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We also evaluate the optimized policy when the agents are being released from the randomly chosen origin, which aims to test the robustness of the learning framework, and possible solutions to improve the success rate are suggested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' This work has implications for long-distance migration problems, such as unmanned aerial vehicles patrolling in a turbulent convective environment, where planning energy-efficient trajectories can be beneficial to increase their endurance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' INTRODUCTION Humans have long been fascinated with flight, and we can learn how to fly efficiently from birds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Some soaring birds can fly long distances during their trips without flapping their wings, and they spend the greatest effort only during the take-off or landing stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' For example, Weimerskirch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' [1] showed that frigate birds can stay aloft for up to 48 days during transoceanic flight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' [2] recorded that an Andean condor flew for over 5 hours without flapping, which covers 172 kilometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Croxall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' [3] revealed that the fastest gray-headed albatrosses can make global circumnavigations in just 46 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' It was not until 1885, when Lancaster published his pioneer observations and deductions [4], that the mystery of flying birds not flapping their wings is gradually solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The secret of birds is that they can utilize warm rising atmospheric currents (also known as thermals) to ∗ Author to whom correspondence should be addressed: axu@nwpu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='cn 2 reduce the expenditure of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Thermals are part of the convection flows that develop in the convective layer of the atmosphere (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=', the troposphere).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' During sunny days, heat from the sun warms the earth and the earth warms the air above it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Warm air expands and lighter air rises, and the resulting column of rising air is called thermals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Not only birds but also gliders and unmanned aerial vehicles (UAVs) can utilize the updrafts of thermals to increase endurance and save energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' For example, MacCready [5] determined the optimal gliding speed to fly between thermals to maximize speed and energy gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Allen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' [6] estimated a UAV with a nominal endurance of 2 hours can achieve a 12-hours increase in the summer and a 6-hours increase in the winter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' To investigate the use of the convective lift, various thermal models have been developed, such as chimney models and bubble models [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Chimney thermals are continuous columns of rising air, which extend from the ground surface to the highest level of the troposphere [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Bubble thermals are closed updraft masses that form a rising vortex ring near the ground, and the updraft at the core of the vortex ring is provided by the buoyancy of the air [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' When the air leaves the bubble core, it cools down and loses buoyancy, thus moving downward on the outside of the vortex ring to complete a cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Both the chimney and bubble thermal models describe simplified situations, where there is no turbulent motion or the fluctuations are modeled as Gaussian white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' However, in the troposphere, the wind field exhibits strong turbulent fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Akos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' [10] found that turbulent fluctuations of the environment bring challenges in identifying effective thermal soaring strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Laurent et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' [11] further pointed out that turbulence leaves an imprint on all modes of flight, and they revealed the analogy between the flight trajectories of a golden eagle and the trajectories of particles carried by turbulent flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' They also reinforced the need to fully incorporate turbulence into understanding the movement and behaviors of the flying object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' To model the flow patterns of the wind in strong convective weather, a paradigmatic turbulent convection system, known as Rayleigh-B´enard (RB) convection, can describe tur- bulent flows driven by buoyancy forces [12–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The control parameters of the RB system mainly include the Prandtl number (Pr, defined later in the paper), the Rayleigh number (Ra), and the cell aspect ratio (Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The Pr describes the thermophysical properties of the fluid and Pr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='71 for the air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The Ra describes the ratio of buoyancy forces relative to the viscous forces due to temperature differences, and 1018 ≤ Ra ≤ 1022 in the atmosphere [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The Γ characterizes the geometric information of the convection system, and Γ ≈ 100 for mesoscale convective system [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' In the RB turbulent convection, important coherent 3 structures include small-scale thermal plumes, large-scale circulation rolls, and the very- large-scale superstructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The thermal plumes are detached from the hot or cold boundary layers, it then collides and merges, further self-organize into large-scale circulation rolls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' If the convection system extends several times the distance in the horizontal direction than that in the vertical direction, thermal plumes form a web of connected ridgelike structures of cold downwelling and hot upwelling fluids, also known as the superstructure of thermal turbulence [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Adopting the RB turbulent environment, Reddy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' [18] numerically trained a glider to rise on thermals using reinforcement learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The trained glider can ascend from low altitude to high altitude in a spiral form, which has a similar pattern to soaring birds in nature [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' They analyzed the changes in the glider’s flight strategy when the turbulent intensity varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Thereafter, they equipped a glider with a two-meter wingspan and trained the glider in the field to navigate atmospheric thermals autonomously [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' In Reddy et al.’s works [18, 20], the main goal is to train the glider to ascend higher;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' whilst for practical application of UAVs, a more frequently encountered scenario is to fly from one position to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' To minimize energy consumption during the point-to-point migration in a thermal turbulent environment, Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' [21] optimized the trajectory for a self-propelling agent in the RB turbulent convection, such that the agent can utilize the kinetic energy of the thermal turbulence as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Compared with the straight-line propelling trajectory, the optimized trajectory allows the agent to save around two-thirds of its energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' In the previous work on soaring within the RB turbulent environment [18, 21], the simu- lated RB convection cells have an aspect ratio of Γ ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' However, migration often occurs in a large-aspect-ratio convection system and covers a long distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' In this work, our motiva- tion is to train the self-propelling agent to migrate in a large-aspect-ratio RB cell that has multiple circulation rolls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' In Section II, we introduce numerical details for the simulation of the turbulent environment, including the mathematical model and the in-house numerical solver for the RB convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' In Section III, we present details of the dynamics of the self-propelling agent and the reinforcement learning algorithm to train the agent to find an energy-efficient trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' In Section IV, we first present general flow patterns in the RB convection, followed by training results for the agent migrating in a Γ = 2 cell, and then test the agent migrating in larger Γ cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' In Section V, the main findings of this work are summarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 4 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' SIMULATION OF THE TURBULENT ENVIRONMENT II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Mathematical model for the RB turbulent thermal convection We simulate the turbulent environment in the RB convection cells based on the Boussinesq approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We assume the fluid flow is incompressible, and we treat the temperature as an active scalar that influences the velocity field through the buoyancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The viscous heat dissipation and compression work are neglected, and all the transport coefficients are assumed to be constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Then, the governing equations for the RB thermal convection can be written as ∇ · u = 0 (1) ∂u ∂t + u · ∇u = − 1 ρ0 ∇P + ν∇2u + gβT(T − T0)ˆy (2) ∂T ∂t + u · ∇T = αT∇2T (3) where u = (u, v), P and T are the velocity, pressure, and temperature of the fluid, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' ρ0 and T0 are reference density and temperature, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' ˆy is the unit parallel to gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' With the scaling x∗ = x/H, t∗ = t/ � H/(βTg∆T), u∗ = u/ � βT gH∆T, P ∗ = P/(ρ0gβT∆TH), T ∗ = (T − T0)/∆T (4) Then, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 1, 2, 3 can be rewritten in dimensionless from as ∇ · u∗ = 0 (5) ∂u∗ ∂t∗ + u∗ · ∇u∗ = −∇P ∗ + � Pr Ra∇2u∗ + T ∗˜y (6) ∂T ∗ ∂t∗ + u∗ · ∇T ∗ = � 1 PrRa∇2T ∗ (7) Here, H is the cell height and it is chosen as the characteristics length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' tf = � H/(βTg∆T) is the free-fall time and it is chosen as the characteristic time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' ∆T is the temperature difference between heating and cooling walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The two dimensionless parameters are the Ra and the Pr, which are defined as Ra = gβT∆T H3 ναT , Pr = ν αT (8) 5 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The lattice Boltzmann method for thermal convection We adopt the lattice Boltzmann (LB) method to simulate thermal convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The advantages of the LB method include easy implementation and parallelization as well as high computing efficiency [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Specifically, we chose a D2Q9 model for the Navier–Stokes equations to simulate fluid flows and a D2Q5 model for the energy equation to simulate heat transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' To enhance the numerical stability, the multi-relaxation-time collision operator is adopted in the evolution equations of both density and temperature distribution functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The evolution equation of the density distribution function is written as fi(x + eiδt, t + δt) − fi(x, t) = − � M−1S � ij � mj(x, t) − m(eq) j (x, t) � + δtF ′ i (9) where fi is the density distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' x is the fluid parcel position, t is the time, δt is the time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' ei is the discrete velocity along the ith direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The forcing term F ′ i on the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 9 is given by F′ = M−1 (I − S/2) M˜F, and the term M˜F is given as [23] M˜F = [0, 6u · F, −6u · F, Fx, −Fx, Fy, −Fy, 2uFx − 2vFy, uFx + vFy]T where F = ρgβT(T − T0)ˆy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The macroscopic density ρ and velocity u are obtained from ρ = �8 i=0 fi, u = ��8 i=0 eifi + F/2 � /ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The evolution equation of the temperature distribution function is written as gi(x + eiδt, t + δt) − gi(x, t) = − � N−1Q � ij � nj(x, t) − n(eq) j (x, t) � (10) where gi is the temperature distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The macroscopic temperature T is ob- tained from T = �4 i=0 gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' More numerical details of the LB method and validation of the in-house solver can be found in our previous work [24–26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Simulation settings for the turbulent thermal convection We consider a two-dimensional RB cell with length L and height H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The top and bot- tom walls of the cell are kept at constant cold and hot temperatures, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' while the other two vertical walls are adiabatic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' all four walls impose no-slip velocity boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We set the cell aspect ratio (Γ = L/H) as 2 ≤ Γ ≤ 32, and we fix the Prandtl number as Pr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='71 (corresponds to the working fluids of air) and the Rayleigh number as Ra = 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Although the Ra is far less than that in the atmosphere because of limited computing resources to simulate ultra-high Ra convection, we note the RB convection at 6 Ra = 108 already exhibits strong turbulent fluctuations and the flows fall in the ’hard turbu- lence’ regime [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We also checked the turbulent database and confirmed that statistically stationary states have been reached and the initial transient effects of the simulations are washed out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' DYNAMICS AND CONTROL OF THE SELF-PROPELLING AGENT III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Kinematic model of the self-propelling agent The dynamics of the self-propelling agent can be described as uagent(t) = ufluid(t) + upropel(t) (11) xagent(t + dt) = xagent(t) + uagent(t) · dt (12) Here, dt is the time step, uagent and xagent denote the velocity and position of the agent, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' ufluid denotes the velocity of the carrier fluids, and upropel denotes the velocity generated by the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We assume that, without control, the velocity of the agent equals that of the carrier flows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' whilst, with control, the velocity of the agent is the superposition of the carrier flow and agent’s propulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Similar dynamics of the agent have been previously adopted by Krishna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' To mimic the limited propelling ability of the agent in real-world scenarios, we restricted the maximum propelling velocity of the agent ∥uagent∥ to be less than one-third of the largest carrier flow velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' On the other hand, a more complex kinematic model for the self-propelling agent, such as the one that includes inertial and rotational dynamics [29–33], fluttering and tumbling [34], multimodal locomotion [35] of the propelling agent can be considered in the future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Optimal control via the reinforcement learning We adopt the RL algorithm to optimize the control of the agent to migrate in an energy- efficient trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The advantages of the RL algorithm include agnostic for control and optimization tasks, easy to be re-used to speed-up optimization in a similar system configu- ration, and robust to the disturbances in the chaotic system [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' In the RL algorithm, the agent observes the state of the environment and decides to take an action interacting with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' If the agent then receives a reward (or a penalty), it is more likely 7 to repeat (or forego) that action in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Overall, the agent learns by trial and er- ror, with the long-term goal to maximize the cumulative expected return, and eventually improve its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Applying the RL algorithm, we can obtain the optimal policy, which advises the favorable action to take for the agent [38–41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The model-free RL algo- rithm can generally be classified into policy-based methods and value-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' In the policy-based method, such as the policy gradient method, the parameter of the policy network θ is optimized to maximize the performance objective J(πθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Here, πθ denotes the parameterized stochastic policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The policy-based methods are inefficient in sampling, thus leading to slow learning, and are not suitable for complex flow problems [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' In the value-based methods, such as the Q-learning method, the agent takes action a that tried to maximize the optimal action-value function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=', a(s) = arg maxa Qθ(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Here, s denotes the state of the environment and Qθ(s, a) approximates the optimal action-value function Q∗(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Using the Q-learning method, Colabrese et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' [29] showed that gravitactic swim- mers can reach high altitudes in steady Taylor-Green vortex flow;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Mui˜nos-Landin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' [43] demonstrated the artificial self-thermophoretic micro-swimmers can navigate under the influence of Brownian motion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Monderkamp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' [44] trained active Brownian particles through complex motility landscapes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Gazzola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' [45] and Verma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' [46] found op- timal swimming strategies that minimize drag and energy consumption in the school of fish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The above five examples adopt the off-policy learning techniques, which means that each update stochastic samples the data collected at any point during training, namely, Q(st, at) = Q(st, at) + α [rt+1 + γ maxa Q(st+1, a) − Q(st, at)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Earlier, Reddy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' [18] adopted the state-action-reward-state-action (SARSA) method, which can be regarded as a variation of the Q-learning method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The main difference is that the SARSA method adopts the on-policy learning technique, which uses the action performed by the current policy to learn the Q-value, namely, Q(st, at) = Q(st, at) + α [rt+1 + γQ(st+1, at+1) − Q(st, at)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' How- ever, the value-based method can only work in discrete state and action spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' whilst in most real-world scenarios, such as training a vehicle to navigate, the continuous state and action spaces are preferred to develop more versatile motion for complex navigation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' In this work, we adopt the soft actor-critic (SAC) algorithm, which is an interpolation between policy-based methods and value-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' In the SAC algorithm, the agent decides the next action via the actor network, whilst that action is further evaluated by the critic network to guide the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' In addition, the actor aims to maximize the expected reward (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=', succeed at the task) while also maximizing entropy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=', acting 8 as randomly as possible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' In entropy regularized reinforcement learning, the optimization problem can be described as π∗(θ) = arg max π Eτ∼π � ∞ � t=0 (R(st, at, st+1) + αH(π(·|st))) � (13) In the above equation, π∗ is the optimal policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The reward function r depends on the current state of the environment st, the current action at, and the next state of the environment st+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' α is the trade-off coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The entropy H of τ is computed from its distribution π as H(π(·|st)) = Eτ∼π[− log π(τ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' More details on the SAC algorithm can be found by Haarnoja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Key ingredients in the RL framework include the environmental cues that the agent can observe (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=', the current state st of the environment), the actions the agent takes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=', at), and the response of the agent to its behavior (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=', the reward rt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' In this work, to migrate in a large-aspect-ratio convection cell, the observation variables for the agent include the carrier flow velocity ufluid, the agent’s spatial coordinate in the vertical direction yagent, and the fluid temperature T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' In Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='2, we provide a detailed discussion on selecting observation variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The action variable is the propelling velocity upropel generated by the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Following the previous work of Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' [21], we assume the rewards received by the agent are simultaneously affected by the current state, energy consumption, and time consumption of the agent r(t) = rs(t) + re(t) + rh(t) (14) Here, the rs denotes the reward affected by the current state of the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' If the agent migrates out of the flow domain through the top or the bottom boundaries, it will receive a penalty of -φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' if the agent moves rightward and gets closer to the right-side boundary, it will receive a basic reward ebasic with an empirical pre-factor ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Thus, rs is written as rs(t) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 − φ, agent is out of the flow domain εebasic, xt agent > xt−1 agent 0, otherwise (15) Here, we adopt φ = 10 and ε = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' A detailed discussion on the sensitivity of the hyper- parameters in the reward function can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 14, the re denotes the reward affected by the energy consumption of the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' If the propelling velocity of the agent upropel is in alignment with that of the background flow ufluid, namely, the angle between these two vectors (denoted by θ) is less than 90◦, the agent will receive a reward 9 of ε[ebasic + (emax − e)], where e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='5||upropel||2 and ebasic = emax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='5(||upropel||)2 max;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' if the angle between upropel and ufluid is greater than 90◦, the agent will receive a penalty of −2ε(ebasic + e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The above designs also imply that, when the agent migrates in alignment with the carrier flow direction, it will receive a lower reward if the propelling velocity is higher;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' when the agent migrates against the carrier flow direction, it will receive a higher penalty if the propelling velocity is higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Thus, re is written as re(t) = \uf8f1 \uf8f2 \uf8f3 ε[ebasic + (emax − e)], 0◦ ≤ θ ≤ 90◦ − 2ε(ebasic + e), 90◦ ≤ θ ≤ 180◦ (16) In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 14, the rh denotes the reward affected by the time consumption of the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' If the agent migrates out of the domain via the right-side boundary, we assume the agent completes the task and it will receive a reward that is inversely proportional to the time taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' This design implies that the sooner the agent reaches the destination, the higher the reward it receives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Thus, rh is written as rh(t) = \uf8f1 \uf8f2 \uf8f3 (tmax − t)/ε, xt agent > L 0, otherwise (17) IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' RESULTS AND DISCUSSION IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' General flow patterns in the RB convection with a large aspect ratio Typical snapshots of the temperature field at Ra = 108, Pr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='71, and 2 ≤ Γ ≤ 32 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We can distinguish small-scale thermal plumes and large-scale circulation rolls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Thin thermal boundary layers appear near the bottom heating wall and top cooling wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Plumes that are released from the boundary layers penetrate upwards (or downwards) towards the opposite wall of lower (or higher temperature), and intense mixing occurs in the central region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Thermals are almost periodically released from relatively fixed locations, and neighboring thermals that move in opposite directions entrain the surrounding fluid and self-organize into circulation rolls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Previously, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' [48] found that depending on the initial conditions, the flow system at a given aspect ratio evolves to different final turbulent states with different roll numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' In our work, the convection roll number n is 2, 4, 9, 18 and 34, in the Γ = 2, 4, 8, 16 and 32 convection cells, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' their corresponding mean aspect ratios are Γr = Γ/n = 1, 1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='889, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='889 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='941, which is consistent with that predicted by the elliptical instability theory [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 10 (a) (b) (c) (d) (e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='8 Temperature FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Typical instantaneous temperature field at Ra = 108, Pr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='71, (a) Γ = 2, (b) Γ = 4, (c) Γ = 8, (d) Γ = 16, (e) Γ = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We then examine the global response parameter of Reynolds number (Re) on the control parameter Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Here, the global flow strength is calculated as Re = � ⟨(u2 + v2)⟩V,tH/ν and ⟨· · · ⟩V,t denotes the spatial and temporal average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The measured Re as a function of Γ is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 2(a), and we can observe enhanced global flow strength with the increase of cell aspect ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' In addition, with the increase of Γ, the Re gradually reaches an asymptotic value, similar to that in the 3D convection cell [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Previously, Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' [21] found the optimized energy-efficient strategy obtained from the reinforcement learning algorithm is not sensitive to small perturbation of the global flow strength, but it changes when the global flow strength increases (or decreases) more than one magnitude of order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Because the self-propelling agent was trained in the Γ = 2 cell, we further checked that even in the Γ = 32 cell, the flow strength increases around 22% compared to that in the Γ = 2 cell, indicating the flow strength in a larger aspect ratio cell does not increase significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' To 11 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='� Re 10 0 10 1 10 2 3500 4000 4500 5000 ||u|| / ||u||rms PDF 0 2 4 6 10 8 10 7 10 6 10 5 10 4 10 3 10 2 10 1 10 0 10 1 � = 2 � = 4 � = 8 � = 16 � = 32 (a) (b) fluid fluid rms / u u 100 10-2 10-4 10-6 10-8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' (a) Reynolds number, and (b) probability density functions (PDFs) of normalized velocity magnitude ∥ufluid∥/∥ufluid∥rms, for various Γ at Ra = 108 and Pr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' quantify the fluctuations of velocity magnitude, we plot the probability density functions (PDFs) of normalized velocity magnitude ∥ufluid∥/∥ufluid∥rms for various Γ, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We can see the PDF heads collapse for different Γ, whilst the PDF tails become slightly extended with the increase of Γ, which implies an increased degree of fluctuations for the velocity magnitude ∥ufluid∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' To extract the coherent flow structure from the turbulent database, we adopt the proper orthogonal decomposition (POD) analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' In the POD, the spatiotemporal vector field X(r, t) is decomposed as a superposition of orthogonal eigenfunctions φi(r) and their am- plitudes ai(t) as X(r, t) = ∞ � i=1 ai(t)φi(r) (18) Here, the vector field X(r, t) is chosen as the flow velocity field X = (u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Practically, we can use the singular value decomposition (SVD) on the dataset X to obtain the flow mode φi(r) and the corresponding mode amplitude ai(t) [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Because the database for turbulent convection in a large-aspect-ratio cell is huge, which consists of fine spatial resolution and long temporal evolution data, the memory consumption to perform standard SVD is intense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Here, we adopt the randomized SVD to reduce the computational load [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The shape of the first POD mode φ1(r) at various Γ is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The most energetic POD mode consists of horizontally stacked circulation primary rolls rotating in either the clockwise direction or the anti-clockwise direction, and these primary rolls exhibit a periodical pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Large values of velocity magnitude appear near the vortex edge, indicating a strong energy barrier for the agent to move across the vortex edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' It is noteworthy that at much higher Ra (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=', Ra > 1010), when the large-scale-circulation is weaker and the flow consists of multiple 12 (a) (b) (c) (d) (e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='8 U / Umax FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Contour of the first proper orthogonal decomposition (POD) mode φ1(r) at Ra = 108, Pr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='71, (a) Γ = 2, (b) Γ = 4, (c) Γ = 8, (d) Γ = 16, (e) Γ = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Here, U = √ u2 + v2 is the velocity magnitude, and Umax denotes the maximum value of U for normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' mobile and orbital small vortices [51–53], whether the current learning framework still works deserves further study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We further calculate the stability of the first POD mode as S1 = � ⟨a1(t)⟩t/σa1, such that a larger value of S1 indicates a more stable pattern of circulation rolls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Similar estimations of the roll stability have been previously used in the Fourier mode decomposition of the turbulent thermal convection [49, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 4(a), we can see the stability of the first POD mode decreases with the increase of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We also analyze the energy contained in the first POD mode and calculate the energy percentage as λ1/ �∞ i=1 λi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Here, we have λiδij = ⟨ai(t)aj(t)⟩t and λi denotes the energy of the ith POD mode, δij is the Kronecker symbol, and ⟨· · ·⟩t denotes the temporal average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 4(b), we can see the energy percentage in the first POD mode also decreases with the increase of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Although the first 13 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='� Stability 10 0 10 1 10 2 10 0 10 1 10 2 � Energy pct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' (%) 10 0 10 1 10 2 80 90 100 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' (a) The stability of the first POD mode, (b) the energy contained in the first POD mode at various Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' mode is still the dominant flow structure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=', in the Γ = 32 cell, the energy contained in the first POD mode accounts for more than 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='8% of the total energy), higher-order flow modes become stronger with the increase of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Thus, in a large-aspect-ratio cell, the horizontally stacked circulation rolls that form a periodical pattern are less stable, and those higher-order modes lead to a more irregular flow pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Because the agent was trained in the Γ = 2 cell, the above-mentioned complex flow features bring challenges for the agent to identify an energy-efficient trajectory in a larger Γ cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Training the agent to migrate in the RB cell with an aspect ratio of 2 We first train the agent to migrate across the turbulent RB cell with Γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The agent starts from the point of (0, 0) at the bottom-left corner of the cell, and its goal is to reach the right-side boundary of the cell (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=', the vertical line of x = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Here, the simple Γ = 2 cell consists of the characteristic large-scale coherent structure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=', a clockwise rotating primary roll and an anti-clockwise rotating primary roll), thus it serves as a paradigm environment for the learning agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 5, we show the instantaneous trajectories of the smart agent after training, and the corresponding video can be viewed in the supplementary movie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Initially, the agent moves upward driven by the clockwise rotating corner roll at the bottom- left corner [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 5(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' When the agent reaches the edge of the primary roll, which is rotating in the anti-clockwise direction, it moves along with the horizontal currents [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 5(b)], until it meets the rising thermals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The agent then rises on the thermals and ascends higher [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 5(c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' After reaching the top layer of the cell, due to the right-directed propelling velocity, the agent moves rightward and utilizes the horizontal currents [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 14 (a) (b) (c) (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='8 Temperature FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Trajectory (black dotted line) of the smart agent in the RB convection at (a) t = 18, (b) t = 36, (c) t = 54, and (d) t = 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The contour shows the typical instantaneous temperature field, and the vectors denote the velocity field of the convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 5(d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The migration task is completed when the agent reaches the right-side boundary of the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Overall, the smart agent tries to follow the carrier currents as much as possible, and it discovers an effective policy of moving along the edge of the rolls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' A comparison between the smart agent and the naive agent is performed to highlight the differences in propelling behaviors and the savings in energy expenditure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Here, the naive agent refers to the agent that moves straight from the origin to the destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We set the naive agent to spend the same amount of total time ttotal as that of the smart agent to complete the migration task, and its destination point is also the same as the point where the smart agent left the right-side boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Thus, the velocity of the naive agent keeps a constant direction pointing from the origin to the destination, and it keeps a constant magnitude of ∥uagent∥ = ∥xgoal − xstart∥/ttotal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 6(a) and 6(b), we show trajectories of the smart agent and the naive agent, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' From the instantaneous velocity magnitude shown on the color-coded trajectories, we can see that the naive agent generally migrates slower than the smart agent, because the naive agent travels a shorter 15 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='distance during the same ttotal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Although the smart agent migrates faster, it does not indicate that the smart agent will consume more energy, since the smart agent can utilize the carrier flow currents to save energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Along with the trajectories, we also plot the propelling velocity vector (denoted by the red arrows) and the fluid velocity vector (denoted by the blue arrows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We calculate the correlation coefficient between the orientation of the propelling velocity vector (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=', θpropel) and the orientation of the fluid velocity vector (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=', θfluid) as C = ⟨[θpropel(t) − ⟨θpropel⟩] [θfluid(t) − ⟨θfluid⟩]⟩/ � σθpropelσθfluid � (19) Here, the orientation is in the range from -180◦ to 180◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The positive orientation is defined as anti-clockwise rotating the x-axis, and the negative orientation is defined as clockwise ro- tating the x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' For the smart agent, the resulting correlation coefficient of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='56 suggests positive statistical relevance between them, revealing that the smart agent adjusts its migra- tion direction in response to the changing carrier flow, enabling energy-efficient migration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' for the naive agent, the resulting correlation coefficient of -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='74 implies negative statistical relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' In addition, to quantitatively describe the angles between the propelling velocity vector and the fluid velocity vector, in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 6(c) and 6(d), we plot the histogram of those angles for the smart agent and the naive agent, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We can see for the smart agent, the angles are generally less than 90◦, and the frequency exhibits a peak around 20◦, which is another evidence that the agent tries to follow the carrier currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' For the naive agent, the frequency of those angles exhibits a peak around 120◦, indicating that the naive agent has to generate propelling velocity against the carrier flow currents to keep the shortest migrating path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We assume that only the propelling velocity upropel affects the propelling energy con- sumption for the agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Thus, the accumulative energy consumption is calculated as Epropel(t) = � t 0 1 2∥upropel(τ)∥2dτ (20) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 7(a), we plot the time series of accumulative energy consumed by the agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' After completing the migration task, the smart agent consumed around 38% of the propelling energy compared to that of the naive agent, meaning migrating in the shortest path does not always save energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We also calculate the instantaneous energy consumption as epropel(t) = 1 2∥upropel(t)∥2 (21) From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 7(b), we can see the epropel for the smart agent is generally lower than that of the naive agent, and the epropel keeps smaller values for the smart agent during the whole 16 Angle Frequency 0 60 120 180 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='2 Angle Frequency 0 60 120 180 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='2 (a) (b) (c) (d) Velocity Angle (�) Angle (�) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' (a, b) Trajectories for the smart agent enabling energy-efficient migration and a naive agent moving straightly, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The red arrows denote the propelling velocity and the blue arrows denote the fluid velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The trajectories are color-coded by the instantaneous velocity magnitude of the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' (c, d) Histogram of angles between the propelling velocity vector and the fluid velocity vector for the smart agent and the naive agent, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' migration process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' For the naive agent, epropel exhibits a first peak around t ≈ 3, when it moves across the edges of the left-bottom corner roll that requires a high energy barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' At around t ≈ 23, epropel drops to the minimum, because the naive agent reaches the location where flow currents are in alignment with the migration direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The second peak of epropel appears at around t ≈ 38, when the naive agent crosses the edge of the primary roll and drifts to the clockwise rotating primary roll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The third peak of epropel appears at around t ≈ 67, when the naive agent approaches the right-side boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We also compare the accumulative total kinetic energy of the agents [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 7(c)], which is calculated as Etotal(t) = � t 0 1 2∥uagent∥2dτ (22) After completing the migration task, the total kinetic energy of the smart agent is almost twice that of the naive agent, mostly contributed by the kinetic energy of the carrier flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 17 t Epropel 0 20 40 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='03 smart naive t epropel 0 20 40 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='0004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='0008 smart naive t Etotal 0 20 40 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='06 smart naive t etotal 0 20 40 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='0006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='0012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='0018 smart naive (a) (b) (c) (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Comparison of the (a) accumulative energy consumption Epropel, (b) instantaneous energy consumption epropel, (c) accumulative total kinetic energy Etotal, and (d) instantaneous total kinetic energy etotal for the smart agent and the naive agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Similarly, we calculate the instantaneous total kinetic energy etotal as etotal(t) = 1 2∥uagent(t)∥2 (23) From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 7(d), we can see the etotal of the smart agent is generally higher than that of the naive agent and keeps larger values during the whole migration process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Three peaks appear when the smart agents migrate in alignment with the flow direction, thus utilizing more kinetic energy of the carrier flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' On the other hand, the etotal of the naive agent keeps constant due to the constant value of ∥uagent(t)∥ in the simulation settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Choosing appropriate environment cues that the agent can observe is crucial in the RL training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Here, we numerically determine the set of observation variables by comparing the evolution of cumulative reward during the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We train five different instances of each observation variable set with different random seeds, and each set performs one evaluation rollout every 1000 environment steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The solid curves correspond to the mean and the shaded region to the minimum and maximum returns over the five trials, and it 18 represents a moving average with a window of 20 timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We first consider the agent is flow-blinded and cannot sense the surrounding flow and temperature information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The agent can have access to its position information, but only the vertical component, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=', s = {y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Here, we do not consider the horizontal position information (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=', x /∈ {s} );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' otherwise, the agent trained in the Γ = 2 cell would fail to find optimal trajectory in a larger Γ cell once its horizontal position is x > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 8(a), the agent performs poorly in the case of flow-blinded, which shows similar behavior as that of navigating through unsteady cylinder flow [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We then consider the agent can also sense the carrier flow velocity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=', s = {y, u, v}, and plot the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 8(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We can see that the agent performs much better, and the cumulative reward is higher than that of the flow-blinded agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' In addition, we consider the agent can sense extra vorticity information (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=', s = {y, u, v, ω}) [29, 56], and plot the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 8(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' With the consideration of additional flow field information, the cumulative reward converges at an earlier time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=', t ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='5 × 105 for s = {y, u, v, ω}) compared to that of velocity information (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=', t ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='0 × 105 for s = {y, u, v}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' On the other hand, in the turbulent RB convection, the temperature acts as an active scalar that influences the velocity, we next consider the agent can sense extra temperature information (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=', s = {y, u, v, T}) [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 8(d), the agent outperforms previous ones and the cumulative reward converges at an earlier time and almost remains steady, suggesting temperature is an important environment cue for the agent to migrate in thermal convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' In the Appendix B, we evaluate more combinations and then choose s = {y, u, v, T} as observation variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' On the other hand, Kubo and Shimizu [57] proposed a framework that can perform fluid flow control with partial observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We expect the extension of that framework to turbulent flows will simplify the selection of observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Testing the agent to migrate in the RB cell with a larger aspect ratio We now apply the obtained policy to test whether the smart agent can migrate in an energy-efficient way in convection cells with larger Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The flow mode analysis presented in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='1 indicates that in a larger Γ cell, the dominant flow modes of horizontally stacked rolls are less stable, and the energy contained in higher-order flow modes increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Despite these challenges brought by the complex flow features, we can still obtain optimized trajectories, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Starting from the origin of (0, 0) point, the smart agent first escapes the corner roll at the left-bottom cell corner, it then ascends higher and rises on 19 (a) (b) (c) (d) �105 �105 Timesteps Timesteps �105 Timesteps Rewards �105 Timesteps Rewards Rewards Rewards FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Evolution of the cumulative reward during training for different combinations of observation variables: (a) s = {y};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' (b) s = {y, u, v};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' (c) s = {y, u, v, ω};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' (d) s = {y, u, v, T}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' the thermal (if the first primary roll is clockwise rotating), or follows the horizontal currents (if the first primary roll is anti-clockwise rotating).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Afterward, the smart agent always tries to migrate along the edges of the primary rolls, where the carrier fluid flows fast and plenty of kinetic energy from the flow is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We then compare the propelling energy consumed by the smart agent and the naive agent in the convection cell with various Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 10(a), we plot the accumulative propelling energy for the agents when they complete the migration task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Generally, for both agents, the energy consumption increases with the increase of Γ, due to longer migration distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' For the smart agent, it enables an energy-efficient migration strategy via migrating along the edges of horizontally stacked multiple primary rolls, thus we have Epropel ∝ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' To quantitatively describe how much propelling energy can be saved, we plot the ratio of energy consumed by the smart agent to that of the naive agent, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 10(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We can see that in a larger Γ cell, the ratio of Esmart/Enaive is smaller, meaning more propelling energy 20 (a) (b) (c) (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='8 Temperature FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Trajectory (black dotted line) of the smart agent in the convection cell with (a) Γ = 4, (b) Γ = 8, (c) Γ = 16, and (d) Γ = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The contour shows the typical instantaneous temperature field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' can be saved by the smart agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The reason is that in a larger Γ cell, the naive agent has to cross more edges of the circulation rolls and overcome higher energy barriers, whilst the smart agent follows the carrier currents in an energy-efficient way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The above results are obtained with the prescribed and fixed origin position, namely, the location at the (0, 0) point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We further test the robustness of the energy-efficient policy concerning random origin position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We first release the agent in the position where the local flow velocity is weak, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=', ∥ufluid∥ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='03, and 100 example trajectories are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We can see regardless of the random origin position, the successful attempts gradually converge to a similar path line, and the agent migrates along the edges of the primary rolls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' It should be noted that we restrict these ’random’ positions to be x < Γ/2, which prevents the agent from being released too close to the outlet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The average success 21 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='� Epropel 10 0 10 1 10 2 10 2 10 1 10 0 smart naive � Esmart / Enaive 10 0 10 1 10 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='4 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' (a) The propelling energy consumed by the smart agent and the naive agent, (b) the ratio of energy consumed by the smart agent to that of the naive agent as a function of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' rate to complete the migration task in the Γ = 4, 8, 16, and 32 cell is 74%, 87%, 92%, and 97%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We then release the agent in the position where the local flow velocity is strong, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=', ∥ufluid∥ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The average success rate is much higher, and it is 100%, 100%, 100%, and 99% in the Γ = 4, 8, 16, and 32 cell, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' For the sake of clarity, we do not repeat plotting the example trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The above results indicate that the success rate to complete the migration task increases with the increase of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' On the other hand, in a larger Γ cell, the flow structure is more complex, and we expect it would be more challenging for the agent to complete the migration task and earn a higher success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' To understand such behaviors, we further consider the following scenarios: (i) the agents being released where carrier flow velocity is ∥ufluid∥ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='01;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' (ii) the agents being released where carrier flow velocity is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='01 < ∥ufluid∥ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='02;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' (iii) the agents being released where carrier flow velocity is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='02 < ∥ufluid∥ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We can see from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 12 that in convection cells with the same Γ, the average success rate is higher if the carrier flow velocity at the origin is stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Because the global flow strength is stronger in a larger Γ cell (see discussion in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='1), the agents are more likely to be released where carrier flow is strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Utilizing stronger carrier flow currents, the agents can complete the migration task within a shorter time and earn a higher reward, thus the agent is more likely to repeat that action in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The higher sampling frequency for the agent in areas with stronger flow strength leads to higher success rate in larger Γ cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' It should be noted that such a trend is only obvious when the origin of agents possesses weak carrier flow velocity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' with strong carrier flow velocity, the success rates are always near 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 22 (a) (b) (c) (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='8 Temperature FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Example trajectories in the (a) Γ = 4, (b) Γ = 8, (c) Γ = 16, and (d) Γ = 32 cell (the agents are released where ∥ufluid∥ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='03).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Green lines represent successful attempts to complete the migration, while red lines represent unsuccessful attempts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The contour shows the typical instantaneous temperature field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' CONCLUSIONS In this work, using the reinforcement learning algorithm, we performed numerical train- ing of the self-propelling agent migrating long-distance in a thermal turbulent environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We choose the paradigmatic turbulent RB convection cell as the flow environment, which can incorporate strong fluctuations of velocity and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' To build up the reinforce- ment learning framework, we designed a reward function that simultaneously considers the current state, energy consumption, and time consumption of the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We also compare the evolution of cumulative reward for different combinations of observation variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We select the position of the agent, as well as the velocity and temperature of the carrier flow 23 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='� Success rate (%) 10 0 10 1 10 2 60 80 100 fluid 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='01 � u fluid 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='02 � � u fluid 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='03 � � u fluid 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='03 � u FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Average success rate as functions of Γ when the agents are randomly released at the positions with the different local carrier flow velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' as appropriate environmental cues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The simulation results in a Γ = 2 RB cell showed that, compared to a naive agent that moves straight from the origin to the destination, the smart agent can learn to utilize the carrier flow currents to save propelling energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We then apply the optimal policy obtained in the Γ = 2 cell and test the smart agent migrating in convection cells with larger Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' From flow mode analysis, we found the dominant flow modes in a larger Γ RB cell consist of less stable horizontally stacked rolls, and the energy contained in higher-order flow modes increases with the increase of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Although these complex flow features bring challenges to optimizing the trajectories for the smart agent, we can still obtain energy-efficient migrating trajectories using the policy trained in the Γ = 2 RB cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' In addition, we found the ratio of propelling energy consumed by the smart agent to that of the naive agent decreases with the increase of Γ, meaning more propelling energy can be saved by the smart agent in a larger Γ cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The reason is that in a larger Γ cell, the naive agent has to cross more edges of the circulation rolls and overcome higher energy barriers, whilst the smart agent always tries to follow the carrier currents as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We also evaluate the optimized policy when the agents are being released from the ran- domly chosen origin, which aims to test the robustness of the learning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We found the success rate increases with the increase of Γ, despite the flow structures being more com- plex in a larger Γ cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The main reason is that, in a larger Γ cell, the global flow strength is stronger (evident by the relationship between Re and Γ), and the agent is more likely 24 to be released in positions where the carrier flow velocity is stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Utilizing stronger carrier flow velocity, the agent can complete the migration task within a shorter time and receive a higher reward, thus leading to a higher success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Our work has implications for long-distance migration problems, for example, the UAVs patrolling in the convective layer of the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Migrating in energy-efficient trajectories, the UAVs can increase endurance and cover a wider range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (NSFC) through Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 12272311 and 12125204, the National Key Project via No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' GJXM92579, and the 111 project of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' B17037).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' The authors acknowledge the Beijing Beilong Super Cloud Computing Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=', Ltd for providing HPC resources that have contributed to the research results reported within this paper (URL: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='blsc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='cn/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Appendix A: Sensitivity of the hyperparameters in the reward function In our designed reward function (see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 14-17), we have two hyperparameters: one is φ, which represents the penalty when the agent is out of the flow domain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' the other is ε, which is the reward scale coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We tuned these two parameters separately to determine the optimal hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' It should be noted that we compared the value of the normalized reward (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=', varied between 0 and 1) rather than the absolute value of the reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We can see from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 13 (a) that, small values of the penalty φ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=', φ = 1 and 5) results in substantial degradation of performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' large values of the penalty (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=', φ ≥ 10) almost lead to the same performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' As for the reward scale coefficient ε, it is almost insensitive for the investigated value and they can all give optimal policy, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 13 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' However, a large value (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=', ε = 100, not shown here for clarity) would result in �(rs + re) > � rh during training, and the agent’s failure to explore the successful trajectory within the given time of tmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 25 Timesteps Normalized reward 0 400000 800000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='8 1 � = 1 � = 5 � = 10 � = 20 � = 100 Timesteps Normalized reward 0 400000 800000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='8 1 � = 1 � = 5 � = 10 � = 20 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Sensitivity of the hyperparameters in the reward function: (a) the penalty φ when the agent is out of the flow domain, (b) the reward scale coefficient ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Appendix B: Evaluation of different combinations of observation variables In addition to the observation variables described in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='2, we also consider the following different combinations: (i) the agent has access to position and velocity informa- tion, and it can sense strain rate [58, 59], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=', s = {y, u, v, sxx} and s = {y, u, v, sxy}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Here, sxx = ∂xu and sxy = (∂yu + ∂xv)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We did not consider the syy = ∂yv component of the strain rate tensor, because flow continuity equation gives sxx + syy = 0 in 2D flows, which means sxx and syy are negatively correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' (ii) the agent has access to position and velocity information, and it can sense temperature gradient, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=', s = {y, u, v, (∇T)x} and s = {y, u, v, (∇T)y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Here, (∇T)x essentially represents the vorticity produced by buoyancy in the 2D convection flow [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' (iii) the agent has access to position, velocity and tempera- ture information, and it can sense additional vorticity, strain rate, or temperature gradient, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=', s = {y, u, v, T, sxx}, s = {y, u, v, T, sxy}, s = {y, u, v, T, ω}, s = {y, u, v, T, (∇T)x}, and s = {y, u, v, T, (∇T)y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 14, we plot the evolution of the cumulative reward during training for the above nine combinations of observation variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' We can see these combinations only slightly changes the converging speed of the training, not the asymptotic accumulative reward value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Among them, the s = {y, u, v, T, (∇T)x} shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 14(h) outperforms other combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' In practical applications, velocity or temperature sensing could be implemented via a variety of methods, such as pitot tubes, hot wire, and so on;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' while vorticity, shear strain component, and temperature gradient should be computed from several velocities or temperature sensors, which increases the complexity that the agent has to sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Thus, as described in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content='2, we deliberately keep simple the environmental 26 (a) (b) Rewards (c) (d) (e) (f ) (g) �105 �105 �105 �105 �105 �105 �105 Rewards Rewards Timesteps Timesteps Timesteps (h) (i) �105 �105 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' Evolution of the cumulative reward during training for different combinations of ob- servation variables: (a) s = {y, u, v, sxx};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' (b) s = {y, u, v, sxy};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' (c) s = {y, u, v, (∇T)x};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' (d) s = {y, u, v, (∇T)y};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' (e) s = {y, u, v, T, sxx};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' (f ) s = {y, u, v, T, sxy};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' (g) s = {y, u, v, T, ω};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' (h) s = {y, u, v, T, (∇T)x};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' (i) s = {y, u, v, T, (∇T)y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' cues of local information s = {y, u, v, T} that the agent can see to guide its migration, such that the amount of data storage by the agent can be reduced in practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE3T4oBgHgl3EQf8wtj/content/2301.04810v1.pdf'} +page_content=' [1] H.' 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0000000000000000000000000000000000000000..b815b471508f7b81ff4abb133610f7338f5473ff --- /dev/null +++ b/tdE1T4oBgHgl3EQfjwQd/content/tmp_files/2301.03265v1.pdf.txt @@ -0,0 +1,330 @@ +arXiv:2301.03265v1 [math.RT] 9 Jan 2023 +THE RANK ONE PROPERTY FOR FREE FROBENIUS EXTENSIONS +GWYN BELLAMY AND ULRICH THIEL +Abstract. A conjecture by the second author, proven by Bonnaf´e-Rouquier, says that the multiplicity +matrix for baby Verma modules over the restricted rational Cherednik algebra has rank one over Q when +restricted to each block of the algebra. +In this paper, we show that if H is a prime algebra that is a free Frobenius extension over a regular +central subalgebra R, and the centre of H is normal Gorenstein, then each central quotient A of H by a +maximal ideal m of R satisfies the rank one property with respect to the Cartan matrix of A. Examples +where the result is applicable include graded Hecke algebras, extended affine Hecke algebras, quantized +enveloping algebras at roots of unity, non-commutative crepant resolutions of Gorenstein domains and 3 +and 4 dimensional PI Skylanin algebras. +In particular, since the multiplicity matrix for restricted rational Cherednik algebras has the rank one +property if and only if its Cartan matrix does, our result provides a different proof of the original conjecture. +1. Introduction +1.1. +It was conjectured by the second author [21, Question. 2(ii)] that baby Verma modules over the +restricted rational Cherednik algebra satisfy a remarkable rank one property. Namely, if ∆(λ), resp. L(λ), +denotes the baby Verma module, resp. irreducible module, associated to λ ∈ IrrW, then the multiplicity +matrix M, given by +Mλ,µ := [∆(λ) : L(µ)], +has rank one over Q when restricted to each block of the algebra. This conjecture was confirmed by Bonnaf´e- +Rouquier [4, Proposition 14.4.2], whose ingenious proof makes use of a splitting extension of the centre of the +(un-restricted) rational Cherednik algebra, together with properties the corresponding decomposition map to +the restricted rational Cherednik algebra. Let C denote the Cartan matrix of the algebra. The baby Verma +modules for the restricted rational Cherednik algebra satisfy BGG-reciprocity, C = M T M, from which it +is easily seen (Section 3) that the rank one property can equivalently be stated as saying that the Cartan +matrix of each block of the restricted rational Cherednik algebra has rank one. This reformulation makes no +explicit mention of baby Verma modules. +1.2. +In this article we give a completely different proof of the rank one property. Our proof applies to a +broader class of algebras, including for instance quantum groups at roots of unity, and makes use of the +fact that these algebras are free Frobenius R-algebras, for an appropriate regular central subalgebra R. In +particular, our result applies to all but two of the families of examples considered in [7]. +In the introduction, we assume for simplicity that K is an algebraically closed field of characteristic zero; +in the body of the paper we prove a more general result that does not require this assumption. We start +with H a (unital) affine K-algebra and central subalgebra R ⊂ H such that H is a free Frobenius extension +of R; see Section 2.1. +1 + +We assume that R is regular, H is prime and the centre Z := Z(H) of H is Gorenstein and integrally +closed. Let m be a maximal ideal of R and A = H/mH. The K-algebra A is finite-dimensional and split. Let +K0(A) be the Grothendieck group of finitely generated projective A-modules and G0(A) the Grothendieck +group of all finitely generated A-modules. The Cartan map is the canonical map C : K0(A) → G0(A). If Λ +is a (finite) set parametrizing isomorphism classes of simple A-modules, with representatives L(λ) for each +λ ∈ Λ, then we can think of C as a |Λ| × |Λ| matrix whose entries are the multiplicities [P(λ) : L(µ)], where +P(λ) is the projective cover of L(λ). If A = A1 ⊕ · · · ⊕ Ak is the block decomposition of A then C admits a +corresponding decomposition C = C1 ⊕ · · · ⊕ Ck, where Ci is the Cartan matrix of Ai. +We say that the algebra A has the rank one property if each matrix Ci has rank one over Q. Though this +appears at first to be a very strong property, our main result is: +Theorem 1.1. A has the rank one property. +The proof of Theorem 1.1 centres around the Higman map τ′ : H → H, which descends to the Higman +map τ : A → A. The key result is Theorem 3(a) of Lorenz-Fitzgerald Tokoly [17] (see also [15]), which says +that the rank of the Cartan matrix C equals the dimension of Im τ as a K-vector space. We show that this +dimension equals the number of blocks of A. We do this by identifying Im τ with the socle of the image Z′ +α +in A of the Nakyama centre Zα of H. +1.3. Examples. In the motivating case, H = Hc(W) is the rational Cherednik algebra at t = 0 associated +to the complex reflection group (h, W). This C-algebra contains R = C[h]W ⊗C[h∗]W as a central subalgebra +and Z is an integrally closed Gorenstein ring [10, Theorem 3.3, Lemma 3.5]. It has been shown in [7] that +H is a (symmetric) free Frobenius extension of R. If m ⊂ C[h]W ⊗ C[h∗]W is the augmentation ideal, then +the central quotient A = H/mH = Hc(W) is the restricted rational Cherednik algebra. Since this algebra +satisfies BGG-reciprocity, the rank one property with respect to the Cartan matrix C is equivalent to the +rank one property with respect to the multiplicity matrix M. +Theorem 1.1 applies to many other examples commonly studied in representation theory. These include +graded Hecke algebras, extended affine Hecke algebras, affine nil-Hecke algebras, quantized enveloping alge- +bras at roots of unity, non-commutative crepant resolutions of Gorenstein domains and 3 and 4 dimensional +PI Skylanin algebras; see section 3.2. +1.4. Acknowledgements. The first author would like to thank Ken Brown and Lewis Topley for fruitful +discussions. The first author was partially supported by a Research Project Grant from the Leverhulme +Trust and by the EPSRC grant EP-W013053-1. +2. The proof +We begin again, dropping any assumptions on the field K, unless specifically stated. Theorem 1.1 will +follow from the more general Theorem 2.4 below. +2.1. The Nakayama centre. We start with H a (unital) ring and a central subring R ⊂ H such that H is +a free Frobenius extension of R. By definition, this means that H is a finite free R-module and there is an +isomorphism +φ: 1Hα−1 +∼ +−→ HomR(H, R) +(2.1) +2 + +of H-bimodules, for some automorphism α of H. Here 1Hα−1 = H as left H-modules, but with right action +given by h·a = hα−1(a). The automorphism α is unique up to inner automorphisms and one can check that +α|Z = IdZ, where Z := Z(H). Let +Zα(H) = {h ∈ H | ha = α(a)h for all a ∈ H}; +this is the Nakayama centre of H. We note that under the isomorphism φ of (2.1), +Zα(H) +∼→ HomR(H/[H, H], R). +(2.2) +Since α|Z = IdZ, we have an identification of Z-modules Zα(H) ∼= HomR(H/[H, H], R). +Assume that H is a prime ring. Then it is a prime PI ring since it is a finite R-module. Let d denote +the PI degree of H. Let T (H) denote the trace ring of H and Tr: H → T (H) the reduced trace; see [19, +Section 9a] for the definition of these. The following result is [5, Proposition 2.3]. +Lemma 2.1. Assume that T (H) = H and d is invertible in H. Then the map +HomR(H/[H, H], R) +∼ +−→ HomR(Z, R), +f �→ f|Z, +is an isomorphism of Z-modules, with inverse g �→ g ◦ (d−1Tr). +Here the map d−1Tr: H → Z is a projection, realizing Z as a direct summand of H as Z-modules. +Lemma 2.2. Assume that T (H) = H and d is invertible in H. Then Zα(H) is a direct summand of H as +Z-modules. +Proof. Since Zα(H) corresponds to HomR(H/[H, H], R) under the isomorphism φ, it suffices to show that +HomR(H/[H, H], R) is a summand of HomR(H, R) as Z-modules. Consider the series of maps +Z ֒→ H → H/[H, H] +d−1Tr +−→ Z +which dualise to +HomR(Z, R) +(d−1Tr)∗ +−→ +HomR(H/[H, H], R) → HomR(H, R) ։ HomR(Z, R), +where we have used the fact that Z is a summand of H to conclude that the last arrow is surjective. Lemma 2.1 +says that the composite map is an automorphism of HomR(Z, R). Since (d−1Tr)∗ is an isomorphism, we +deduce that HomR(H/[H, H], R) → HomR(H, R) is a split injection. +□ +Lemma 2.3. Let m ⊂ R be a maximal ideal, contained in the regular locus and let Z′ denote the image of +Z in H/mH. Assume that: +(1) The integer d is invertible as an element of H. +(2) Z is Gorenstein and integrally closed. +Then Z/mZ is a Frobenius algebra, the morphism Z/mZ → Z′ is an isomorphism and Zα(H)/mZα(H) is a +free rank one Z′-module. +Proof. First we note that the fact that Z integrally closed implies that the trace ring T (H) of H equals H; +see [19, Theorem 10.1]. Then, as noted previously, the fact that the integer d is invertible as an element of +H means that Z is a direct summand of H as a Z-module. It follows that the canonical map Z/mZ → Z′ +is an isomorphism. +3 + +Since H is a free R-module, it follows that Z is a projective R-module. Let Rm denote the localization of R +at m and Zm = Z ⊗R Rm. Then Zm is a free Rm-module. Therefore, we need to show that Z/mZ = Zm/mZm +is Frobenius. +Since Rm is assumed to be a regular local ring, mRm is generated by a regular sequence +f1, . . . , fk. This sequence is also regular in Zm. It follows from Proposition 3.1.19(b) of [8] that Zm/mZm is +zero-dimensional Gorenstein. This means it is a Frobenius algebra. +The hypothesis of Lemma 2.1 hold and hence HomR(H/[H, H], R) ∼= HomR(Z, R) as Z-modules. Now, +HomR(Z, R)m ∼= HomRm(Zm, Rm) as Zm-modules. +Since H is prime, Z is a domain and hence equi- +dimensional. Also, R is a regular ring and thus Gorenstein. Under these hypothesis, [5, Proposition 2.6] says +that Zm being Gorenstein is equivalent to HomRm(Zm, Rm) ∼= Zm as Zm-modules. Since HomR(H/[H, H], R) +is isomorphic to Zα(H) by (2.2), we deduce that Zα(H)m ∼= Zm. This implies that Zα(H)/mZα(H) is iso- +morphic to Z/mZ. +□ +2.2. The main theorem. Let m be a maximal ideal of R and A = H/mH. Let K denote the residue field +of R at m. Then A is a finite-dimensional K-algebra. We assume that A is split over K. Let p ≥ 0 denote the +characteristic of K. Lemma 2.2 implies that the map Zα(H) → A realises the Z′-module Zα(H)/mZα(H) +as a direct summand of A. +The centre of A is denoted Z(A). Clearly Z′ ⊂ Z(A), but in general the inclusion is strict. As in the +introduction, A = A1 ⊕ · · · ⊕ Ak is the block decomposition of A and C = C1 ⊕ · · · ⊕ Ck the corresponding +decomposition of the Cartan matrix. We set CK : K0(A) ⊗Z K → G0(A) ⊗Z K with decomposition CK = +CK,1 ⊕ · · · ⊕ CK,k. We call the rank of CK over K the p-rank of C (abuse of language). +Theorem 2.4. If the p-rank of every CK,i is non-zero then the p-rank of each CK,i is exactly one. +We note that if K has characteristic zero then it is automatic that the rank (= p-rank) of each block of +C is at least one. Therefore, Theorem 1.1 is a direct consequence of Theorem 2.4. +2.3. Proof of Theorem 2.4. Recall that we have assumed the following hold. +(1) H is a free Frobenius R-algebra. +(2) R is a regular ring. +(3) Z is Gorenstein and integrally closed. +(4) A := H/mH is split over K := R/m, for m ⊳ R maximal. +(5) The p-rank of every block Ci of C is non-zero. +(6) H is a prime ring whose PI degree is invertible in H. +The isomorphsim (2.1) defines an R-linear pairing on H by (g, h) := φ(1)(gh). We fix a pair {gj : j ∈ I} +and {hj : j ∈ I} of dual R-bases for H, meaning that (gi, hj) = δi,j. This allows us to define the Higman +map +τ ′ : H → H, +τ ′(x) = +� +j∈I +gjxhj. +By [16, Lemma 2.13], the image of τ ′ is a Z-submodule of Zα(H). +Then the K-algebra A is also Frobenius with {gj : j ∈ I} and {hj : j ∈ I} giving dual bases. Again, there +is a Higman map τ : A → A, τ(x) = � +j∈I gjxhj, with image Im τ contained in Zα(A). In the case where A +is symmetric, this image is called the Higman ideal or projective centre of A. +4 + +By construction, there is a commutative diagram +H +A +Z +Z(A). +τ ′ +τ +ι +(2.3) +Note that the image of ι is Z′ by definition. +Lemma 2.5. Im τ ⊂ SocZ′Z′ +α. +Proof. First we note that (SocAA) ∩ Z′ +α ⊂ SocZ′Z′ +α. Next, by diagram (2.3), the image of τ is contained in +Z′ +α since the image of τ ′ is contained in Zα(H). On the other hand, it is explained in Section 2.4.2 of [17] +(see also [16, Proposition 2.20]) that the image of τ is contained in SocAA. The lemma follows. +□ +Let Z′ = B1 ⊕ · · ·⊕ Bl denote the blocks of Z′. Since Z′ is Gorenstein, Lemma 2.3 implies that the socles +of the indecomposable summands BiZ′ +α of Z′ +α are simple. +We note that Z(A) is K-split because we have assumed in (4) that A is K-split; see [13, Exercise 7.5]. +Since Z′ ⊂ Z(A), the algebra Z′ is also K-split [13]. Thus, every simple Z′-module is one-dimensional over +K and it follows from Lemma 2.5 that dimK Im τ ≤ ℓ(SocZ′Z′ +α), the length of SocZ′Z′ +α. By Lemma 2.3, the +latter equals ℓ(SocZ′Z′). Since we have assumed by (3) that Z′ is Frobenius, the number of blocks of Z′ +equals the number of simple modules in the socle of Z′. Thus, we have shown that +dimK Im τ ≤ |Bl(Z′)|. +Using once again (4) that A is split over K, the key result Theorem 3(a) of [17] says that dimK Im τ equals +the p-rank of the Cartan map CK. Assumption (5) implies that the p-rank of C (the rank of the map CK) +is at least as big as |Bl(A)|. M¨uller’s Theorem [6, Proposition 2.7] implies that |Bl(Z′)| = |Bl(A)|. Thus, +dimK Im τ ≥ |Bl(A)| and hence +dimK Im τ ≤ |Bl(Z′)| = |Bl(A)| ≤ dimK Im τ. +We deduce that |Bl(A)| = dimK Im τ equals the p-rank of A. Since the p-rank of each block is at least one, +we have proven Theorem 2.4. +We note that a consequence of the proof is that Im τ = SocZ′Z′ +α. +2.4. The Casimir map. We also have the Casimir map q: A → A given by q(a) = � +j hjagj. By [16, +Lemma 2.13], the image of q is contained in Z(A). It is a consequence of [16, Proposition 2.20] that both τ +and q vanish on rad A and have image in SocAA. +If H is the free Frobenius extension C[x]⋊ S2 of C[x2] and m = (x2) then A = C[x]/(x2)⋊ S2 is precisely +the example considered in Exercise 2.2.3 of [16]. This is also a special case of the graded Hecke algebras +considered in [7]. In this example, Im τ is one-dimensional whilst q is the zero map. Thus, the rank of τ +does not equal the rank of q in general for a Frobenius algebra. This example also shows that the image of +τ need not be central. +5 + +3. Examples +3.1. Triangular decompositions. In this section, we use freely the notation from [3]. We begin by justi- +fying the claim made in Section 1.1 that if H is a free Frobenius extension such that A is a graded algebra +with triangular decomposition then the multiplicity matrix M has the rank one property if and only if the +Cartan matrix C does. +To be precise, we assume that H is Z-graded, R a graded (central) subalgebra and m a homogeneous +maximal ideal of R such that A = H/mH admits a triangular decomposition A = A− ⊗ T ⊗ A+ as in [3, +Definition 3.1]. Let B± be the subalgebras of A generated by A± and T . +Lemma 3.1. Assume that T is semi-simple and B− ∼= (B+)⊛ as graded T -bimodules. Then A satisfies +the rank one property with respect to the multiplicity matrix M if and only if it does so with respect to the +Cartan matrix C. +Indeed, in this situation, C = M T M by BGG-reciprocity [3, Theorem 1.3]. Since M is integer (and hence +real) valued, the rank of C equals the rank of M and it follows that each block of M has rank one if and +only if each block of C does. +Thus, for graded algebras with a triangular decomposition, the rank one property can be encoded as +[∆(λ) : L(ρ)] dimK ∆(µ) = [∆(µ) : L(ρ)] dimK ∆(λ) +(3.1) +for all λ, ρ, µ ∈ IrrT . Indeed, the fact that each block of M has rank one implies that there exist rational +numbers aλ, bλ such that [∆(λ) : L(ρ)] = aλbρ. If d := � +µ bµ dimK L(µ), then dimK ∆(λ) = aλd. Hence, +both sides of (3.1) equal aλaµbρd. +In particular, Lemma 3.1 applies to H = Hc(W), the rational Cherednik algebra and m ⊳ R := C[h]W ⊗ +C[h∗]W the augmentation ideal. The quotient A = H/mH = Hc(W) is the restricted rational Cherednik +algebra. +3.2. Other examples. In this section, we assume all algebras are defined over an algebraically closed field of +characteristic zero. In addition to rational Cherednik algebras (I), most examples in [7] satisfy the hypothesis +of Theorem 2.4. Numbered as in [7], they are: +(II) Graded Hecke algebras with R = Z regular. +(III) The extended affine Hecke algebra with R = Z. +(IV) Affine nil-Hecke algebra with R = Z. +(V) Quantized enveloping algebra at an ℓth root of unity with R = Z0 the ℓ-centre. The centre is a +complete intersection ring (and hence Gorenstein) by [9, Theorem 21.3]. Moreover, it is shown in +the proof of [9, Theorem 21.5] that it is an integrally closed domain. +It is not clear to us when the remaining algebras (VI) Quantized function algebras and (VII) Quantum +Borels considered in [7] satisfy the hypothesis of Theorem 2.4; see for instance [11, Remark 5.4]. +We note that example (V) is Z-graded and the restricted quantum group A is equal to H/mH for a graded +maximal ideal m⊳R. Then A admits a triangular decomposition, implying that it has the rank one property +with respect to baby Verma modules. This was also shown in [2, Proposition 4.16(2)] using results from +the literature on the multiplicities [∆(λ) : L(µ)]. As explained in the proof of [2, Proposition 4.16(2)], this +implies that Lusztig’s small quantum group also satisfies the rank one property. +6 + +The results of [5] provide other important examples satisfying the hypothesis of Theorem 2.4. Firstly, we +may take H to be a non-commutative crepant resolution of an integrally closed Gorenstein domain over an +algebraically closed field of characteristic zero; see [5, Example 2.22]. Secondly, we may take H to be a 3 or +4 dimensional PI Sklyanin algebra [5, Example 2.24]. +The main result of [14] says that, for a filtered algebra, being a free Frobenius extension lifts from the +associated graded. This provides an effective way of checking the property for a large class of examples. +Finally, we expect that quiver Hecke algebras (KLR algebras) [12, 20] also satisfy the hypothesis of +Theorem 1.1. +3.3. Positive characteristic. It is natural to ask to what extent our result extends to algebras in positive, +or mixed, characteristic. In these situations, assumption (5) of Section 2.3 can fail. However, in the case +H = U(g), for g a simple Lie algebra over an algebraically closed field of characteristic p > 0, the second +Kac-Weisfeiler conjecture [18] provides an effective way of showing that (5) holds for regular characters χ. +For rational Cherednik algebras at t = 1, the result [1, Proposition 6.8] is similarly applicable. +Rather, it is assumption (6) that causes difficulties since the PI degree is always zero in K in these +examples. Recall that (6) is used to argue that Z and Zα are summand of H. It raises the question: is Z a +direct summand of H in either of these two examples? +References +[1] G. Bellamy and M. Martino. On the smoothness of centres of rational Cherednik algebras in positive characteristic. Glasg. +Math. J., 55(A):27–54, 2013. +[2] G. Bellamy and U. Thiel. Cores of graded algebras with triangular decomposition. arXiv, 1711.00780v1, 2017. +[3] G. Bellamy and U. Thiel. Highest weight theory for finite-dimensional graded algebras with triangular decomposition. Adv. +Math., 330:361–419, 2018. +[4] C. Bonnaf´e and R. Rouquier. Cherednik algebras and Calogero-Moser cells. arXiv, 1708.09764v3, 2017. +[5] A. Braun. On symmetric, smooth and Calabi-Yau algebras. J. Algebra, 317(2):519–533, 2007. +[6] K. A. Brown and I. G. Gordon. The ramification of centres: Lie algebras in positive characteristic and quantised enveloping +algebras. Math. Z., 238(4):733–779, 2001. +[7] K. A. Brown, I. G. Gordon, and C. H. Stroppel. Cherednik, Hecke and quantum algebras as free Frobenius and Calabi-Yau +extensions. J. Algebra, 319(3):1007–1034, 2008. +[8] W. Bruns and J. Herzog. Cohen-Macaulay rings, volume 39 of Cambridge Studies in Advanced Mathematics. Cambridge +University Press, Cambridge, 1993. +[9] C. De Concini and C. Procesi. Quantum groups. In D-modules, representation theory, and quantum groups (Venice, 1992), +volume 1565 of Lecture Notes in Math., pages 31–140. Springer, Berlin, 1993. +[10] P. Etingof and V. Ginzburg. Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homo- +morphism. Invent. Math., 147(2):243–348, 2002. +[11] I. Gordon. Representations of semisimple Lie algebras in positive characteristic and quantum groups at roots of unity. In +Quantum groups and Lie theory (Durham, 1999), volume 290 of London Math. Soc. Lecture Note Ser., pages 149–167. +Cambridge Univ. Press, Cambridge, 2001. +[12] M. Khovanov and A. D. Lauda. A diagrammatic approach to categorification of quantum groups II. Trans. Amer. Math. +Soc., 363(5):2685–2700, 2011. +[13] T. Y. Lam. A first course in noncommutative rings, volume 131 of Graduate Texts in Mathematics. Springer-Verlag, New +York, second edition, 2001. +[14] S. Launois and L. Topley. Transfer results for Frobenius extensions. J. Algebra, 524:35–58, 2019. +[15] M. Lorenz. Representations of finite-dimensional Hopf algebras. J. Algebra, 188(2):476–505, 1997. +7 + +[16] M. Lorenz. A tour of representation theory, volume 193 of Graduate Studies in Mathematics. American Mathematical +Society, Providence, RI, 2018. +[17] M. Lorenz and L. Fitzgerald Tokoly. Projective modules over Frobenius algebras and Hopf comodule algebras. Comm. +Algebra, 39(12):4733–4750, 2011. +[18] A. Premet. Irreducible representations of Lie algebras of reductive groups and the Kac-Weisfeiler conjecture. Invent. Math., +121(1):79–117, 1995. +[19] I. Reiner. Maximal orders. London Mathematical Society Monographs, No. 5. Academic Press, London-New York, 1975. +[20] R. Rouquier. Quiver Hecke algebras and 2-Lie algebras. Algebra Colloq., 19(2):359–410, 2012. +[21] U. Thiel. Champ: a Cherednik algebra Magma package. LMS J. Comput. Math., 18(1):266–307, 2015. +School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow, G12 8QQ. +Email address: gwyn.bellamy@glasgow.ac.uk +Department of Mathematics, University of Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany +Email address: thiel@mathematik.uni-kl.de +8 + diff --git a/tdE1T4oBgHgl3EQfjwQd/content/tmp_files/load_file.txt b/tdE1T4oBgHgl3EQfjwQd/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..75ac1814b2a8369ff9a0ff64e730e1a80784b66b --- /dev/null +++ b/tdE1T4oBgHgl3EQfjwQd/content/tmp_files/load_file.txt @@ -0,0 +1,457 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf,len=456 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='03265v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='RT] 9 Jan 2023 THE RANK ONE PROPERTY FOR FREE FROBENIUS EXTENSIONS GWYN BELLAMY AND ULRICH THIEL Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' A conjecture by the second author, proven by Bonnaf´e-Rouquier, says that the multiplicity matrix for baby Verma modules over the restricted rational Cherednik algebra has rank one over Q when restricted to each block of the algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' In this paper, we show that if H is a prime algebra that is a free Frobenius extension over a regular central subalgebra R, and the centre of H is normal Gorenstein, then each central quotient A of H by a maximal ideal m of R satisfies the rank one property with respect to the Cartan matrix of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Examples where the result is applicable include graded Hecke algebras, extended affine Hecke algebras, quantized enveloping algebras at roots of unity, non-commutative crepant resolutions of Gorenstein domains and 3 and 4 dimensional PI Skylanin algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' In particular, since the multiplicity matrix for restricted rational Cherednik algebras has the rank one property if and only if its Cartan matrix does, our result provides a different proof of the original conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' It was conjectured by the second author [21, Question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' 2(ii)] that baby Verma modules over the restricted rational Cherednik algebra satisfy a remarkable rank one property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Namely, if ∆(λ), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' L(λ), denotes the baby Verma module, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' irreducible module, associated to λ ∈ IrrW, then the multiplicity matrix M, given by Mλ,µ := [∆(λ) : L(µ)], has rank one over Q when restricted to each block of the algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' This conjecture was confirmed by Bonnaf´e- Rouquier [4, Proposition 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='2], whose ingenious proof makes use of a splitting extension of the centre of the (un-restricted) rational Cherednik algebra, together with properties the corresponding decomposition map to the restricted rational Cherednik algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Let C denote the Cartan matrix of the algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' The baby Verma modules for the restricted rational Cherednik algebra satisfy BGG-reciprocity, C = M T M, from which it is easily seen (Section 3) that the rank one property can equivalently be stated as saying that the Cartan matrix of each block of the restricted rational Cherednik algebra has rank one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' This reformulation makes no explicit mention of baby Verma modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' In this article we give a completely different proof of the rank one property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Our proof applies to a broader class of algebras, including for instance quantum groups at roots of unity, and makes use of the fact that these algebras are free Frobenius R-algebras, for an appropriate regular central subalgebra R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' In particular, our result applies to all but two of the families of examples considered in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' In the introduction, we assume for simplicity that K is an algebraically closed field of characteristic zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' in the body of the paper we prove a more general result that does not require this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' We start with H a (unital) affine K-algebra and central subalgebra R ⊂ H such that H is a free Frobenius extension of R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' 1 We assume that R is regular, H is prime and the centre Z := Z(H) of H is Gorenstein and integrally closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Let m be a maximal ideal of R and A = H/mH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' The K-algebra A is finite-dimensional and split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Let K0(A) be the Grothendieck group of finitely generated projective A-modules and G0(A) the Grothendieck group of all finitely generated A-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' The Cartan map is the canonical map C : K0(A) → G0(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' If Λ is a (finite) set parametrizing isomorphism classes of simple A-modules, with representatives L(λ) for each λ ∈ Λ, then we can think of C as a |Λ| × |Λ| matrix whose entries are the multiplicities [P(λ) : L(µ)], where P(λ) is the projective cover of L(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' If A = A1 ⊕ · · · ⊕ Ak is the block decomposition of A then C admits a corresponding decomposition C = C1 ⊕ · · · ⊕ Ck, where Ci is the Cartan matrix of Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' We say that the algebra A has the rank one property if each matrix Ci has rank one over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Though this appears at first to be a very strong property, our main result is: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' A has the rank one property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='1 centres around the Higman map τ′ : H → H, which descends to the Higman map τ : A → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' The key result is Theorem 3(a) of Lorenz-Fitzgerald Tokoly [17] (see also [15]), which says that the rank of the Cartan matrix C equals the dimension of Im τ as a K-vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' We show that this dimension equals the number of blocks of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' We do this by identifying Im τ with the socle of the image Z′ α in A of the Nakyama centre Zα of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' In the motivating case, H = Hc(W) is the rational Cherednik algebra at t = 0 associated to the complex reflection group (h, W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' This C-algebra contains R = C[h]W ⊗C[h∗]W as a central subalgebra and Z is an integrally closed Gorenstein ring [10, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='3, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' It has been shown in [7] that H is a (symmetric) free Frobenius extension of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' If m ⊂ C[h]W ⊗ C[h∗]W is the augmentation ideal, then the central quotient A = H/mH = Hc(W) is the restricted rational Cherednik algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Since this algebra satisfies BGG-reciprocity, the rank one property with respect to the Cartan matrix C is equivalent to the rank one property with respect to the multiplicity matrix M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='1 applies to many other examples commonly studied in representation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' These include graded Hecke algebras, extended affine Hecke algebras, affine nil-Hecke algebras, quantized enveloping alge- bras at roots of unity, non-commutative crepant resolutions of Gorenstein domains and 3 and 4 dimensional PI Skylanin algebras;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' see section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' The first author would like to thank Ken Brown and Lewis Topley for fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' The first author was partially supported by a Research Project Grant from the Leverhulme Trust and by the EPSRC grant EP-W013053-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' The proof We begin again, dropping any assumptions on the field K, unless specifically stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='1 will follow from the more general Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='4 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' The Nakayama centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' We start with H a (unital) ring and a central subring R ⊂ H such that H is a free Frobenius extension of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' By definition, this means that H is a finite free R-module and there is an isomorphism φ: 1Hα−1 ∼ −→ HomR(H, R) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='1) 2 of H-bimodules, for some automorphism α of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Here 1Hα−1 = H as left H-modules, but with right action given by h·a = hα−1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' The automorphism α is unique up to inner automorphisms and one can check that α|Z = IdZ, where Z := Z(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Let Zα(H) = {h ∈ H | ha = α(a)h for all a ∈ H};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' this is the Nakayama centre of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' We note that under the isomorphism φ of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='1), Zα(H) ∼→ HomR(H/[H, H], R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='2) Since α|Z = IdZ, we have an identification of Z-modules Zα(H) ∼= HomR(H/[H, H], R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Assume that H is a prime ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Then it is a prime PI ring since it is a finite R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Let d denote the PI degree of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Let T (H) denote the trace ring of H and Tr: H → T (H) the reduced trace;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' see [19, Section 9a] for the definition of these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' The following result is [5, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Assume that T (H) = H and d is invertible in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Then the map HomR(H/[H, H], R) ∼ −→ HomR(Z, R), f �→ f|Z, is an isomorphism of Z-modules, with inverse g �→ g ◦ (d−1Tr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Here the map d−1Tr: H → Z is a projection, realizing Z as a direct summand of H as Z-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Assume that T (H) = H and d is invertible in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Then Zα(H) is a direct summand of H as Z-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Since Zα(H) corresponds to HomR(H/[H, H], R) under the isomorphism φ, it suffices to show that HomR(H/[H, H], R) is a summand of HomR(H, R) as Z-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Consider the series of maps Z ֒→ H → H/[H, H] d−1Tr −→ Z which dualise to HomR(Z, R) (d−1Tr)∗ −→ HomR(H/[H, H], R) → HomR(H, R) ։ HomR(Z, R), where we have used the fact that Z is a summand of H to conclude that the last arrow is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='1 says that the composite map is an automorphism of HomR(Z, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Since (d−1Tr)∗ is an isomorphism, we deduce that HomR(H/[H, H], R) → HomR(H, R) is a split injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Let m ⊂ R be a maximal ideal, contained in the regular locus and let Z′ denote the image of Z in H/mH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Assume that: (1) The integer d is invertible as an element of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' (2) Z is Gorenstein and integrally closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Then Z/mZ is a Frobenius algebra, the morphism Z/mZ → Z′ is an isomorphism and Zα(H)/mZα(H) is a free rank one Z′-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' First we note that the fact that Z integrally closed implies that the trace ring T (H) of H equals H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' see [19, Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Then, as noted previously, the fact that the integer d is invertible as an element of H means that Z is a direct summand of H as a Z-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' It follows that the canonical map Z/mZ → Z′ is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' 3 Since H is a free R-module, it follows that Z is a projective R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Let Rm denote the localization of R at m and Zm = Z ⊗R Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Then Zm is a free Rm-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Therefore, we need to show that Z/mZ = Zm/mZm is Frobenius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Since Rm is assumed to be a regular local ring, mRm is generated by a regular sequence f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' , fk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' This sequence is also regular in Zm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' It follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='19(b) of [8] that Zm/mZm is zero-dimensional Gorenstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' This means it is a Frobenius algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' The hypothesis of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='1 hold and hence HomR(H/[H, H], R) ∼= HomR(Z, R) as Z-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Now, HomR(Z, R)m ∼= HomRm(Zm, Rm) as Zm-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Since H is prime, Z is a domain and hence equi- dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Also, R is a regular ring and thus Gorenstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Under these hypothesis, [5, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='6] says that Zm being Gorenstein is equivalent to HomRm(Zm, Rm) ∼= Zm as Zm-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Since HomR(H/[H, H], R) is isomorphic to Zα(H) by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='2), we deduce that Zα(H)m ∼= Zm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' This implies that Zα(H)/mZα(H) is iso- morphic to Z/mZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' The main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Let m be a maximal ideal of R and A = H/mH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Let K denote the residue field of R at m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Then A is a finite-dimensional K-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' We assume that A is split over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Let p ≥ 0 denote the characteristic of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='2 implies that the map Zα(H) → A realises the Z′-module Zα(H)/mZα(H) as a direct summand of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' The centre of A is denoted Z(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Clearly Z′ ⊂ Z(A), but in general the inclusion is strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' As in the introduction, A = A1 ⊕ · · · ⊕ Ak is the block decomposition of A and C = C1 ⊕ · · · ⊕ Ck the corresponding decomposition of the Cartan matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' We set CK : K0(A) ⊗Z K → G0(A) ⊗Z K with decomposition CK = CK,1 ⊕ · · · ⊕ CK,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' We call the rank of CK over K the p-rank of C (abuse of language).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' If the p-rank of every CK,i is non-zero then the p-rank of each CK,i is exactly one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' We note that if K has characteristic zero then it is automatic that the rank (= p-rank) of each block of C is at least one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Therefore, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='1 is a direct consequence of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Recall that we have assumed the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' (1) H is a free Frobenius R-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' (2) R is a regular ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' (3) Z is Gorenstein and integrally closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' (4) A := H/mH is split over K := R/m, for m ⊳ R maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' (5) The p-rank of every block Ci of C is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' (6) H is a prime ring whose PI degree is invertible in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' The isomorphsim (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='1) defines an R-linear pairing on H by (g, h) := φ(1)(gh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' We fix a pair {gj : j ∈ I} and {hj : j ∈ I} of dual R-bases for H, meaning that (gi, hj) = δi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' This allows us to define the Higman map τ ′ : H → H, τ ′(x) = � j∈I gjxhj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' By [16, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='13], the image of τ ′ is a Z-submodule of Zα(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Then the K-algebra A is also Frobenius with {gj : j ∈ I} and {hj : j ∈ I} giving dual bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Again, there is a Higman map τ : A → A, τ(x) = � j∈I gjxhj, with image Im τ contained in Zα(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' In the case where A is symmetric, this image is called the Higman ideal or projective centre of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' 4 By construction, there is a commutative diagram H A Z Z(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' τ ′ τ ι (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='3) Note that the image of ι is Z′ by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Im τ ⊂ SocZ′Z′ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' First we note that (SocAA) ∩ Z′ α ⊂ SocZ′Z′ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Next, by diagram (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='3), the image of τ is contained in Z′ α since the image of τ ′ is contained in Zα(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' On the other hand, it is explained in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='2 of [17] (see also [16, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='20]) that the image of τ is contained in SocAA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' The lemma follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' □ Let Z′ = B1 ⊕ · · ·⊕ Bl denote the blocks of Z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Since Z′ is Gorenstein, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='3 implies that the socles of the indecomposable summands BiZ′ α of Z′ α are simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' We note that Z(A) is K-split because we have assumed in (4) that A is K-split;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' see [13, Exercise 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Since Z′ ⊂ Z(A), the algebra Z′ is also K-split [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Thus, every simple Z′-module is one-dimensional over K and it follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='5 that dimK Im τ ≤ ℓ(SocZ′Z′ α), the length of SocZ′Z′ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='3, the latter equals ℓ(SocZ′Z′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Since we have assumed by (3) that Z′ is Frobenius, the number of blocks of Z′ equals the number of simple modules in the socle of Z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Thus, we have shown that dimK Im τ ≤ |Bl(Z′)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Using once again (4) that A is split over K, the key result Theorem 3(a) of [17] says that dimK Im τ equals the p-rank of the Cartan map CK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Assumption (5) implies that the p-rank of C (the rank of the map CK) is at least as big as |Bl(A)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' M¨uller’s Theorem [6, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='7] implies that |Bl(Z′)| = |Bl(A)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Thus, dimK Im τ ≥ |Bl(A)| and hence dimK Im τ ≤ |Bl(Z′)| = |Bl(A)| ≤ dimK Im τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' We deduce that |Bl(A)| = dimK Im τ equals the p-rank of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Since the p-rank of each block is at least one, we have proven Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' We note that a consequence of the proof is that Im τ = SocZ′Z′ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' The Casimir map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' We also have the Casimir map q: A → A given by q(a) = � j hjagj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' By [16, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='13], the image of q is contained in Z(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' It is a consequence of [16, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='20] that both τ and q vanish on rad A and have image in SocAA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' If H is the free Frobenius extension C[x]⋊ S2 of C[x2] and m = (x2) then A = C[x]/(x2)⋊ S2 is precisely the example considered in Exercise 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='3 of [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' This is also a special case of the graded Hecke algebras considered in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' In this example, Im τ is one-dimensional whilst q is the zero map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Thus, the rank of τ does not equal the rank of q in general for a Frobenius algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' This example also shows that the image of τ need not be central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Examples 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Triangular decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' In this section, we use freely the notation from [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' We begin by justi- fying the claim made in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='1 that if H is a free Frobenius extension such that A is a graded algebra with triangular decomposition then the multiplicity matrix M has the rank one property if and only if the Cartan matrix C does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' To be precise, we assume that H is Z-graded, R a graded (central) subalgebra and m a homogeneous maximal ideal of R such that A = H/mH admits a triangular decomposition A = A− ⊗ T ⊗ A+ as in [3, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Let B± be the subalgebras of A generated by A± and T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Assume that T is semi-simple and B− ∼= (B+)⊛ as graded T -bimodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Then A satisfies the rank one property with respect to the multiplicity matrix M if and only if it does so with respect to the Cartan matrix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Indeed, in this situation, C = M T M by BGG-reciprocity [3, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Since M is integer (and hence real) valued, the rank of C equals the rank of M and it follows that each block of M has rank one if and only if each block of C does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Thus, for graded algebras with a triangular decomposition, the rank one property can be encoded as [∆(λ) : L(ρ)] dimK ∆(µ) = [∆(µ) : L(ρ)] dimK ∆(λ) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='1) for all λ, ρ, µ ∈ IrrT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Indeed, the fact that each block of M has rank one implies that there exist rational numbers aλ, bλ such that [∆(λ) : L(ρ)] = aλbρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' If d := � µ bµ dimK L(µ), then dimK ∆(λ) = aλd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Hence, both sides of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='1) equal aλaµbρd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' In particular, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='1 applies to H = Hc(W), the rational Cherednik algebra and m ⊳ R := C[h]W ⊗ C[h∗]W the augmentation ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' The quotient A = H/mH = Hc(W) is the restricted rational Cherednik algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Other examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' In this section, we assume all algebras are defined over an algebraically closed field of characteristic zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' In addition to rational Cherednik algebras (I), most examples in [7] satisfy the hypothesis of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Numbered as in [7], they are: (II) Graded Hecke algebras with R = Z regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' (III) The extended affine Hecke algebra with R = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' (IV) Affine nil-Hecke algebra with R = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' (V) Quantized enveloping algebra at an ℓth root of unity with R = Z0 the ℓ-centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' The centre is a complete intersection ring (and hence Gorenstein) by [9, Theorem 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Moreover, it is shown in the proof of [9, Theorem 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='5] that it is an integrally closed domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' It is not clear to us when the remaining algebras (VI) Quantized function algebras and (VII) Quantum Borels considered in [7] satisfy the hypothesis of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' see for instance [11, Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' We note that example (V) is Z-graded and the restricted quantum group A is equal to H/mH for a graded maximal ideal m⊳R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Then A admits a triangular decomposition, implying that it has the rank one property with respect to baby Verma modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' This was also shown in [2, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='16(2)] using results from the literature on the multiplicities [∆(λ) : L(µ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' As explained in the proof of [2, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='16(2)], this implies that Lusztig’s small quantum group also satisfies the rank one property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' 6 The results of [5] provide other important examples satisfying the hypothesis of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Firstly, we may take H to be a non-commutative crepant resolution of an integrally closed Gorenstein domain over an algebraically closed field of characteristic zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' see [5, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Secondly, we may take H to be a 3 or 4 dimensional PI Sklyanin algebra [5, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' The main result of [14] says that, for a filtered algebra, being a free Frobenius extension lifts from the associated graded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' This provides an effective way of checking the property for a large class of examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Finally, we expect that quiver Hecke algebras (KLR algebras) [12, 20] also satisfy the hypothesis of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Positive characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' It is natural to ask to what extent our result extends to algebras in positive, or mixed, characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' In these situations, assumption (5) of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='3 can fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' However, in the case H = U(g), for g a simple Lie algebra over an algebraically closed field of characteristic p > 0, the second Kac-Weisfeiler conjecture [18] provides an effective way of showing that (5) holds for regular characters χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' For rational Cherednik algebras at t = 1, the result [1, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='8] is similarly applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Rather, it is assumption (6) that causes difficulties since the PI degree is always zero in K in these examples.' 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+page_content=' School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow, G12 8QQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content=' Email address: gwyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='bellamy@glasgow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='uk Department of Mathematics, University of Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany Email address: thiel@mathematik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='uni-kl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} +page_content='de 8' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfjwQd/content/2301.03265v1.pdf'} diff --git a/vdFPT4oBgHgl3EQf-DWo/content/tmp_files/2301.13214v1.pdf.txt b/vdFPT4oBgHgl3EQf-DWo/content/tmp_files/2301.13214v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6924aa2377651d038a2fb82690735dba6d67adb3 --- /dev/null +++ b/vdFPT4oBgHgl3EQf-DWo/content/tmp_files/2301.13214v1.pdf.txt @@ -0,0 +1,2351 @@ +MNRAS 000, 1–18 (2023) +Preprint 1 February 2023 +Compiled using MNRAS LATEX style file v3.0 +The extended "stellar halo" of the Ursa Minor dwarf galaxy +Federico Sestito1⋆, Daria Zaremba1,2, Kim A. Venn1, Lina D’Aoust1, Christian Hayes3, +Jaclyn Jensen1, Julio F. Navarro1, Pascale Jablonka4,5, Emma Fernández-Alvar6,7, +Jennifer Glover1, Alan W. McConnachie3,1 , and André-Nicolas Chené8,9 +1 Department of Physics and Astronomy, University of Victoria, PO Box 3055, STN CSC, Victoria BC V8W 3P6, Canada +2 National University of Kyiv-Mohyla Academy, 04655 Kyiv, Ukraine +3 NRC Herzberg Astronomy & Astrophysics, 5071 West Saanich Road, Victoria, BC V9E 2E7, Canada +4 Laboratoire d’astrophysique, École Polytechnique Fédérale de Lausanne (EPFL), Observatoire, CH-1290 Versoix, Switzerland +5 GEPI, Observatoire de Paris, Université PSL, CNRS, 5 Place Jules Janssen, F-92195 Meudon, France +6 Instituto de Astrofısica de Canarias, E-38200 La Laguna, Tenerife, Spain +7 Dept. Astrofısica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain +8 Gemini Observatory/NSF’s NOIRLab, 670 N. A’ohoku Place, Hilo, Hawai’i, 96720, USA +9 Visiting astronomer at the Université de Montréal, Complexe des Sciences, Montréal, QC H2V 0B3, Canada +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +Five stars in the extreme outskirts (from ∼ 5 to ∼ 12 elliptical half-light radii, rh) of +the Ursa Minor (UMi) dwarf galaxy have been identified as potential new members +using a Bayesian algorithm applied to Gaia EDR3 data. These targets were observed +with the GRACES spectrograph, resulting in precise radial velocities and metallic- +ities that confirm their association with UMi. For the brightest and outermost star +(Target 1, at ∼ 12 rh), the chemical abundances of α- (Mg, Ca, Ti), odd-Z (Na, K, +Sc), Fe-peak (Fe, Ni, Cr), and neutron-capture process (Ba) elements have also been +determined. We also discuss data from the literature and from APOGEE DR17. We +find the chemical patterns in UMi are consistent with a star formation history that +includes yields from core collapse supernovae, asymptotic giant branch stars, and su- +pernovae Ia. Evidence for a knee in the [α/Fe] ratios near [Fe/H] ∼ −2.1 indicates a +low star formation efficiency similar to that in other dwarf galaxies. Given the dis- +tance of Target 1 from the centre of UMi (R∼4.5 kpc), we show that UMi has a more +extended structure than previously thought. This "stellar halo" around UMi could +be a secondary feature resulting from tidal stripping after multiple orbits around the +Galaxy, or maybe a primary UMi feature due to early hierarchical accretion activity +or to strong gravitational fluctuations prompted by feedback in the early star forma- +tion phase. Also consistent with observations is a late-time merger-free scenario where +outside-in star formation is accompanied by gradual supernovae Ia enrichment. +Key words: stars: abundances – stars: Population II – galaxies : formation – galaxies: +dwarf – galaxies: individual: Ursa Minor – galaxies: evolution +1 +INTRODUCTION +Dwarf satellites of the Milky Way (MW) are amongst the +oldest and most metal-poor galaxies known (e.g., +Tolstoy +et al. 2009). They are at the low-mass end of the hierar- +chical formation process, just massive enough to form very +metal-poor stars (VMP, [Fe/H] ≤ −2.0, Simon 2019). The +mass of faint dwarf galaxies is dominated by dark matter +(e.g., +Simon 2019). In fact, their dynamical mass-to-light +ratios (M/L) can exceed 1000. They remain one of the best +⋆ Email: sestitof@uvic.ca +targets for studies seeking to understand the properties of +dark matter and early events in the formation our Galaxy +(e.g., Bullock & Boylan-Kolchin 2017). +Hierarchical +formation +of +Λ−Cold +Dark +Matter +(Λ−CDM) cosmology (e.g., White & Rees 1978; Frenk et al. +1988; Navarro et al. 1997) predicts that haloes grow from +the accretion of smaller systems. Therefore, galaxies should +possess an extended stellar halo built from disrupted sys- +tems. A stellar halo is clearly observed in large galaxies as +the Milky Way, but it remain elusive and poorly studied +in dwarf galaxies (e.g., Deason et al. 2022, and references +therein). One reason is because the fraction of mass assem- +© 2023 The Authors +arXiv:2301.13214v1 [astro-ph.GA] 30 Jan 2023 + +2 +F. Sestito et al. +bled through mergers is reduced at the dwarf galaxy mass +scales, while ‘smooth’ accretion dominates at this regime +(e.g., Genel et al. 2010). Second, while at the Milky Way- +size the stellar mass to halo mass ratio is well modeled, on +the dwarf size is not the case (e.g., Moster et al. 2013, and +references therein). +Given their shallow gravitational potential, faint dwarf +galaxies are extremely susceptible to internal processes, such +as star formation and the subsequent stellar feedback (e.g., +El-Badry et al. 2018); and external, such as mergers (e.g., +Deason et al. 2014), ram pressure stripping (e.g., +Grebel +et al. 2003) and stirring (e.g., Kazantzidis et al. 2011), tidal +interaction (e.g., Fattahi et al. 2018), and reionization (e.g., +Wheeler et al. 2019). All of these processes may act to influ- +ence their individual morphologies (e.g., Higgs et al. 2021, +and references therein). Signatures of these gravitational in- +teractions will be most evident in the outskirts of the dwarf +galaxy, where accreted remnants can show up as an excess +of stars over and above expectations from a simple single- +component model (akin to a stellar halo in a more massive +galaxy). +Stars in the extreme outskirts of dwarf galaxies, and yet +which are not clearly associated with prominent tidal tails, +have been discovered only relatively recently. Chiti et al. +(2021) spectroscopically identified member stars up to ∼9 +half-light radii (rh), or physical distances up to 1 kpc, away +from the centre of the faint dwarf galaxy, Tucana II. Dy- +namical analysis, as well as chemical abundances, were used +to distinguish between a tidal origin, where stars were re- +moved from the main body due to tidal effects with the +MW, and an accreted, i.e., dwarf-dwarf merger, origin. The +stars identified by Chiti et al. (2021) were found to be ex- +tremely metal deficient compared to the main body, suggest- +ing that the outskirts had a different origin from the bulk of +stars in Tucana II, perhaps due to an early merger with a +low-mass, metal-poor companion. Recently, Longeard et al. +(2022) analysed the chemo-dynamical properties of Boötes I, +suggesting that the system could have been more massive +than nowadays and that tidal stripping is largely affecting +the satellite. +Inspired by Chiti et al. (2021), we have examined other +MW satellites to help constrain the frequency of such stellar +halos around dwarfs. McConnachie & Venn (2020b,a) de- +veloped a Bayesian method, and updated by Jensen et al. +(prep), to estimate the probability that a star in the vicinity +of a dwarf galaxy is a member of the dwarf, using the full +astrometric and photometric data from Gaia EDR3 (Gaia +Collaboration et al. 2021). Jensen et al. (prep) report that +only a few dwarf galaxies out of nearly 60 examined suggest +evidence for an extended stellar halo. The systems already +in the literature include Tucana II, as examined by Chiti +et al. (2021), Sculptor (Sestito et al. prep), and also Coma +Berenices, Ursa Major I, and Boötes I (see also Longeard +et al. 2022), recently analysed by Waller et al. (2023). +Waller et al. (2023) showed that stars in Coma +Berenices have been polluted by supernovae type Ia, in con- +trast to previous views of this system. Waller et al. (2023) +discussed that the chemistry of the outermost stars in these +systems is consistent with their formation in the central re- +gions, then moving them to their current locations through +tidal stripping and/or supernova feedback, although in the +case of Boötes I the lower metallicities and lack of strong +carbon enrichment of its outermost stars could also be ev- +idence of a late dwarf-dwarf merger. Although the detailed +and precise chemical abundance analysis a firmer conclusion +on the origin of the outermost stars is hard to pinpoint. +In this work, we use this Bayesian algorithm to search +for member stars in the outermost regions of the dwarf +galaxy Ursa Minor (UMi). We make use of recent updates +to the algorithm by Jensen et al. (prep), which allow for the +presence of a secondary, extended component (i.e., an outer +stellar halo). Previously, Piatek et al. (2005) has suggested +the presence of tidal effects on the substructure of UMi. +Ursa Minor is historically a well-studied system. Some +controversies remain regarding the star formation history +(SFH) and its efficiency. For example, Carrera et al. (2002) +suggested that up to ∼95 per cent of UMi stars are older +than 10 Gyr, invoking an episodic SFH at early times. This +is based on studies of its colour-magnitude diagram (e.g., +Mighell & Burke 1999; Bellazzini et al. 2002). Other models +interpreted the chemical properties of UMi as due to ex- +tended SFH, from 3.9 and 6.5 Gyr (Ikuta & Arimoto 2002; +Ural et al. 2015). Kirby et al. (2011, 2013) matched the +wide metallicity distribution function (MDF) of UMi with +a chemical evolution model that includes infall of gas. On +the other hand, Ural et al. (2015) developed three chemical +evolution models, showing that winds from supernovae are +needed to describe UMi’s MDF, especially to reproduce stars +at higher metallicities. The authors underline that winds +help to explain the absence of gas at the present time. In +agreement with Ikuta & Arimoto (2002), their models use +an extended low-efficiency SFH duration (5 Gyr, Ural et al. +2015). +Ural et al. (2015) argued that is not easy to discern if the +[α/Fe] displays a plateau up to [Fe/H]∼ −2.0, or whether +it shows a gradual decrease. However, they conclude that +a slow decrease is present above this metallicity, pointing +to the contribution of supernovae type Ia (SNe Ia). On the +other hand, Cohen & Huang (2010) noted that a very short +duration of star formation (∼ 2 Gyr) implies that SNe Ia +did not have enough time to contribute to the chemical evo- +lution of UMi. More recent studies discovered that SNe Ia +can occur in the very first 2 Gyr of the Universe (e.g., Maoz +et al. 2012, 2014; de los Reyes et al. 2020; Kobayashi et al. +2020). The Λ−CDM cosmological zoom-in simulations de- +veloped by Revaz & Jablonka (2018) that incorporate gas +cooling found that the star formation and chemical evolu- +tion of UMi can be explained. In particular, when SNe Ia and +II events are taken into account with thermal blastwave-like +feedback (Revaz & Jablonka 2018, and references therein), +then they can reproduce the observed distribution in metal- +licity, [Mg/Fe], and the radial velocity dispersion with a +short star formation of only 2.4 Gyr. +In this paper, we present a chemo-dynamical inves- +tigation of stars in the extreme outskirts of UMi ob- +served with high-resolution GRACES spectrograph at Gem- +ini North/CFHT. Our results combined with spectroscopic +results for additional stars in the literature are used to dis- +cuss the extended chemical and dynamical evolution of UMi. +The target selection, the observations, and the spectral re- +duction are reported in Section 2. Stellar parameters are +inferred in Section 3. The model atmosphere and chemi- +cal abundance analysis for Target 1 are reported in Sec- +tion 4 and 5, respectively. Section 6 describes the measure- +MNRAS 000, 1–18 (2023) + +Extreme outskirt of Ursa Minor +3 +ment of [Fe/H] using Ca Triplet lines for Target 2–5. The +inference of the orbital parameters of UMi is described in +Section 7. The chemo-dynamical properties of Ursa Minor +are discussed in Section 8. +2 +DATA +2.1 +Target selection +Using the Bayesian algorithm from McConnachie & Venn +(2020b), with updates from Jensen et al. (prep), we have +searched for stars that inhabit the extended stellar halo of +Ursa Minor. Briefly, this algorithm provides the probability +for any star in Gaia to be a member of a given MW satel- +lite or to belong to the MW halo. The total likelihood is a +function of the position of the star on the sky, on the colour- +magnitude diagram, and in proper motion space (thus, no +radial velocity or metallicity information is used). This al- +gorithm has proved useful to identify new members in the +extreme outskirt of some ultra-faint dwarf galaxies (Waller +et al. 2023; Sestito et al. prep) and performs excellently to +remove Milky Way foreground contamination (Jensen et al. +prep). +In this work, we further validate this identification +method by examining the extreme outskirts of UMi. All +stars with a high probability (> 80%) of being associated +to UMi, and at a distance greater than 5 half-light radii +(≳ 85 arcmin or ≳ 900 pc) from the centre of the dwarf, +were selected. This included five red giants with magnitudes +in the range 17.4 ≤ G ≤ 18.3 mag in the Gaia EDR3 G +band. The brightest target is also the farthest in projection, +reaching an extreme distance of 11.7 half-light radii from +the centre of UMi. Our other four targets, at a distance of +5.2 − 6.3 rh, are also listed as highly likely UMi candidates +by Qi et al. (2022, with a probability > 90 percent). The +main properties of UMi and our five targets are reported in +Tables 1 and 2, respectively. +The position of our five candidates together with other +known UMi members are shown in Figure 1 in projected +sky coordinates, on the colour-magnitude diagram, and in +proper motion space. This figure shows that even if the can- +didates are located far from the centre of UMi, the algo- +rithm is very efficient in selecting new candidate members +in the very outskirts of the system. We gather UMi members +from Spencer et al. (2018), Pace et al. (2020), and APOGEE +data release 17 (DR17, Abdurro’uf et al. 2022) and then +cross-match with Gaia EDR3 to retrieve coordinates, proper +motion, and photometry. When we examine the APOGEE +DR17 targets, we have applied our selection algorithm to +select the stars with high membership probability (> 70 %) +and with high signal-to-noise in their spectra (SNR > 70). +Surprisingly, two stars from APOGEE DR17 have an ellip- +tical distance of ∼ 7 rh. We note that the [Fe/H] values +for these two stars are at the edge of the metallicity grid +of APOGEE; thus, while their radial velocity measurements +are precise, their true [Fe/H] could be lower, in turn affecting +their [X/Fe] ratios. +Table 1. Galactic parameters of Ursa Minor. The coordinates +α, δ, the mean metallicity, the mean radial velocity, the velocity +dispersion, the heliocentric distance D⊙, the ellipticity, the posi- +tion angle φ, and the half-light radius rh in arcmin and pc, the +mean proper motion from Gaia EDR3, the dynamical mass, the +mass density, and the luminosity are reported with the respective +references. (a) refers to McConnachie (2012), (b) to McConnachie +& Venn (2020b), (c) to McConnachie & Venn (2020a), (d) to Qi +et al. (2022), and (e) to Mateo (1998). +Property +Value +Reference +α +227.2854 deg +(b) +δ +67.2225 deg +(b) +[Fe/H] +−2.13 ± 0.01 +(b) +RV +246.9 ± 0.1 km s−1 +(b) +σV +9.5 ± 1.2 km s−1 +(b) +D⊙ +76 ± 10 kpc +(a) +ellipticity +0.55 ± 0.01 +(b) +φ +50 ± 1 deg +(b) +rh +17.32 ± 0.11 arcmin +(b) +rh +382 ± 53 pc +(b) +rh,plummer +407 pc +(d) +µαcosδ +−0.124 ± 0.004 mas yr−1 +(c) +µδ +0.078 ± 0.004 mas yr−1 +(c) +Mdyn(≤ rhalf) +9.5 × 106 M⊙ +(a) +Mass density +0.35 M⊙ pc−3 +(e) +L +0.29 × 106 L⊙ +(e) +2.2 +GRACES observations +Targets were observed with the Gemini Remote Access to +CFHT ESPaDOnS Spectrograph (GRACES, Chene et al. +2014; Pazder et al. 2014) using the 2-fibre (object+sky) +mode with a resolution of R∼ 40000. GRACES consists a +270-m optical fibre that connects the Gemini North tele- +scope to the Canada–France–Hawaii Telescope ESPaDOnS +cross-dispersed high resolution échelle spectrograph (Donati +et al. 2006). The targets were observed within the GN- +2022A-Q-128 program (P.I. F. Sestito). +For the brightest target (Target 1, G= 17.4 mag), which +is also the farthest one from the centre (∼ 11.7 +rh), we +achieved a spectrum with SNR per resolution element of +∼ 30 at the Ba ii 6141 Å region. This spectrum has suf- +ficient SNR to measure the abundances for additional ele- +ments, specifically the α− (Mg, Ca, Ti), odd−Z (Na, K, Sc), +Fe−peak (Fe, Cr, Ni), and neutron−capture process (Ba) +elements across the entire GRACES spectral coverage. We +refer to this observational set-up as the “high-SNR mode”. +For the remaining four targets, which have distances from 5 +− 7 rh, a SNR per resolution element of ∼ 20 in the Ca ii +T region (∼8550 Å) was obtained for precise radial veloci- +ties and metallicities. In this “low-SNR mode”, the metallic- +ities are derived from the equivalent width (EW) of the NIR +Ca ii T, as described in Section 6. Observing information +is summarized in Table 3, including the signal-to-noise ratio +measured at the Mg i b, Ba ii 614nm, and Ca ii T regions. +2.3 +Spectral reductions +The GRACES spectra were first reduced using the Open +source Pipeline for ESPaDOnS Reduction and Analysis +MNRAS 000, 1–18 (2023) + +4 +F. Sestito et al. +Table 2. The Gaia EDR3 source ID, the coordinates (α, δ), the projected coordinates (ξ, η), the elliptical radius distance rell in rh +unit, the probability to be a member from Jensen et al. (prep), the Gaia EDR3 photometry G and BP−RP, and the reddening AV from +Schlafly & Finkbeiner (2011) are reported for each target. +Target +source id +α +δ +ξ +η +rell +Psat +G +BP−RP +AV +(deg) +(deg) +(deg) +(deg) +(rh) +(mag) +(mag) +(mag) +Target 1 +1647329728514964480 +234.45303 +69.29204 +2.53226 +2.21888 +11.67 +0.80 +17.39 +1.29 +0.08 +Target 2 +1693464785444020224 +224.67731 +67.35983 +−1.00378 +0.15842 +6.34 +0.97 +17.83 +1.19 +0.06 +Target 3 +1693573430936780032 +226.08983 +67.77965 +−0.45214 +0.56153 +5.55 +0.96 +17.91 +1.19 +0.05 +Target 4 +1669324938936435200 +224.50756 +66.21361 +−1.12033 +−0.98413 +5.17 +0.94 +18.25 +1.17 +0.06 +Target 5 +1645948119139534336 +230.43949 +68.29581 +1.16629 +1.10328 +5.60 +0.92 +18.29 +1.17 +0.06 +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +(BP-RP)0 (mag) +16.0 +16.5 +17.0 +17.5 +18.0 +18.5 +19.0 +19.5 +20.0 +G0 (mag) +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 + (mas yr +1) +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 + (mas yr +1) +3 +2 +1 +0 +1 +2 +3 + (deg) +3 +2 +1 +0 +1 +2 +3 + (deg) +3 +2 +1 +0 +1 +2 +3 +D (kpc) +3 +2 +1 +0 +1 +2 +3 +D (kpc) +Figure 1. Ursa Minor seen through Gaia EDR3. All the panels: Target 1 is marked with a red diamond, while black diamonds are +Target 2–5. Magenta circles are UMi literature stars from Spencer et al. (2018) and Pace et al. (2020). Blue squares are UMi stars selected +from APOGEE DR17. MW foreground stars are marked with grey small dots. These are selected from Gaia EDR3 in the direction of +UMi and within the field of view of the η − ξ panel. Left panel: Projected sky coordinates and projected distance from UMi centre. The +orange ellipses denotes the elliptical distances from UMi centre of 3, 5, 7, and 11 rh. The arrow points in the direction of UMi proper +motion. Central panel: Colour-magnitude diagram. Dark green dashed lines is a Padova isochrone at [Fe/H] = −2.0 and age of 12 Gyr +(Bressan et al. 2012). Right panel: Proper motion space. +Table 3. Total exposure time, number of exposures, signal-to- +noise ratio (SNR) measured at the Mg i 518nm, Ba ii 614nm, and +Ca ii 850nm regions, and the observation dates are reported for +each target. The SNR is defined as the ratio between the median +flux and its standard deviation in given spectral region. +Target +texp +Nexp +SNR +SNR +SNR +Obs. date +(s) +@Mg ib +@Ba ii +@Ca iiT +YY/MM/DD +Target 1 +14400 +6 +9 +27 +37 +22/06/18 +Target 2 +1800 +1 +5 +12 +17 +22/03/14 +Target 3 +1800 +1 +1 +6 +8 +22/03/14 +Target 4 +2400 +1 +2 +6 +11 +22/06/17 +Target 5 +2400 +1 +1 +5 +10 +22/06/17 +(OPERA, Martioli et al. 2012) tool, which also corrects for +heliocentric motion. Then the reduced spectra were post- +processed following an updated procedure of the pipeline +described in Kielty et al. (2021). The latter pipeline allows +us to measure the radial velocity of the observed star, to +co-add multiple observations, to check for possible radial ve- +locity variations, to correct for the motion of the star, and +to eventually re-normalise the flux. This procedure also im- +proves the signal-to-noise ratio in the overlapping spectral +order regions without downgrading the spectral resolution. +Radial velocities are reported in Table 4. +This procedure failed for one of the spectral orders of +Target 1 covering the Mg i b region for reasons that we +could not overcome within the scope of this project. We +therefore extracted the data for Target 1 ourselves using +DRAGraces1 IDL code (Chené et al. 2021). +The final spectra for all five targets near the Na i Dou- +blet (left) and in the NIR Ca ii Triplet (right) regions are +shown in Figure 2. The quality of the spectra indicates that +the adopted exposure time were sufficient for the requested +science, i.e., chemical abundances for Target 1, and [Fe/H] +and RV only for Targets 2−5. +3 +STELLAR PARAMETERS +Given the low SNR of our spectra, we use the InfraRed +flux method (IRFM) from González Hernández & Bonifa- +cio (2009) with photometry from Gaia EDR3 to find the ef- +1 https://github.com/AndreNicolasChene/DRAGRACES/ +releases/tag/v1.4 +MNRAS 000, 1–18 (2023) + +- +..2 +: +- +- +*+ += +- +- +- +:3 += +.:-. +i. +- ++** +: +1 ++ +** +-. +" +. +.-: +-=Extreme outskirt of Ursa Minor +5 +5888 +5890 +5892 +5894 +5896 +wavelength (A) +0 +1 +2 +3 +4 +5 +6 +Flux +8536 +8538 +8540 +8542 +8544 +8546 +8548 +wavelength (A) +0 +1 +2 +3 +4 +5 +6 +Flux +Target 1 +Target 2 +Target 3 +Target 4 +Target 5 +Figure 2. GRACES spectra for the five new UMi member stars. Left panel: Na i Doublet region. Chemical abundance ratios are +measurable only for Target 1 given the low SNR of Targets 2–5. Right panel: The second component of the Ca ii Triplet. This spectral +line is used to infer [Fe/H] (see Section 6). +fective temperatures, adopting the Mucciarelli et al. (2021) +colour-temperature relationship for giants. The input pa- +rameters are the Gaia EDR3 (BP − RP) de-reddened colour +and a metallicity estimate. The 2D Schlafly & Finkbeiner +(2011) map2 has been used to correct the photometry +for extinction3. As input metallicities, we adopt the value +[Fe/H] = −2.0 ± 0.5, compatible with the metallicity distri- +bution in UMi. +Surface +gravities +were +found +using +the +Stefan- +Boltzmann equation4. This step required the effective tem- +perature, the distance of the object, the Gaia EDR3 G de- +reddened photometry, and the bolometric corrections on the +flux (Andrae et al. 2018) as input. A Monte Carlo algorithm +has been applied to the input parameters with their uncer- +tainties to estimate the total uncertainties on the stellar pa- +rameters. The input quantities were randomised within 1σ +using a Gaussian distribution, except for the stellar mass. +The latter is treated with a flat prior from 0.5 to 0.8 M⊙, +2 https://irsa.ipac.caltech.edu/applications/DUST/ +3 To convert from the E(B-V) map to Gaia extinction coeffi- +cients, the AV/E(B − V) = 3.1 (Schultz & Wiemer 1975) and +the AG/AV = 0.85926, ABP/AV = 1.06794, ARP/AV = 0.65199 +relations (Marigo et al. 2008; Evans et al. 2018) are used. +4 L⋆ = 4πR2 +⋆σT 4 +⋆ ; the radius of the star can be calculated from +this equation, then the surface gravity is inferred assuming the +mass. +Table 4. Stellar parameters of the five targets. [Fe/H] for Target 1 +is from Fe i and Fe ii lines, while for the other stars is from Ca ii +Triplet lines. +Target +RV +Teff +log g +[Fe/H] +(km s−1) +(K) +Target 1 +−256.91 ± 0.05 +4604 ± 94 +1.15 ± 0.08 +−2.09 ± 0.09 +Target 2 +−265.26 ± 1.89 +4771 ± 93 +1.43 ± 0.07 +−2.79 ± 0.15 +Target 3 +−218.78 ± 1.82 +4760 ± 100 +1.45 ± 0.08 +−2.67 ± 0.08 +Target 4 +−245.63 ± 1.78 +4795 ± 85 +1.60 ± 0.07 +−2.85 ± 0.10 +Target 5 +−247.29 ± 1.63 +4814 ± 100 +1.61 ± 0.08 +−2.30 ± 0.20 +which is consistent with the mass of long-lived very metal- +poor stars. The mean uncertainty on the effective tempera- +ture is ∼ 94 K, while on the surface gravity it is ∼ 0.08 dex. +This method has been shown to provide reliable stellar pa- +rameters suitable for spectroscopic studies of very metal- +poor stars (e.g., Kielty et al. 2021; Sestito et al. 2023; Waller +et al. 2023). The stellar parameters are reported in Table 4. +4 +MODEL ATMOSPHERE ANALYSIS +In this Section, we describe the model atmospheres, the +method, and the atomic data for our spectral line list +adopted to determine detailed chemical abundances for Tar- +get 1. +MNRAS 000, 1–18 (2023) + +6 +F. Sestito et al. +4.1 +Model atmospheres +Model atmospheres are generated from the MARCS5 mod- +els (Gustafsson et al. 2008; Plez 2012); in particular, we +selected the OSMARCS spherical models as Target 1 is a +giant with log(g)< 3.5. An initial model atmosphere was +generated using the derived stellar parameters, a metallicity +[Fe/H] = −2.0, and microturbulence velocity scaled by the +surface gravity from the calibration by Mashonkina et al. +(2017) for giants. +4.2 +The lines list and the atomic data +Spectral lines were selected from our previous analyses of +very metal-poor stars in the Galactic halo and other nearby +dwarf galaxies observed with GRACES (Kielty et al. 2021; +Sestito et al. 2023; Waller et al. 2023). Atomic data is taken +from linemake6 (Placco et al. 2021), with the exception of +K i lines taken from the National Institute of Standards and +Technology (NIST, Kramida et al. 2021)7. +4.3 +Spectral line measurements +Spectral line measurements are made using spectrum syn- +thesis, broadened with a Gaussian smoothing kernel of +FWHM = 0.15, which matches the resolution of the +GRACES 2-fibre mode spectra) in a four-step process: (1) +the synthesis of the [Fe/H] lines in our initial line list (see +above) is carried out using an initial model atmosphere and +invoking the MOOG8 spectrum synthesis program (Sneden +1973; Sobeck et al. 2011); (2) a new [Fe/H] is determined by +removing noisy lines; (3) the model atmosphere is updated +with the new [Fe/H] as metallicity; (4) the chemical abun- +dances are derived using the updated model atmosphere and +our full line list. The final chemical abundance is given by +the average measurement in case of multiple spectral lines. +4.4 +Checking the stellar parameters +Excitation equilibrium in the line abundances of Fe i is a +check on the quality of the effective temperature. For Tar- +get 1, the slope in A(Fe i) − Excitation potential (EP) from +the linear fit has a value of −0.027 dex eV−1. This value +is smaller than the dispersion in the measurements of the +chemical abundances (∼ 0.2 dex) over the range in EP (∼4 +eV). Thus, we conclude our effective temperature estimates +are sufficient from the IRFM. +Ionization balance between Fe i − Fe ii is widely used +as a sanity check on the surface gravity estimates (e.g., +Mashonkina et al. 2017). However, Karovicova et al. (2020) +have strongly advised against using this method for very +metal-poor giants. They used interferometric observations of +metal-poor stars to find radii, and subsequently precise stel- +lar parameters for a set of metal-poor benchmark stars. With +their stellar parameters, they have found that deviations in +Fe i − Fe ii can reach up to ∼ 0.8 dex. This effect is the +5 https://marcs.astro.uu.se +6 https://github.com/vmplacco/linemake +7 NIST database at https://physics.nist.gov/asd +8 https://www.as.utexas.edu/~chris/moog.html +strongest in very metal-poor cool giants (e.g., [Fe/H]< −2.0, +log(g)< 3, and Teff ≲ 5500 K), such as UMi Target 1 (see +Table 4). If we examine A(Fe i) and A(Fe ii) in UMi Tar- +get 1, we find they differ by only 1.43σ or 0.16 ± 0.11 dex. +This value is consistent with ionization equilibrium, and also +within the range in the discrepancies found by Karovicova +et al. (2020) for cool giants. For these reasons, we refrain +from tuning the surface gravity based on the Fe lines. +5 +CHEMICAL ABUNDANCE ANALYSIS +This section describes the chemical abundances that we de- +termine from the spectrum of Target 1. This includes an +application of non-local thermodynamic equilibrium correc- +tions, and a comparison with other UMi members and MW +halo stars in the literature. +5.1 +α−elements +α-elements are primarily formed in the cores of massive stars +and during the explosive phases of core-collapse supernovae +(e.g., Timmes et al. 1995; Kobayashi et al. 2020). There are +only three α-elements which produce measurable lines in our +GRACES spectrum of Target 1; Mg, Ca, Ti. The A(Mg i) is +from two lines of the Mg i Triplet (λλ5172.684, 5183.604Å) +and the weaker 5528.405Å line. The A(Ca i) is inferred from +13 spectral lines, from 5588 Å to 6500 Å. Up to 12 and 9 +lines of Ti i and Ti ii are useful to infer A(Ti i) and A(Ti ii), +respectively (Lawler et al. 2013; Wood et al. 2013). The first +row of panels in Figure 3 display the [Mg, Ca, Ti/Fe] ratios +as a function of the [Fe/H]. Both the LTE and NLTE analysis +are reported (see Section 5.5). Since both Ti i and Ti ii lines +are present in the spectrum, [Ti/Fe] is the average weighted +by the number of lines of each species. +To highlight the strong Mg lines in UMi Target 1, we +compare it to the metal-poor benchmark giant HD 122563 +([Fe/H]=−2.7, Lind et al. (2022); Kielty et al. (2021)) in +Figure 4. +5.2 +Odd-Z elements +Odd-Z elements are excellent tracers of metal-poor core- +collapse supernovae due to the odd-even effect in the pre- +dicted yields (Heger & Woosley 2010; Nomoto et al. 2013; +Kobayashi et al. 2020; Ebinger et al. 2020). Three odd-Z +elements are observable in our spectrum of Target 1; Na, +K, Sc. A(Na i) is measurable from the spectral lines of the +Na i Doublet (λλ5889.951, 5895.924 Å). K i is observable +with two lines at λλ7664.899, 7698.965 Å (Falke et al. 2006; +Trubko et al. 2017). These lines are very close to water +vapour lines of the Earth’s atmosphere; however, the radial +velocity for Target 1 places these lines in clear windows. Sc is +measured from only one Sc ii line at λλ5526.785 Å (Lawler +et al. 2019). The abundances of K and Sc have been mea- +sured with the synth configuration in MOOG, taking into +account hyperfine splitting effects for Sc. The second row of +panels of Figure 3 shows [Na, K, Sc/Fe] (LTE for all and +also NLTE for Na). +MNRAS 000, 1–18 (2023) + +Extreme outskirt of Ursa Minor +7 +3.5 +3.0 +2.5 +2.0 +1.5 +1.0 +[Fe/H] +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +[Mg/Fe] +3.5 +3.0 +2.5 +2.0 +1.5 +1.0 +[Fe/H] +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +[Ca/Fe] +3.5 +3.0 +2.5 +2.0 +1.5 +1.0 +[Fe/H] +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +[Ti/Fe] +3.5 +3.0 +2.5 +2.0 +1.5 +1.0 +[Fe/H] +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +[Na/Fe] +3.5 +3.0 +2.5 +2.0 +1.5 +1.0 +[Fe/H] +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +[K/Fe] +3.5 +3.0 +2.5 +2.0 +1.5 +1.0 +[Fe/H] +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +[Sc/Fe] +3.5 +3.0 +2.5 +2.0 +1.5 +1.0 +[Fe/H] +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +[Cr/Fe] +3.5 +3.0 +2.5 +2.0 +1.5 +1.0 +[Fe/H] +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +[Ni/Fe] +3.5 +3.0 +2.5 +2.0 +1.5 +1.0 +[Fe/H] +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +[Ba/Fe] +Figure 3. Chemical abundances for stars in UMi. Target 1 is marked with a red diamond (LTE) and with an orange diamond (NLTE). +UMi stars from the high-resolution observations from literature are denoted with magenta diamonds. The literature compilation is from +Shetrone et al. (2001), Sadakane et al. (2004), Cohen & Huang (2010), Kirby & Cohen (2012), and Ural et al. (2015) and it is in LTE. +Grey open circles mark MW halo stars compiled from Aoki et al. (2013), Yong et al. (2013), Kielty et al. (2021), and Buder et al. (2021). +The black cross at the corner of each panel represents the typical uncertainty on the UMi literature chemical abundances. +5.3 +Fe-peak elements +Fe-peak elements are important tracers of stellar evolu- +tion. At early times, they were produced primarily in core +collapse supernovae (e.g., +Heger & Woosley 2010), and +then later in supernova Ia events (e.g., Nomoto et al. +2013). The Fe-peak elements observable in our GRACES +spectra include Fe, Cr and Ni. The A(Fe i) is from +29 lines, while A(Fe ii) is from only 3 lines. Our fi- +nal [Fe/H] values are the average measurements weighted +by the number of lines per star. A(Cr i) is measured +from +3 +spectral +lines +(λλ5296.691, 5345.796, 5409.783Å, +Sobeck +et +al. +2007), +while +Ni +i +is +found +from +four +lines (λλ5476.904, 5754.656, 6586.31, 6643.63 Å, Wood et al. +2014). The left and centre panels of the third row of Fig- +ure 3 show [Cr/Fe] (LTE and NLTE) and [Ni/Fe] (LTE) as +a function of [Fe/H]. +5.4 +Neutron-capture process elements +Neutron-capture elements are primarily synthesised through +two main channels, the rapid and the slow neutron captures +processes. If the neutron capture timescale is shorter than +the β− decay time, then rapid-process elements are formed. +Conditions where this is most likely to happen are found +in core collapse supernovae and neutron-star mergers. Oth- +erwise, as in the stellar atmospheres of AGB stars, where +neutron fluxes are lower and have weaker energies, then the +beta-decay timescale is shorter, leading to the production +MNRAS 000, 1–18 (2023) + +8 +F. Sestito et al. +5526.0 +5526.5 +5527.0 +5527.5 +5528.0 +5528.5 +5529.0 +5529.5 +wavelength (A) +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +1.1 +Flux +Figure 4. Mg i 5528Å region. The Mg-rich spectrum of Tar- +get 1 (black solid line) is compared with the standard VMP star +HD122563 (black dashed line, [Fe/H] ∼ −2.7, [Mg/Fe] ∼ +0.3 +Lind et al. 2022; Kielty et al. 2021) and three synthetic spec- +tra with [Mg/Fe] = +0.5, +0.8, +1.0 (light blue, yellow, and pink +shaded areas, respectively). Synthetic spectra have been gener- +ated using the synth mode in MOOG (Sneden 1973) with the +line list from linemake (Placco et al. 2021). Model atmosphere +are from MARCS (Gustafsson et al. 2008; Plez 2012). The syn- +thetics are created at the same resolution of GRACES and with +the stellar parameters and metallicity as Target 1. +via the slow-neutron capture processes. The only neutron- +capture process element present in our GRACES spectra is +Ba, with two Ba ii lines (λλ6141.73, 6496.91 Å. To infer the +A(Ba ii), MOOG has been run with the synthetic config- +uration to account for the hyperfine structure corrections. +Bottom right panel of Figure 3 displays [Ba/Fe] (LTE and +NLTE) as a function of [Fe/H]. +5.5 +NLTE corrections +The elemental abundances in the atmospheres of very metal- +poor stars are affected by departures from Local Thermody- +namic Equilibrium (LTE). Thus, the statistical equilibrium +solutions need to be corrected for radiative effects (non-LTE +effects, or “NLTE”), which can be large for some species. To +correct for NLTE effects in Fe (Bergemann et al. 2012) and +Na i (Lind et al. 2012), we adopted the results compiled in +the INSPECT9 database. The NLTE corrections for Mg i +(Bergemann et al. 2017), Ca i (Mashonkina et al. 2017), Ti i +and Ti ii (Bergemann 2011), and Cr i (Bergemann & Ces- +cutti 2010) are from the MPIA webtool database10. For Ba ii +lines, we adopted the NLTE corrections from Mashonkina & +Belyaev (2019), also available online11. +9 http://inspect-stars.com +10 http://nlte.mpia.de +11 http://www.inasan.ru/~lima/pristine/ba2/ +Table 5. Chemical abundances of Target 1. The LTE and NLTE +ratios are reported together with the σ and the number of lines +for each measured species. For Fe and ti we report the number of +lines relative to both the neutral and the single-ionised states. +Ratio +LTE +σ +Nlines +NLTE +(dex) +(dex) +(dex) +[Fe/H] +−2.09 +0.09 +29+3 +−1.98 +[Mg/Fe] +0.86 +0.20 +3 +0.75 +[Ca/Fe] +0.12 +0.11 +13 +0.07 +[Ti/Fe] +0.21 +0.12 +12+9 +0.27 +[Na/Fe] +−0.44 +0.24 +2 +−0.82 +[K/Fe] +0.40 +0.10 +2 +−− +[Sc/Fe] +0.15 +0.10 +1 +−− +[Cr/Fe] +−0.06 +0.24 +3 +0.14 +[Ni/Fe] +−0.04 +0.18 +4 +−− +[Ba/Fe] +−1.00 +0.15 +2 +−1.13 +5.6 +Uncertainty on the chemical abundances +The uncertainty on element X is given by σA(X) += +δA(X)/√NX if the number of the measured spectral lines +is NX > 5, or σA(X) = δA(Fe i)/√NX otherwise. Given the +SNR across the observed combined spectrum of Target 1, +the uncertainty on the chemical abundance ratios is in the +range 0.10 ≤ σ[X/Fe] ≤ 0.24. This range for the uncertainty +is compatible with the ones measured by Kielty et al. (2021) +and Waller et al. (2023), in which they use a similar obser- +vational setup with GRACES to study chemical abundances +of very metal-poor giant stars. +5.7 +Elemental abundance compilation from the +literature +UMi is an interesting and nearby dwarf galaxy that has had +extensive observations of stars in its inner regions. We have +gathered the elemental abundance results from optical high- +resolution observations of stars in Ursa Minor from the lit- +erature. This compilation is composed of 21 stars in total, +including Shetrone et al. (2001, 4 stars), Sadakane et al. +(2004, 3 stars), Cohen & Huang (2010, 10 stars), Kirby & +Cohen (2012, 1 star), and Ural et al. (2015, 3 stars). All of +these studies provide 1D LTE chemical abundances. +We also compare the chemistry of the stars in UMi with +those in the MW halo from a compilation including Aoki +et al. (2013); Yong et al. (2013); Kielty et al. (2021); Buder +et al. (2021). The stars from Buder et al. (2021) are from the +third data release of GALactic Archaeology with HERMES +(GALAH, De Silva et al. 2015; Buder et al. 2021) collab- +oration. We select GALAH stars to be in the halo, with +reliable metallicities (flag_fe = 0), chemical abundances +(flag_X_fe = 0), and stellar parameters (flag_sp = +0). +MNRAS 000, 1–18 (2023) + +Extreme outskirt of Ursa Minor +9 +6 +METALLICITIES FROM THE NIR CA ii T +LINES +For our UMi Targets 2–5 observed in low-SNR mode, metal- +licities are derived from the NIR Ca ii T lines. We follow the +method described in Starkenburg et al. (2010) with some mi- +nor modifications. Starting with their Equation A.1: +[Fe/H] = a+b·MV +c·EW2+3 +d·EW−1.5 +2+3 +e·EW2+3 ·MV, +(1) +where MV +is the absolute V magnitude of the star, +EW2+3 is the sum of the equivalent width of the Ca ii +λλ8542.09, 8662.14 Å lines, and a, b, c, d are the coefficients +listed in Table A.1 of Starkenburg et al. (2010). MV is de- +rived converting the Gaia EDR3 magnitudes to the Johnson- +Cousin filter following the relation from Riello et al. (2021, +see their Table C.2 for the coefficients) and adopting a helio- +centric distance of 76 ± 10 kpc (e.g., McConnachie 2012). +Our minor modification is due to the fact that the third +component of our Ca ii T spectra is contaminated by sky +lines. Therefore, EW2+3 is derived assuming that the EW +ratio between the second and the third Ca ii T lines is +EW8542/EW8662 = 1.21 ± 0.03, in agreement with Starken- +burg et al. (2010, see their Figure B.1). The EW of the Ca ii +8542 Å line is measured using the splot routine in IRAF +(Tody 1986, 1993), fitting the line with multiple profiles. +The median and the standard deviation have been adopted +as final values for the EW and its uncertainty. We perform a +Monte Carlo test with 106 randomisations on the heliocen- +tric distance, the EW8542, the EW8542/EW8662 ratio, and +the de-reddened magnitudes assuming a Gaussian distribu- +tion. The final [Fe/H] and its uncertainty are the median +and the standard deviation from the randomisations, respec- +tively. +Although Starkenburg et al. (2010) proved that this +metallicity calibration is reliable and compatible with high- +resolution studies, we use Target 1 to check for a possible +offset in [Fe/H]. Given the different SNR between Target 1 +(∼ 35 at Ca ii T) and the other targets (∼ 8 − 15 at Ca ii +T), the spectrum of Target 1 has been degraded to match +the SNR of the other targets. Its metallicity from Ca ii T is +[Fe/H]CaT = −2.34 ± 0.26, compatible within 0.9σ with the +metallicity inferred from Fe lines ([Fe/H] = −2.09 ± 0.09). +The SNR of the Ca ii T region in the observed spectra is +sufficient to obtain an uncertainty on the metallicity in the +range 0.08 ≤ σ[Fe/H] ≤ 0.20. +Table 4 reports the inferred metallicities together with +the stellar parameters and radial velocities. Figure 5 dis- +plays the metallicities and radial velocities of our targets and +known UMi members (Spencer et al. 2018; Pace et al. 2020, +APOGEE DR17) as a function of their elliptical distances +(left panels); the [Fe/H] vs. RV space and their histograms +(central and right panels). The five targets have metallicities +and radial velocities compatible with the UMi distributions, +therefore we identify them as new members of UMi. +7 +ORBITAL PARAMETERS +In this section, we want to test the gravitational potential so +far used for kinematical studies in the disk and the halo of +the Milky Way (e.g., Sestito et al. 2019, 2020; Lucchesi et al. +2022). We make use of Galpy12 (Bovy 2015) to infer the +pericentric, apocentric, and galactocentric distances of Ursa +Minor. The choice on the isolated gravitational potential and +on all the other assumptions (e.g., distance and motion of +the Sun etc.), the orbital integration time, and the deriva- +tion of the uncertainties mirror the method fully described +in Sestito et al. (2019). The code is run on the sample of +stars from Spencer et al. (2018), Pace et al. (2020), and our +five new targets. The system’s orbital parameters are ob- +tained from the median of the sample. The uncertainties on +the system parameters are derived dividing the dispersion by +the square root of the number or stars in the sample. The +inferred quantities are compared with the values from the +literature (Li et al. 2021; Battaglia et al. 2022; Pace et al. +2022), in which a variety of MW gravitational potentials +were adopted. In particular, Li et al. (2021) make use of four +isolated MW gravitational potential, one NFW dark mat- +ter halo (PNFW) and three with Einasto profiles (PEHM, +PEIM, and PELM). Battaglia et al. (2022) adopted two iso- +lated MW profiles (LMW and HMW) and one perturbed +by the the passage of the Large Magellanic Cloud (PMW). +Pace et al. (2022) used two gravitational potentials, one in +which the MW is isolated (MW), and the other perturbed +by the LMC (MW+LMC). Both Battaglia et al. (2022) and +Pace et al. (2022) make use of NFW dark matter profiles. +Figure 6 displays our results for the apocentric (red +shaded area), pericentric (green shaded area), and Galac- +tocentric (blue shaded area) distances in comparison with +values from the aforementioned gravitational potentials from +the literature. For the literature, we report the pericentric +(green diamonds) and apocentric (red diamonds) distances. +The Galactocentric position of UMi is closer to the apoc- +entre, while the blue arrow indicates the system is moving +towards its pericentre. +The inferred orbital parameters are in broad agreement +with the results from the variety of gravitational potentials +adopted in the literature so far. In particular, the apocentre +(Rapo = 92.67+2.17 +−0.41 kpc) is similar to the ones inferred as- +suming a more massive dark matter halo, such as the PEHM +from Li et al. (2021), HMW from Battaglia et al. (2022), +or MW+LMC and MW from Pace et al. (2022). While the +pericentric distance (Rperi = 57.23+0.48 +−0.83 kpc) is very differ- +ent from the one inferred with the PMW from Battaglia +et al. (2022), PEIM and PELM from Li et al. (2021). The +pericentre variation is narrower among different potentials, +although we can observe our inference is much less in agree- +ment with HMW and PMW from Battaglia et al. (2022). +8 +DISCUSSION +In this section, we discuss the membership of the five targets +observed with GRACES and the chemo-dynamical proper- +ties of the dwarf galaxy, Ursa Minor. +12 http://github.com/jobovy/galpy +MNRAS 000, 1–18 (2023) + +10 +F. Sestito et al. +0 +2 +4 +6 +8 +10 +12 +290 +270 +250 +230 +210 +RV (km s +1) +0 +2 +4 +6 +8 +10 +12 +R/rh +3.5 +3.0 +2.5 +2.0 +1.5 +1.0 +[Fe/H] +300 +280 +260 +240 +220 +RV (km s +1) +3.5 +3.0 +2.5 +2.0 +1.5 +1.0 +[Fe/H] +10 +30 +50 +70 +90 +110 +130 +10 +30 +50 +70 +90 +Figure 5. Distribution of UMi stars. Left panels: Radial velocities (top) and metallicities (bottom) as a function of the elliptical distance. +Central panel: distribution of UMi stars in the [Fe/H] vs. RV space. Corner plots: histograms of the RV (top) and metallicities (right) +distributions of UMi star. Target 1 is marked with a red diamond, while Target 2–5 are displayed with black diamonds. Magenta dots +are the compilation of stars from Spencer et al. (2018) and Pace et al. (2020). Blue squares are UMi members selected from APOGEE +DR17. +8.1 +The stellar halo of a dwarf: five new far +outlying members of UMi +Radial velocities and metallicities for five new targets were +measured above, where [Fe/H] is from Fe i and Fe ii lines +in case of Target 1 (see Section 5), yet [Fe/H] is inferred +through the NIR Ca ii Triplet lines for Target 2–5 (see Sec- +tion 6). Figure 5 clearly shows that these five stars lie within +the distributions in metallicity and radial velocity of Ursa +Minor. The chemical properties of Target 1 shown in Fig- +ure 3 support its membership to UMi. The slightly higher +[Mg/Fe] and the low [Ba/Fe] are interesting, however their +values are in agreement with at least one other member of +UMi. +To further exclude the possibility that these 5 targets +are halo interlopers, we run the Besançon simulation (Robin +et al. 2003, 2017) of the MW halo. We select all the stellar +particles in the direction of UMi. Out of the 300 stellar par- +ticles produced, only 21 inhabit the same proper motion +region as in Figure 1. Within this sample, only 4 stellar +particles lie in the highest RV range of UMi, −250 < RV +< −210 km s−1. All of them have [Fe/H] > −1.5, and 2 +with [Fe/H] > −1.0. The latter 2 stellar particles are outside +the metallicity range of UMi. While the former two stellar +particles, however, have a photometry that differs by 2 mag- +nitudes in the G band from UMi stars at the same colour +BP − RP. Therefore, none of our target, or more in general, +known UMi members are reproduced by the Besançon MW +halo simulation. This is another indication that Target 1–5 +are not foreground stars, but rather new UMi members. +Previously, it was shown that UMi is more elongated +(ϵUMi = 0.55) than other classical satellites (ϵ < 0.45, Muñoz +et al. 2018). The most distant member had been located +near ∼ 5.5rh. With our results, Ursa Minor extends out +to a projected elliptical distance of ∼ 12rh, or ∼ 4.5 kpc +(projected) from its centre. This distance is close to the tidal +radius inferred by Pace et al. (2020), 5 − 6 kpc. +Errani et al. (2022) analysed the dynamical properties +of many satellites of the MW in terms of their dark matter +content and distribution. The authors show that the dynam- +ical properties of UMi are compatible with Λ−CDM model +if tidal stripping effects are taken into account. The find- +ing of a member at ∼ 12rh the multiple apocentric and +pericentric passages reinforce the idea that UMi is strongly +dominated by tidal stripping. In fact, as shown in the left +panel of Figure 1, the proper motion of UMi is almost par- +allel to the semi-major axis of the system. In addition, su- +pernovae feedback can play a role in pushing members to +the extreme outskirt of their host galaxy. These scenarios +have also been proposed to explain the extended structure +of Tucana II ultra-faint dwarf galaxy (Chiti et al. 2021). +The authors discuss a third possible scenario which involves +mergers of UFDs. We discuss and rule out the merger hy- +pothesis for UMi in Section 8.6. +MNRAS 000, 1–18 (2023) + +Extreme outskirt of Ursa Minor +11 +20 +40 +60 +80 +100 120 140 160 180 +d (kpc) +PEHM L21 +PNFW L21 +PEIM L21 +PELM L21 +LMW B22 +HMW B22 +PMW B22 +MW+LMC P22 +MW P22 +Figure 6. Orbital parameters for Ursa minor. The green, red, +and blue vertical bands are the pericentric (Rperi = 57.23+0.48 +−0.83 +kpc), apocentric (Rapo = 92.67+2.17 +−0.41 kpc), and Galactocentric +distances (RGC = 77.55+0.02 +−0.03 kpc) inferred in this work. To infer +the orbital parameters, we use the Spencer et al. (2018); Pace +et al. (2020) compilation. Vertical lines are their median values, +while shaded area are the interval between the 0.16 and 0.84 +quantiles. The blue horizontal arrow departing from the verti- +cal line of the Galactocentric distance represents the direction +of the Galactocentric radial velocity. Pericentric and apocentric +distances from the literature are represented by green and red +points, respectively. Tick labels in the y axis indicate the studies +from which the parameters have been taken: the L21 potentials +are from Li et al. (2021), the B22 are from Battaglia et al. (2022), +and the P22 are from Pace et al. (2022). +8.2 +Contributions from Supernovae Type Ia +The contribution of SNe Ia in UMi is still under debate (e.g., +Ural et al. 2015, and references therein). The flat distribu- +tion in the α− and Fe−peak elements shown in Figure 3 +are consistent with no contributions from SN Ia, with the +exception for the most metal-rich star, COS171 (Cohen & +Huang 2010). While this lone star might draw the eye to +the conclusion of a possible α−knee, i.e., the rapid change +in the slope of the α−elements from a plateau to a steep +decrease, it is really the [Na, Ni/Fe] (and likely [Ti, Sc/Fe]) +ratios that favour the steep decrease, and suggest contri- +butions from SN Ia. In support, McWilliam et al. (2018) +re-analysed COS171 showing that its [Mn, Ni/Fe] ratios do +indicate SN Ia contributions, but from sub-Chandrasekhar- +mass degenerate stars, i.e., ∼ 0.95 M⊙. +Alternatively, one of the more metal-rich star, COS347 +([Fe/H] = −1.63, Sadakane et al. 2004), is slightly enriched +in Mg, Ca, Ti, and Na compared to the stars at the same +metallicity. This may suggest that at higher metallicities +there is a large scatter in chemical abundance ratios, rather +than a decrease with metallicity as expected from enrich- +ment by SN Ia. +To investigate more thoroughly the contribution of +SNe Ia above [Fe/H] ≳ −2, we explore APOGEE DR17 (Ab- +durro’uf et al. 2022). The selection of UMi members from +this dataset is described in Section 2.1. We choose Mg and +O as amongst the most reliable species13. Spectral lines of +O are well-measured in the infrared (APOGEE) spectra, +while in the optical they are hard to measure (e.g., weak +lines, [O i] λλ6300, 6363 Å) or strong lines also suffer from +large NLTE effects (e.g., the O i T λλ7772, 7774, 7775 Å). +The optical and APOGEE chemical abundance results are +shown in Figure 7, and compared with MW halo stars from +APOGEE and GALAH (optical, Buder et al. 2021). With +the addition of reliable [O/Fe] from APOGEE, the presence +of a plateau to [Fe/H] ≲ −2.1 and then a steeper decrease, +i.e., a knee, is more clearly seen. This decrease, now ob- +served in several α-elements, indicates contributions from +SNe Ia. A deeper analysis of the APOGEE spectra in terms +of the chemo-dynamical analyses of dwarf galaxies is cur- +rently under investigation, Shetrone et al. (2023, in prep.). +This study will also quantify any offsets between optical and +infrared measurements, as seen in Figure 7 for [Mg/Fe]. +The metallicity at which the knee occurs ([Fe/H]knee), +is correlated with the time when SNe Ia begin to contribute +to the chemical evolution of a galaxy. This time is also de- +pendent on the star formation efficiency, which is expected +to be lower in dwarf galaxies (e.g., Matteucci 2003; Tolstoy +et al. 2009). Recently, Theler et al. (2020) discussed that +the slope of the knee-decrease is governed by the balance +between the amount of metals ejected by SNe Ia vs. SNe II. +Therefore, a smaller slope indicates an extended star forma- +tion rather than a sharply quenching galaxy (Theler et al. +2020). On the theoretical side, Revaz & Jablonka (2018) de- +veloped cosmological zoom-in simulations that are able to +reproduce most of the observable quantities of dwarf galax- +ies, e.g., velocity dispersion profiles, star formation histories, +stellar metallicity distributions, and [Mg/Fe] abundance ra- +tios. Similarly, the FIRE simulations (e.g., Hopkins et al. +2014) have been used to (a) reproduce the star formation +histories of the MW satellites (Escala et al. 2018), and (b) +reproduce the properties and numbers of ultra-faint dwarf +galaxies (Wheeler et al. 2015). These models suggest that +a higher [Fe/H]knee is attained when the star formation is +more efficient and the system can retain the metals. Given +the value of [Fe/H]knee ∼ −2.1, then the low star formation +efficiency of UMi appears to be similar to measurements in +other dwarf galaxies (e.g., Reichert et al. 2020; Tolstoy et al. +2009; Simon 2019), and much less efficient than in the MW, +where [Fe/H]knee ∼ −0.5, (e.g., Venn et al. 2004; Haywood +et al. 2013; Buder et al. 2021; Recio-Blanco et al. 2022). +8.3 +Presence of rapid- and slow-neutron capture +processes +To examine the contributions from SNe II in UMi, we exam- +ine the distribution in [Ba/Mg] vs. [Mg/H] in Figure 8. At +very low-metallicities, if Ba is produced by the r-processes +(see the review by Cowan et al. 2021, and references therein), +then a tight and flat distribution will be visible, i.e., a Ba- +floor, also shown in Mashonkina et al. (2022). This seems to +be the case for UMi stars with [Mg/H]< −2.0, including Tar- +get 1. A spread in [Ba/Mg] that is significantly larger than +a 3σ error, and subsequent rise from a presumed Ba-floor, +13 https://www.sdss4.org/dr17/irspec/abundances +MNRAS 000, 1–18 (2023) + +12 +F. Sestito et al. +2.6 +2.4 +2.2 +2.0 +1.8 +1.6 +1.4 +1.2 +1.0 +[Fe/H] +1.00 +0.75 +0.50 +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +[Mg/Fe] +2.6 +2.4 +2.2 +2.0 +1.8 +1.6 +1.4 +1.2 +1.0 +[Fe/H] +1.00 +0.75 +0.50 +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +[O/Fe] +Figure 7. UMi chemical abundances from APOGEE DR17 (Ab- +durro’uf et al. 2022). Blue squares are stars from APOGEE with +high SNR (> 70) and very likely to be UMi members (Psat> 70 +percent) according to our algorithm. UMi stars from the literature +are marked with magenta squares, while magenta triangles denote +their upper limits. Target 1 is marked with a red (LTE) and or- +ange (NLTE) diamond. Cyan open circles are MW stars from +APOGEE with high SNR (> 70) and good Gaia EDR3 paral- +lax measurements (ϖ/δϖ > 15). Grey open circles are MW stars +from GALAH (Buder et al. 2021) selected as in Figure 3. Typi- +cal uncertainties are denoted with blue and magenta crosses for +APOGEE (infrared NLTE) and literature stars (high-resolution +optical LTE), respectively. An offset in [Mg/Fe] between the opti- +cal LTE and infrared NLTE measurements is under investigation +by the APOGEE team (Shetrone et al. 2023, in prep.). +is interpreted as Ba contributions from metal-poor asymp- +totic giant branch stars (AGB), via slow neutron-captures +(s-process, e.g., Pignatari et al. 2008; Cescutti & Chiappini +2014). This chemical behaviour is also visible in the bottom +panel of Figure 8, in which we report the [Ba/Fe] vs. [Fe/H] +(as in Figure 3) as a check that our interpretation is not +biased by measurements of Mg. +Based on an overabundance of [Y/Ba] observed in UMi +stars at very low metallicities, [Fe/H]< −2.5, Ural et al. +(2015) have suggested that there are also contributions from +spinstars (e.g., Cescutti et al. 2013) at the earliest epochs. +Spinstars are fast rotating massive stars (25–40 M⊙) that +produce s-process elements from neutron rich isotopes in +their atmospheres (e.g., +Cescutti & Chiappini 2014). Un- +fortunately, our GRACES spectra are insufficient (SNR too +low for the weak Y ii lines) to determine an abundance for +[Y/Ba], including our spectrum of Target 1. +8.4 +No trace of pair-instability supernovae +Pair-instability supernovae (PISNe) are produced by very +metal-poor, very massive stars (> 120 M⊙), predicted to be +amongst the first stars. PISN produce a strong odd-even ef- +fect in the yields, with no neutron-capture process elements +above the mass cut (Takahashi et al. 2018). The odd-even ef- +fect leads to a high [Ca/Mg] and low [Na/Mg] (green shaded +area in Figure 9). Yields of PISNe, coupled with other SNe II +predicted from a normal initial mass function, have been +estimated by Salvadori et al. (2019), and are shown by a +slightly higher [Na/Mg] ratio (red shaded area Figure 9). +There is no trace of PISNe, nor PISNe + SNe II, yields in +Ursa Minor. +8.5 +The Chemistry of Target 1 +The detailed chemistry of Target 1 may provide a glimpse +into the early star formation events in UMi. It stands out +in [Ba/Mg] with unusually low Ba for a stars in UMi or the +MW (see Figure 8). It also appears to be lower in [Na/Mg] +and [Ca/Mg] than the other stars in UMi and the MW; see +Figure 9. This is partially due to the higher [Mg/Fe] com- +pared to other UMi stars. These low abundances relative to +Mg in combination with the little amount of Ba even at rel- +atively higher metallicities ([Fe/H]∼ −2) have been found +in some stars of Coma Berenices (Frebel & Bromm 2012), +Segue 1 (Frebel et al. 2014), Hercules (Koch et al. 2008, +2013; François et al. 2016), and in the Milky Way (e.g., Sit- +nova et al. 2019; Kielty et al. 2021; Sestito et al. 2023). This +particular chemical pattern has been interpreted as contri- +bution from only one or a few low-mass core-collapse SNe II +(CCSNe), known as the "one-shot" model (Frebel & Bromm +2012). We explore a variety of core collapse supernovae yields +to compare to our chemical abundances in Target 1 to test +this "one shot" model hypothesis. +Various yields of SNe II are on the market. We choose +to compare the chemistry of Target 1 against the widely +used faint SNe II yields from Nomoto et al. (2013) and the +recent ones from Ebinger et al. (2020). We included this ad- +ditional comparison as the yields from Nomoto et al. (2013) +are predicted only up to proton number 32, whereas the +yields from Ebinger et al. (2020) reach heavier elements up +MNRAS 000, 1–18 (2023) + +Extreme outskirt of Ursa Minor +13 +3.5 +3.0 +2.5 +2.0 +1.5 +1.0 +[Mg/H] +2.5 +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +[Ba/Mg] +3.5 +3.0 +2.5 +2.0 +1.5 +1.0 +[Fe/H] +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +[Ba/Fe] +Figure 8. Top panel: [Ba/Mg] vs. [Mg/H] space. Bottom panel: +[Ba/Fe] vs. [Fe/H] as in Figure 3. Target 1 is denoted with a +red (LTE) and a orange (NLTE) diamond. Literature UMi stars +(magenta diamonds) are from Shetrone et al. (2001), Sadakane +et al. (2004), Cohen & Huang (2010), Kirby & Cohen (2012), and +Ural et al. (2015). Literature MW halo compilation (grey open +circles) from Aoki et al. (2013), Yong et al. (2013), Kielty et al. +(2021), and Buder et al. (2021). The black cross at the upper left +corner represents the typical uncertainty on the UMi literature +chemical abundances. +1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00 +[Ca/Mg] +1.5 +1.0 +0.5 +0.0 +0.5 +[Na/Mg] +PISNe + SNe +PISNe +Figure 9. PISNe yields space. Target 1 is marked with a red +and a orange diamond when LTE and NLTE, respectively. The +green band is the region of stars polluted by PISNe alone (Taka- +hashi et al. 2018). The red zone is the locus in which the stars +would have been polluted by PISNe and SN II as in Salvadori +et al. (2019). For the latter case, we show the yields relative to +a PISNe to SN II ratio between 0.5 and 0.9 (see Figure 6 from +Salvadori et al. 2019). Literature UMi stars (magenta diamonds) +from Shetrone et al. (2001), Sadakane et al. (2004), Cohen & +Huang (2010), Kirby & Cohen (2012), and Ural et al. (2015). +Literature MW halo compilation (grey open circles) from Aoki +et al. (2013), Yong et al. (2013), Kielty et al. (2021), and Buder +et al. (2021). The black cross at the corner represents the typical +uncertainty on the UMi literature chemical abundances. +to proton number 60. Another difference is how the energy +of the supernovae explosion is parametrized. While Nomoto +et al. (2013) fixed the energy to the value of 1051 erg, this is +treated as a free parameter by Ebinger et al. (2020), in which +it spans from 0.2 to 2.0 ×1051 erg, and varies with the pro- +genitor mass. Both of them uses non-rotating models. The +spatial symmetry of the explosion is also modelled differ- +ently. Nomoto et al. (2013) employed the so-called mixing +and fallback model, which implies the presence of polar jets +and fallback materials around the equatorial plane. On the +other hand, Ebinger et al. (2020) adopted spherical symme- +try. When comparing the yields from Nomoto et al. (2013) +with Target 1, the chemistry of this star is well described +by pollution from a low-mass faint CCSNe (∼ 30 M⊙). Al- +ternatively, we are not able to reproduce the chemistry of +Target 1 when comparing to the yields from Ebinger et al. +(2020). Their predictions at all masses are higher than our +observations for the majority of elements, and we cannot re- +produce their strong odd-even effect, with the exception of +[Ba/Mg]. This is the only ratio we can reproduce adopting +a progenitor mass 25 ≤ Mprog ≤ 30 M⊙. +As Target 1 is very far from the UMi central body, we +suggest it may have formed just after the contributions from +MNRAS 000, 1–18 (2023) + +14 +F. Sestito et al. +low-mass SN II and was exiled by supernova feedback and/or +tidal forces by pericentric passage(s) with the Galaxy. A +deeper analysis of chemistry (heavy elements) of the newly +discovered members in the APOGEE survey, i.e., those lo- +cated between the central body and Target 1 and, more +generally the kinematical characterisation of the UMi halo, +could help to clarify this picture. +8.6 +Outside-in star formation vs. late-time merger +Pace et al. (2020) measured radial velocities and metallici- +ties of likely UMi members selected from Gaia DR2 within +2 half-light radii. They interpreted the spatial distribution +of the stars as composed of two populations with different +chemo-dynamical properties. A more metal-rich ([Fe/H] = +−2.05 ± 0.03) kinematically colder (σRV = 4.9 ± 0.8 km s−1) +and centrally concentrated (rh = 221 ± 17 pc) population. +And a metal-poor hotter and more extended ([Fe/H] = +−2.29 ± 0.05, σRV = 11.5 ± 0.9 km s−1, rh = 374 ± 49 pc) +population. Pace et al. (2020) discussed that the two metal- +licity distributions in UMi are much closer than in other +dwarf spheroidal galaxies (dSphs) found so far. +Benítez-Llambay et al. (2016) and Genina et al. (2019) +proposed that dwarf-dwarf mergers are the cause of the mul- +tiple populations in dSphs. Therefore, Pace et al. (2020) +concluded that UMi underwent a late-time merger event +between two dwarfs with very similar chemical and phys- +ical properties. However, Genina et al. (2019) also pointed +out that kinematic and spatial information alone are insuf- +ficient to disentangle the formation mechanisms of multi- +populations. Additional evidence from precise chemical +abundances and star formation histories are needed, data +that was not included in the study by Pace et al. (2020). +In this paper, we propose an alternative scenario to ex- +plain the chemo-dynamical properties of the two populations +in Ursa Minor. An outside-in star formation history can also +be used to describe the properties of low mass systems, such +as dwarf galaxies (Zhang et al. 2012). Briefly, the extended +metal-poor population ([Fe/H] ≲ −2.0) formed everywhere +in the dwarf, such that the relatively younger stars popu- +late the centre of the galaxy at times when SNe Ia begin to +contribute (e.g., Hidalgo et al. 2013; Benítez-Llambay et al. +2016). This enhances the metallicity only in the central re- +gion, giving the galaxy a non-linear metallicity gradient. +In support of our simpler interpretation, the distribu- +tions in the chemical elements over a wide range in metal- +licity suggests a common path amongst the stars in UMi. +UMi stars are polluted by low mass CCSNe (e.g., their low +[Ba/Fe, Mg] and [Na, Ca/Mg]), they show a SNe Ia knee at +[Fe/H] ∼ −2.1 and a contribution from AGB is also visible in +the more metal-rich stars, and they display a low dispersion +in [Ca/Mg] from star to star over 2 dex in metallicity. +Furthermore, Revaz & Jablonka (2018) used a cosmo- +logical zoom-in simulation to show that the kinematics in +UMi are consistent with secular heating in the central region +of the satellite without invoking late-time mergers. Thus, a +more simple scenario of outside-in star formation is consis- +tent with the chemical, structural, and kinematic properties +of UMi, and we suggest these do not necessarily require a +late-time merger event. +10 +15 +20 +25 +30 +35 +Proton number +3.0 +2.5 +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +[X/Mg] +[Fe/H]=-3.0 CCSNe (Nomoto et al. 2013) +M13 +M15 +M18 +M20 +M25 +M30 +M40 +10 +15 +20 +25 +30 +35 +Proton number +3.0 +2.5 +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +[X/Mg] +[Fe/H]=-4.0 CCSNe (Ebinger et al. 2020) +u11 +u12 +u13 +u14 +u15 +u16 +u17 +u18 +u19 +u20 +u24 +u25 +u26 +u27 +u28 +u30 +10 +20 +30 +40 +50 +60 +70 +Proton number +2 +1 +0 +1 +2 +3 +[X/Mg] +[Fe/H]=-4.0 CCSNe (Ebinger et al. 2020) +u11 +u12 +u13 +u14 +u15 +u16 +u17 +u18 +u19 +u20 +u24 +u25 +u26 +u27 +u28 +u30 +Figure 10. Chemistry of Target 1 in the CCSne yields space. +Top panel: EMP ([Fe/H] = −3.0) CCSNe yields from Nomoto +et al. (2013). Central panel: UMP ([Fe/H] = −4.0) CCSNe from +Ebinger et al. (2020) in the proton number range as top panel. +Bottom panel: same as the central panel but for all the species pre- +dicted by Ebinger et al. (2020). The legend indicates the model’s +name, in which the number is the progenitor’s mass in M⊙ at +its ZAMS. The darker the line, the heavier the mass. Progenitor +masses for models from Ebinger et al. (2020) are predicted up +to 30 M⊙, while Nomoto et al. (2013) modeled the yields up to +100 M⊙. +MNRAS 000, 1–18 (2023) + +Extreme outskirt of Ursa Minor +15 +9 +CONCLUSIONS +A new Bayesian algorithm was used to find new members in +the very extreme outskirts of the ultra faint dwarf galaxy, +Ursa Minor. Five targets were selected for high-resolution +spectroscopy with GRACES at Gemini North. For all five +stars, we determine precise radial velocities and metallicities; +for the brightest and farthest target in projection (Target 1), +the higher SNR of our GRACES spectrum also permitted a +detailed chemical abundance analysis. With the use of data +from th eliterature and APOGEE DR17, we find that: +(i) The Bayesian algorithm is very efficient in finding new +members, even at very large elliptical distances. All five can- +didates are new members of UMi, according to their radial +velocities and metallicities (see Figure 5). +(ii) Ursa Minor extends at least out to a projected ellip- +tical distance of ∼ 12rh, which corresponds to ∼ 4.5 kpc for +an adopted distance of 76 kpc. +(iii) The orbital properties of UMi indicate that the sys- +tem has recently passed apocentre and it is moving towards +pericentre (see Figure 6). Tidal stripping is one scenario that +can explain UMi’s elongated shape. +(iv) The chemical properties of Target 1 (see Figure 3), +the most distant member discovered so far, are compatible +with the overall distribution of the known UMi members +from high-resolution spectral analysis. +(v) The low [Ca, Na/Mg] and the low [Ba/Fe] of Tar- +get 1 suggest that the star formed in an environment pol- +luted by low-mass supernovae type II (Mprog ∼ 30 M⊙, see +Figures 9 and 10). The star is likely exiled by supernovae +feedback or tidal forces. +(vi) Looking at all the UMi stars with high-resolution +chemical analyses, including those from APOGEE DR17, +we conclude there is evidence of pollution by supernovae +type Ia. There is a knee at [Fe/H]knee ∼ −2.1 in the [Mg, O, +Na, Ni/Fe] distributions (see Figures 3 and 7). +(vii) Ursa Minor is also clearly polluted by supernovae +type II and AGB stars given the distribution of [Ba/Mg, Fe] +as a function of [Mg, Fe/H] (see Figure 8). +(viii) There is no trace of yields from pair-instability su- +pernovae, either alone or combined with type II (see Fig- +ure 9). +(ix) The chemo-dynamical properties of UMi can be ex- +plained by an outside-in star formation and the following +SNe Ia enrichment. We propose this as a simpler scenario +than a late-time merger event between two very similar sys- +tems. +(x) We have found two new UMi members at a distance +of ∼ 7rh in APOGEE DR17 (Section 2.1 and Figure 1). +As their metallicities are at the edge of the APOGEE grid +(∼ −2.4), their true [Fe/H] may be lower and their chemical +ratios might be affected. +In the very near future, the Gemini High resolution Op- +tical SpecTrograph (GHOST, Pazder et al. 2016) will be +operative at Gemini South. It will cover a wider spectral re- +gion than GRACES, especially towards the blue where many +spectral lines of heavy elements are found. In synergy with +Gaia satellite and the powerful Bayesian algorithm for tar- +get selections, it should be possible to discover a plethora +of new members in the centre and extreme outskirts of this +and many other ultra-faint and classical dwarf galaxies to +study their star formation histories. This will be a giant +leap forward for detailed studies of low mass systems, and +both observational and theoretical near field cosmological +investigations. +ACKNOWLEDGEMENTS +We acknowledge and respect the l@IJkw@ŋ@n peoples on whose +traditional territory the University of Victoria stands and +the Songhees, Esquimalt and ¯WSÁNEĆ peoples whose his- +torical relationships with the land continue to this day. +The authors wish to recognize and acknowledge the very +significant cultural role and reverence that the summit of +Maunakea has always had within the Native Hawaiian com- +munity. We are very fortunate to have had the opportunity +to conduct observations from this mountain. +We want to thank the supporter astronomers, Joel +Roediger and Hyewon Suh, for their help during Phase II +and the observational runs. +FS thanks the Dr. Margaret "Marmie" Perkins Hess +postdoctoral fellowship for funding his work at the Univer- +sity of Victoria. KAV, LDA, and JG thank the National +Sciences and Engineering Research Council of Canada for +funding through the Discovery Grants and CREATE pro- +grams. DZ thanks the Mitacs Globalink program for sum- +mer funding. The authors thanks the International Space +Science Institute (ISSI) in Bern, Switzerland, for funding +the "The Early Milky Way" Team led by Else Starkenburg. +Based on observations obtained through the Gem- +ini Remote Access to CFHT ESPaDOnS Spectrograph +(GRACES), as part of the Gemini Program GN-2022A-Q- +128. ESPaDOnS is located at the Canada-France-Hawaii +Telescope (CFHT), which is operated by the National Re- +search Council of Canada, the Institut National des Sci- +ences de l’Univers of the Centre National de la Recherche +Scientifique of France, and the University of Hawai’i. ES- +PaDOnS is a collaborative project funded by France (CNRS, +MENESR, OMP, LATT), Canada (NSERC), CFHT and +ESA. ESPaDOnS was remotely controlled from the inter- +national Gemini Observatory, a program of NSF’s NOIR- +Lab, which is managed by the Association of Universi- +ties for Research in Astronomy (AURA) under a coop- +erative agreement with the National Science Foundation +on behalf of the Gemini partnership: the National Science +Foundation (United States), the National Research Coun- +cil (Canada), Agencia Nacional de Investigación y Desar- +rollo (Chile), Ministerio de Ciencia, Tecnología e Innovación +(Argentina), Ministério da Ciência, Tecnologia, Inovações +e Comunicações (Brazil), and Korea Astronomy and Space +Science Institute (Republic of Korea). +This work has made use of data from the European +Space Agency (ESA) mission Gaia (https://www.cosmos. +esa.int/gaia), processed by the Gaia Data Processing and +Analysis Consortium (DPAC, https://www.cosmos.esa. +int/web/gaia/dpac/consortium). Funding for the DPAC +has been provided by national institutions, in particular the +institutions participating in the Gaia Multilateral Agree- +ment. +Funding for the Sloan Digital Sky Survey IV has been +provided by the Alfred P. Sloan Foundation, the U.S. De- +MNRAS 000, 1–18 (2023) + +16 +F. Sestito et al. +partment of Energy Office of Science, and the Participating +Institutions. +SDSS-IV acknowledges support and resources from the +Center for High Performance Computing at the University +of Utah. The SDSS website is www.sdss4.org. +SDSS-IV is managed by the Astrophysical Research +Consortium for the Participating Institutions of the SDSS +Collaboration including the Brazilian Participation Group, +the Carnegie Institution for Science, Carnegie Mellon Uni- +versity, Center for Astrophysics | Harvard & Smithsonian, +the Chilean Participation Group, the French Participation +Group, Instituto de Astrofísica de Canarias, The Johns Hop- +kins University, Kavli Institute for the Physics and Math- +ematics of the Universe (IPMU) / University of Tokyo, +the Korean Participation Group, Lawrence Berkeley Na- +tional Laboratory, Leibniz Institut für Astrophysik Potsdam +(AIP), Max-Planck-Institut für Astronomie (MPIA Heidel- +berg), Max-Planck-Institut für Astrophysik (MPA Garch- +ing), Max-Planck-Institut für Extraterrestrische Physik +(MPE), National Astronomical Observatories of China, New +Mexico State University, New York University, University +of Notre Dame, Observatário Nacional / MCTI, The Ohio +State University, Pennsylvania State University, Shanghai +Astronomical Observatory, United Kingdom Participation +Group, Universidad Nacional Autónoma de México, Univer- +sity of Arizona, University of Colorado Boulder, University +of Oxford, University of Portsmouth, University of Utah, +University of Virginia, University of Washington, University +of Wisconsin, Vanderbilt University, and Yale University. +This research has made use of the SIMBAD database, +operated at CDS, Strasbourg, France (Wenger et al. 2000). +This work made extensive use of TOPCAT (Taylor 2005). +DATA AVAILABILITY +GRACES spectra will be available at the Gemini Archive +web page https://archive.gemini.edu/searchform after +the proprietary time. 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J., +2020, ApJ, 891, 85 +This paper has been typeset from a TEX/LATEX file prepared by +the author. +MNRAS 000, 1–18 (2023) + diff --git a/vdFPT4oBgHgl3EQf-DWo/content/tmp_files/load_file.txt b/vdFPT4oBgHgl3EQf-DWo/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cf191ae546aa94fdae9c3e506161396b8d24982d --- /dev/null +++ b/vdFPT4oBgHgl3EQf-DWo/content/tmp_files/load_file.txt @@ -0,0 +1,2292 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf,len=2291 +page_content='MNRAS 000, 1–18 (2023) Preprint 1 February 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 The extended "stellar halo" of the Ursa Minor dwarf galaxy Federico Sestito1⋆, Daria Zaremba1,2, Kim A.' metadata={'source': 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and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' University of Victoria,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' PO Box 3055,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' STN CSC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Victoria BC V8W 3P6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Canada 2 National University of Kyiv-Mohyla Academy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 04655 Kyiv,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Observatoire,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' CH-1290 Versoix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Switzerland 5 GEPI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Observatoire de Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Université PSL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 5 Place Jules Janssen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' F-92195 Meudon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' France 6 Instituto de Astrofısica de Canarias,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' E-38200 La Laguna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Tenerife,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Spain 7 Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Astrofısica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain 8 Gemini Observatory/NSF’s NOIRLab, 670 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' A’ohoku Place, Hilo, Hawai’i, 96720, USA 9 Visiting astronomer at the Université de Montréal, Complexe des Sciences, Montréal, QC H2V 0B3, Canada Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' in original form ZZZ ABSTRACT Five stars in the extreme outskirts (from ∼ 5 to ∼ 12 elliptical half-light radii, rh) of the Ursa Minor (UMi) dwarf galaxy have been identified as potential new members using a Bayesian algorithm applied to Gaia EDR3 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' These targets were observed with the GRACES spectrograph, resulting in precise radial velocities and metallic- ities that confirm their association with UMi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' For the brightest and outermost star (Target 1, at ∼ 12 rh), the chemical abundances of α- (Mg, Ca, Ti), odd-Z (Na, K, Sc), Fe-peak (Fe, Ni, Cr), and neutron-capture process (Ba) elements have also been determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' We also discuss data from the literature and from APOGEE DR17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' We find the chemical patterns in UMi are consistent with a star formation history that includes yields from core collapse supernovae, asymptotic giant branch stars, and su- pernovae Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Evidence for a knee in the [α/Fe] ratios near [Fe/H] ∼ −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='1 indicates a low star formation efficiency similar to that in other dwarf galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Given the dis- tance of Target 1 from the centre of UMi (R∼4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 kpc), we show that UMi has a more extended structure than previously thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' This "stellar halo" around UMi could be a secondary feature resulting from tidal stripping after multiple orbits around the Galaxy, or maybe a primary UMi feature due to early hierarchical accretion activity or to strong gravitational fluctuations prompted by feedback in the early star forma- tion phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Also consistent with observations is a late-time merger-free scenario where outside-in star formation is accompanied by gradual supernovae Ia enrichment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Key words: stars: abundances – stars: Population II – galaxies : formation – galaxies: dwarf – galaxies: individual: Ursa Minor – galaxies: evolution 1 INTRODUCTION Dwarf satellites of the Milky Way (MW) are amongst the oldest and most metal-poor galaxies known (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', Tolstoy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' They are at the low-mass end of the hierar- chical formation process, just massive enough to form very metal-poor stars (VMP, [Fe/H] ≤ −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0, Simon 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The mass of faint dwarf galaxies is dominated by dark matter (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', Simon 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' In fact, their dynamical mass-to-light ratios (M/L) can exceed 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' They remain one of the best ⋆ Email: sestitof@uvic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='ca targets for studies seeking to understand the properties of dark matter and early events in the formation our Galaxy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', Bullock & Boylan-Kolchin 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Hierarchical formation of Λ−Cold Dark Matter (Λ−CDM) cosmology (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', White & Rees 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Frenk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 1997) predicts that haloes grow from the accretion of smaller systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Therefore, galaxies should possess an extended stellar halo built from disrupted sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' A stellar halo is clearly observed in large galaxies as the Milky Way, but it remain elusive and poorly studied in dwarf galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', Deason et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2022, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' One reason is because the fraction of mass assem- © 2023 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='13214v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='GA] 30 Jan 2023 2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Sestito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' bled through mergers is reduced at the dwarf galaxy mass scales, while ‘smooth’ accretion dominates at this regime (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', Genel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Second, while at the Milky Way- size the stellar mass to halo mass ratio is well modeled, on the dwarf size is not the case (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', Moster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2013, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Given their shallow gravitational potential, faint dwarf galaxies are extremely susceptible to internal processes, such as star formation and the subsequent stellar feedback (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', El-Badry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' and external, such as mergers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', Deason et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2014), ram pressure stripping (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', Grebel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2003) and stirring (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', Kazantzidis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2011), tidal interaction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', Fattahi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2018), and reionization (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', Wheeler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' All of these processes may act to influ- ence their individual morphologies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', Higgs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2021, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Signatures of these gravitational in- teractions will be most evident in the outskirts of the dwarf galaxy, where accreted remnants can show up as an excess of stars over and above expectations from a simple single- component model (akin to a stellar halo in a more massive galaxy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Stars in the extreme outskirts of dwarf galaxies, and yet which are not clearly associated with prominent tidal tails, have been discovered only relatively recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Chiti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2021) spectroscopically identified member stars up to ∼9 half-light radii (rh), or physical distances up to 1 kpc, away from the centre of the faint dwarf galaxy, Tucana II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Dy- namical analysis, as well as chemical abundances, were used to distinguish between a tidal origin, where stars were re- moved from the main body due to tidal effects with the MW, and an accreted, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', dwarf-dwarf merger, origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The stars identified by Chiti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2021) were found to be ex- tremely metal deficient compared to the main body, suggest- ing that the outskirts had a different origin from the bulk of stars in Tucana II, perhaps due to an early merger with a low-mass, metal-poor companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Recently, Longeard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2022) analysed the chemo-dynamical properties of Boötes I, suggesting that the system could have been more massive than nowadays and that tidal stripping is largely affecting the satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Inspired by Chiti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2021), we have examined other MW satellites to help constrain the frequency of such stellar halos around dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' McConnachie & Venn (2020b,a) de- veloped a Bayesian method, and updated by Jensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (prep), to estimate the probability that a star in the vicinity of a dwarf galaxy is a member of the dwarf, using the full astrometric and photometric data from Gaia EDR3 (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Jensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (prep) report that only a few dwarf galaxies out of nearly 60 examined suggest evidence for an extended stellar halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The systems already in the literature include Tucana II, as examined by Chiti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2021), Sculptor (Sestito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' prep), and also Coma Berenices, Ursa Major I, and Boötes I (see also Longeard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2022), recently analysed by Waller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Waller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2023) showed that stars in Coma Berenices have been polluted by supernovae type Ia, in con- trast to previous views of this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Waller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2023) discussed that the chemistry of the outermost stars in these systems is consistent with their formation in the central re- gions, then moving them to their current locations through tidal stripping and/or supernova feedback, although in the case of Boötes I the lower metallicities and lack of strong carbon enrichment of its outermost stars could also be ev- idence of a late dwarf-dwarf merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Although the detailed and precise chemical abundance analysis a firmer conclusion on the origin of the outermost stars is hard to pinpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' In this work, we use this Bayesian algorithm to search for member stars in the outermost regions of the dwarf galaxy Ursa Minor (UMi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' We make use of recent updates to the algorithm by Jensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (prep), which allow for the presence of a secondary, extended component (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', an outer stellar halo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Previously, Piatek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2005) has suggested the presence of tidal effects on the substructure of UMi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Ursa Minor is historically a well-studied system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Some controversies remain regarding the star formation history (SFH) and its efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' For example, Carrera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2002) suggested that up to ∼95 per cent of UMi stars are older than 10 Gyr, invoking an episodic SFH at early times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' This is based on studies of its colour-magnitude diagram (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', Mighell & Burke 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Bellazzini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Other models interpreted the chemical properties of UMi as due to ex- tended SFH, from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='9 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 Gyr (Ikuta & Arimoto 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Ural et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Kirby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2011, 2013) matched the wide metallicity distribution function (MDF) of UMi with a chemical evolution model that includes infall of gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' On the other hand, Ural et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2015) developed three chemical evolution models, showing that winds from supernovae are needed to describe UMi’s MDF, especially to reproduce stars at higher metallicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The authors underline that winds help to explain the absence of gas at the present time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' In agreement with Ikuta & Arimoto (2002), their models use an extended low-efficiency SFH duration (5 Gyr, Ural et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Ural et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2015) argued that is not easy to discern if the [α/Fe] displays a plateau up to [Fe/H]∼ −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0, or whether it shows a gradual decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' However, they conclude that a slow decrease is present above this metallicity, pointing to the contribution of supernovae type Ia (SNe Ia).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' On the other hand, Cohen & Huang (2010) noted that a very short duration of star formation (∼ 2 Gyr) implies that SNe Ia did not have enough time to contribute to the chemical evo- lution of UMi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' More recent studies discovered that SNe Ia can occur in the very first 2 Gyr of the Universe (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', Maoz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2012, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' de los Reyes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The Λ−CDM cosmological zoom-in simulations de- veloped by Revaz & Jablonka (2018) that incorporate gas cooling found that the star formation and chemical evolu- tion of UMi can be explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' In particular, when SNe Ia and II events are taken into account with thermal blastwave-like feedback (Revaz & Jablonka 2018, and references therein), then they can reproduce the observed distribution in metal- licity, [Mg/Fe], and the radial velocity dispersion with a short star formation of only 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='4 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' In this paper, we present a chemo-dynamical inves- tigation of stars in the extreme outskirts of UMi ob- served with high-resolution GRACES spectrograph at Gem- ini North/CFHT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Our results combined with spectroscopic results for additional stars in the literature are used to dis- cuss the extended chemical and dynamical evolution of UMi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The target selection, the observations, and the spectral re- duction are reported in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Stellar parameters are inferred in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The model atmosphere and chemi- cal abundance analysis for Target 1 are reported in Sec- tion 4 and 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Section 6 describes the measure- MNRAS 000, 1–18 (2023) Extreme outskirt of Ursa Minor 3 ment of [Fe/H] using Ca Triplet lines for Target 2–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The inference of the orbital parameters of UMi is described in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The chemo-dynamical properties of Ursa Minor are discussed in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2 DATA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='1 Target selection Using the Bayesian algorithm from McConnachie & Venn (2020b), with updates from Jensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (prep), we have searched for stars that inhabit the extended stellar halo of Ursa Minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Briefly, this algorithm provides the probability for any star in Gaia to be a member of a given MW satel- lite or to belong to the MW halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The total likelihood is a function of the position of the star on the sky, on the colour- magnitude diagram, and in proper motion space (thus, no radial velocity or metallicity information is used).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' This al- gorithm has proved useful to identify new members in the extreme outskirt of some ultra-faint dwarf galaxies (Waller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Sestito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' prep) and performs excellently to remove Milky Way foreground contamination (Jensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' prep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' In this work, we further validate this identification method by examining the extreme outskirts of UMi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' All stars with a high probability (> 80%) of being associated to UMi, and at a distance greater than 5 half-light radii (≳ 85 arcmin or ≳ 900 pc) from the centre of the dwarf, were selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' This included five red giants with magnitudes in the range 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='4 ≤ G ≤ 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='3 mag in the Gaia EDR3 G band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The brightest target is also the farthest in projection, reaching an extreme distance of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='7 half-light radii from the centre of UMi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Our other four targets, at a distance of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='2 − 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='3 rh, are also listed as highly likely UMi candidates by Qi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2022, with a probability > 90 percent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The main properties of UMi and our five targets are reported in Tables 1 and 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The position of our five candidates together with other known UMi members are shown in Figure 1 in projected sky coordinates, on the colour-magnitude diagram, and in proper motion space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' This figure shows that even if the can- didates are located far from the centre of UMi, the algo- rithm is very efficient in selecting new candidate members in the very outskirts of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' We gather UMi members from Spencer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2018), Pace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2020), and APOGEE data release 17 (DR17, Abdurro’uf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2022) and then cross-match with Gaia EDR3 to retrieve coordinates, proper motion, and photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' When we examine the APOGEE DR17 targets, we have applied our selection algorithm to select the stars with high membership probability (> 70 %) and with high signal-to-noise in their spectra (SNR > 70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Surprisingly, two stars from APOGEE DR17 have an ellip- tical distance of ∼ 7 rh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' We note that the [Fe/H] values for these two stars are at the edge of the metallicity grid of APOGEE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' thus, while their radial velocity measurements are precise, their true [Fe/H] could be lower, in turn affecting their [X/Fe] ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Galactic parameters of Ursa Minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The coordinates α, δ, the mean metallicity, the mean radial velocity, the velocity dispersion, the heliocentric distance D⊙, the ellipticity, the posi- tion angle φ, and the half-light radius rh in arcmin and pc, the mean proper motion from Gaia EDR3, the dynamical mass, the mass density, and the luminosity are reported with the respective references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (a) refers to McConnachie (2012), (b) to McConnachie & Venn (2020b), (c) to McConnachie & Venn (2020a), (d) to Qi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2022), and (e) to Mateo (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Property Value Reference α 227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='2854 deg (b) δ 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='2225 deg (b) [Fe/H] −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='01 (b) RV 246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='1 km s−1 (b) σV 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='2 km s−1 (b) D⊙ 76 ± 10 kpc (a) ellipticity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='01 (b) φ 50 ± 1 deg (b) rh 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='11 arcmin (b) rh 382 ± 53 pc (b) rh,plummer 407 pc (d) µαcosδ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='124 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='004 mas yr−1 (c) µδ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='078 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='004 mas yr−1 (c) Mdyn(≤ rhalf) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 × 106 M⊙ (a) Mass density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='35 M⊙ pc−3 (e) L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='29 × 106 L⊙ (e) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='2 GRACES observations Targets were observed with the Gemini Remote Access to CFHT ESPaDOnS Spectrograph (GRACES, Chene et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Pazder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2014) using the 2-fibre (object+sky) mode with a resolution of R∼ 40000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' GRACES consists a 270-m optical fibre that connects the Gemini North tele- scope to the Canada–France–Hawaii Telescope ESPaDOnS cross-dispersed high resolution échelle spectrograph (Donati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The targets were observed within the GN- 2022A-Q-128 program (P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Sestito).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' For the brightest target (Target 1, G= 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='4 mag), which is also the farthest one from the centre (∼ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='7 rh), we achieved a spectrum with SNR per resolution element of ∼ 30 at the Ba ii 6141 Å region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' This spectrum has suf- ficient SNR to measure the abundances for additional ele- ments, specifically the α− (Mg, Ca, Ti), odd−Z (Na, K, Sc), Fe−peak (Fe, Cr, Ni), and neutron−capture process (Ba) elements across the entire GRACES spectral coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' We refer to this observational set-up as the “high-SNR mode”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' For the remaining four targets, which have distances from 5 − 7 rh, a SNR per resolution element of ∼ 20 in the Ca ii T region (∼8550 Å) was obtained for precise radial veloci- ties and metallicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' In this “low-SNR mode”, the metallic- ities are derived from the equivalent width (EW) of the NIR Ca ii T, as described in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Observing information is summarized in Table 3, including the signal-to-noise ratio measured at the Mg i b, Ba ii 614nm, and Ca ii T regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='3 Spectral reductions The GRACES spectra were first reduced using the Open source Pipeline for ESPaDOnS Reduction and Analysis MNRAS 000, 1–18 (2023) 4 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Sestito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The Gaia EDR3 source ID, the coordinates (α, δ), the projected coordinates (ξ, η), the elliptical radius distance rell in rh unit, the probability to be a member from Jensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (prep), the Gaia EDR3 photometry G and BP−RP, and the reddening AV from Schlafly & Finkbeiner (2011) are reported for each target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Target source id α δ ξ η rell Psat G BP−RP AV (deg) (deg) (deg) (deg) (rh) (mag) (mag) (mag) Target 1 1647329728514964480 234.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='45303 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='29204 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='53226 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='21888 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='80 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='39 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='08 Target 2 1693464785444020224 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='67731 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='35983 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='00378 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='15842 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='97 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='83 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='06 Target 3 1693573430936780032 226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='08983 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='77965 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='45214 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='56153 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='96 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='91 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='05 Target 4 1669324938936435200 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='50756 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='21361 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='12033 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='98413 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='94 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='06 Target 5 1645948119139534336 230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='43949 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='29581 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='16629 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='10328 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='92 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='29 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='8 (BP-RP)0 (mag) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 G0 (mag) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 (mas yr 1) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 (mas yr 1) 3 2 1 0 1 2 3 (deg) 3 2 1 0 1 2 3 (deg) 3 2 1 0 1 2 3 D (kpc) 3 2 1 0 1 2 3 D (kpc) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Ursa Minor seen through Gaia EDR3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' All the panels: Target 1 is marked with a red diamond, while black diamonds are Target 2–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Magenta circles are UMi literature stars from Spencer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2018) and Pace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Blue squares are UMi stars selected from APOGEE DR17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' MW foreground stars are marked with grey small dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' These are selected from Gaia EDR3 in the direction of UMi and within the field of view of the η − ξ panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Left panel: Projected sky coordinates and projected distance from UMi centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The orange ellipses denotes the elliptical distances from UMi centre of 3, 5, 7, and 11 rh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The arrow points in the direction of UMi proper motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Central panel: Colour-magnitude diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Dark green dashed lines is a Padova isochrone at [Fe/H] = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 and age of 12 Gyr (Bressan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Right panel: Proper motion space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Total exposure time, number of exposures, signal-to- noise ratio (SNR) measured at the Mg i 518nm, Ba ii 614nm, and Ca ii 850nm regions, and the observation dates are reported for each target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The SNR is defined as the ratio between the median flux and its standard deviation in given spectral region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Target texp Nexp SNR SNR SNR Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' date (s) @Mg ib @Ba ii @Ca iiT YY/MM/DD Target 1 14400 6 9 27 37 22/06/18 Target 2 1800 1 5 12 17 22/03/14 Target 3 1800 1 1 6 8 22/03/14 Target 4 2400 1 2 6 11 22/06/17 Target 5 2400 1 1 5 10 22/06/17 (OPERA, Martioli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2012) tool, which also corrects for heliocentric motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Then the reduced spectra were post- processed following an updated procedure of the pipeline described in Kielty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The latter pipeline allows us to measure the radial velocity of the observed star, to co-add multiple observations, to check for possible radial ve- locity variations, to correct for the motion of the star, and to eventually re-normalise the flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' This procedure also im- proves the signal-to-noise ratio in the overlapping spectral order regions without downgrading the spectral resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Radial velocities are reported in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' This procedure failed for one of the spectral orders of Target 1 covering the Mg i b region for reasons that we could not overcome within the scope of this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' We therefore extracted the data for Target 1 ourselves using DRAGraces1 IDL code (Chené et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The final spectra for all five targets near the Na i Dou- blet (left) and in the NIR Ca ii Triplet (right) regions are shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The quality of the spectra indicates that the adopted exposure time were sufficient for the requested science, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', chemical abundances for Target 1, and [Fe/H] and RV only for Targets 2−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 3 STELLAR PARAMETERS Given the low SNR of our spectra, we use the InfraRed flux method (IRFM) from González Hernández & Bonifa- cio (2009) with photometry from Gaia EDR3 to find the ef- 1 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='com/AndreNicolasChene/DRAGRACES/ releases/tag/v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='4 MNRAS 000, 1–18 (2023) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='.2 : + = :3 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' :-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' +** : 1 + +** .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' " .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='-: =Extreme outskirt of Ursa Minor 5 5888 5890 5892 5894 5896 wavelength (A) 0 1 2 3 4 5 6 Flux 8536 8538 8540 8542 8544 8546 8548 wavelength (A) 0 1 2 3 4 5 6 Flux Target 1 Target 2 Target 3 Target 4 Target 5 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' GRACES spectra for the five new UMi member stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Left panel: Na i Doublet region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Chemical abundance ratios are measurable only for Target 1 given the low SNR of Targets 2–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Right panel: The second component of the Ca ii Triplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' This spectral line is used to infer [Fe/H] (see Section 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' fective temperatures, adopting the Mucciarelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2021) colour-temperature relationship for giants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The input pa- rameters are the Gaia EDR3 (BP − RP) de-reddened colour and a metallicity estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The 2D Schlafly & Finkbeiner (2011) map2 has been used to correct the photometry for extinction3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' As input metallicities, we adopt the value [Fe/H] = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5, compatible with the metallicity distri- bution in UMi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Surface gravities were found using the Stefan- Boltzmann equation4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' This step required the effective tem- perature, the distance of the object, the Gaia EDR3 G de- reddened photometry, and the bolometric corrections on the flux (Andrae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2018) as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' A Monte Carlo algorithm has been applied to the input parameters with their uncer- tainties to estimate the total uncertainties on the stellar pa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The input quantities were randomised within 1σ using a Gaussian distribution, except for the stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The latter is treated with a flat prior from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='8 M⊙, 2 https://irsa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='ipac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='edu/applications/DUST/ 3 To convert from the E(B-V) map to Gaia extinction coeffi- cients, the AV/E(B − V) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='1 (Schultz & Wiemer 1975) and the AG/AV = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='85926, ABP/AV = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='06794, ARP/AV = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='65199 relations (Marigo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2018) are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 4 L⋆ = 4πR2 ⋆σT 4 ⋆ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' the radius of the star can be calculated from this equation, then the surface gravity is inferred assuming the mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Stellar parameters of the five targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' [Fe/H] for Target 1 is from Fe i and Fe ii lines, while for the other stars is from Ca ii Triplet lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Target RV Teff log g [Fe/H] (km s−1) (K) Target 1 −256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='05 4604 ± 94 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='08 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='09 Target 2 −265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='26 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='89 4771 ± 93 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='43 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='07 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='15 Target 3 −218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='78 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='82 4760 ± 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='08 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='08 Target 4 −245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='63 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='78 4795 ± 85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='07 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='10 Target 5 −247.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='29 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='63 4814 ± 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='61 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='08 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='20 which is consistent with the mass of long-lived very metal- poor stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The mean uncertainty on the effective tempera- ture is ∼ 94 K, while on the surface gravity it is ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='08 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' This method has been shown to provide reliable stellar pa- rameters suitable for spectroscopic studies of very metal- poor stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', Kielty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Sestito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Waller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The stellar parameters are reported in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 4 MODEL ATMOSPHERE ANALYSIS In this Section, we describe the model atmospheres, the method, and the atomic data for our spectral line list adopted to determine detailed chemical abundances for Tar- get 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' MNRAS 000, 1–18 (2023) 6 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Sestito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='1 Model atmospheres Model atmospheres are generated from the MARCS5 mod- els (Gustafsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Plez 2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' in particular, we selected the OSMARCS spherical models as Target 1 is a giant with log(g)< 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' An initial model atmosphere was generated using the derived stellar parameters, a metallicity [Fe/H] = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0, and microturbulence velocity scaled by the surface gravity from the calibration by Mashonkina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2017) for giants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='2 The lines list and the atomic data Spectral lines were selected from our previous analyses of very metal-poor stars in the Galactic halo and other nearby dwarf galaxies observed with GRACES (Kielty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Sestito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Waller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Atomic data is taken from linemake6 (Placco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2021), with the exception of K i lines taken from the National Institute of Standards and Technology (NIST, Kramida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2021)7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='3 Spectral line measurements Spectral line measurements are made using spectrum syn- thesis, broadened with a Gaussian smoothing kernel of FWHM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='15, which matches the resolution of the GRACES 2-fibre mode spectra) in a four-step process: (1) the synthesis of the [Fe/H] lines in our initial line list (see above) is carried out using an initial model atmosphere and invoking the MOOG8 spectrum synthesis program (Sneden 1973;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Sobeck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2) a new [Fe/H] is determined by removing noisy lines;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (3) the model atmosphere is updated with the new [Fe/H] as metallicity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (4) the chemical abun- dances are derived using the updated model atmosphere and our full line list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The final chemical abundance is given by the average measurement in case of multiple spectral lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='4 Checking the stellar parameters Excitation equilibrium in the line abundances of Fe i is a check on the quality of the effective temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' For Tar- get 1, the slope in A(Fe i) − Excitation potential (EP) from the linear fit has a value of −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='027 dex eV−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' This value is smaller than the dispersion in the measurements of the chemical abundances (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='2 dex) over the range in EP (∼4 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Thus, we conclude our effective temperature estimates are sufficient from the IRFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Ionization balance between Fe i − Fe ii is widely used as a sanity check on the surface gravity estimates (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', Mashonkina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' However, Karovicova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2020) have strongly advised against using this method for very metal-poor giants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' They used interferometric observations of metal-poor stars to find radii, and subsequently precise stel- lar parameters for a set of metal-poor benchmark stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' With their stellar parameters, they have found that deviations in Fe i − Fe ii can reach up to ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='8 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' This effect is the 5 https://marcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='uu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='se 6 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='com/vmplacco/linemake 7 NIST database at https://physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='nist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='gov/asd 8 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='utexas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='edu/~chris/moog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='html strongest in very metal-poor cool giants (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', [Fe/H]< −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0, log(g)< 3, and Teff ≲ 5500 K), such as UMi Target 1 (see Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' If we examine A(Fe i) and A(Fe ii) in UMi Tar- get 1, we find they differ by only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='43σ or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='11 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' This value is consistent with ionization equilibrium, and also within the range in the discrepancies found by Karovicova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2020) for cool giants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' For these reasons, we refrain from tuning the surface gravity based on the Fe lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 5 CHEMICAL ABUNDANCE ANALYSIS This section describes the chemical abundances that we de- termine from the spectrum of Target 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' This includes an application of non-local thermodynamic equilibrium correc- tions, and a comparison with other UMi members and MW halo stars in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='1 α−elements α-elements are primarily formed in the cores of massive stars and during the explosive phases of core-collapse supernovae (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', Timmes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' There are only three α-elements which produce measurable lines in our GRACES spectrum of Target 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Mg, Ca, Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The A(Mg i) is from two lines of the Mg i Triplet (λλ5172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='684, 5183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='604Å) and the weaker 5528.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='405Å line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The A(Ca i) is inferred from 13 spectral lines, from 5588 Å to 6500 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Up to 12 and 9 lines of Ti i and Ti ii are useful to infer A(Ti i) and A(Ti ii), respectively (Lawler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Wood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The first row of panels in Figure 3 display the [Mg, Ca, Ti/Fe] ratios as a function of the [Fe/H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Both the LTE and NLTE analysis are reported (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Since both Ti i and Ti ii lines are present in the spectrum, [Ti/Fe] is the average weighted by the number of lines of each species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' To highlight the strong Mg lines in UMi Target 1, we compare it to the metal-poor benchmark giant HD 122563 ([Fe/H]=−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='7, Lind et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Kielty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2021)) in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='2 Odd-Z elements Odd-Z elements are excellent tracers of metal-poor core- collapse supernovae due to the odd-even effect in the pre- dicted yields (Heger & Woosley 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Nomoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Ebinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Three odd-Z elements are observable in our spectrum of Target 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Na, K, Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' A(Na i) is measurable from the spectral lines of the Na i Doublet (λλ5889.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='951, 5895.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='924 Å).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' K i is observable with two lines at λλ7664.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='899, 7698.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='965 Å (Falke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Trubko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' These lines are very close to water vapour lines of the Earth’s atmosphere;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' however, the radial velocity for Target 1 places these lines in clear windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Sc is measured from only one Sc ii line at λλ5526.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='785 Å (Lawler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The abundances of K and Sc have been mea- sured with the synth configuration in MOOG, taking into account hyperfine splitting effects for Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The second row of panels of Figure 3 shows [Na, K, Sc/Fe] (LTE for all and also NLTE for Na).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' MNRAS 000, 1–18 (2023) Extreme outskirt of Ursa Minor 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 [Fe/H] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 [Mg/Fe] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 [Fe/H] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 1.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Target 1 is marked with a red diamond (LTE) and with an orange diamond (NLTE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' UMi stars from the high-resolution observations from literature are denoted with magenta diamonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The literature compilation is from Shetrone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2001), Sadakane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2004), Cohen & Huang (2010), Kirby & Cohen (2012), and Ural et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2015) and it is in LTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Grey open circles mark MW halo stars compiled from Aoki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2013), Yong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2013), Kielty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2021), and Buder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The black cross at the corner of each panel represents the typical uncertainty on the UMi literature chemical abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='3 Fe-peak elements Fe-peak elements are important tracers of stellar evolu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' At early times, they were produced primarily in core collapse supernovae (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', Heger & Woosley 2010), and then later in supernova Ia events (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', Nomoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The Fe-peak elements observable in our GRACES spectra include Fe, Cr and Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The A(Fe i) is from 29 lines, while A(Fe ii) is from only 3 lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Our fi- nal [Fe/H] values are the average measurements weighted by the number of lines per star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' A(Cr i) is measured from 3 spectral lines (λλ5296.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='691, 5345.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='796, 5409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='783Å, Sobeck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2007), while Ni i is found from four lines (λλ5476.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='904, 5754.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='656, 6586.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='31, 6643.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='63 Å, Wood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The left and centre panels of the third row of Fig- ure 3 show [Cr/Fe] (LTE and NLTE) and [Ni/Fe] (LTE) as a function of [Fe/H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='4 Neutron-capture process elements Neutron-capture elements are primarily synthesised through two main channels, the rapid and the slow neutron captures processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' If the neutron capture timescale is shorter than the β− decay time, then rapid-process elements are formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Conditions where this is most likely to happen are found in core collapse supernovae and neutron-star mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Oth- erwise, as in the stellar atmospheres of AGB stars, where neutron fluxes are lower and have weaker energies, then the beta-decay timescale is shorter, leading to the production MNRAS 000, 1–18 (2023) 8 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Sestito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 5526.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 5526.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 5527.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 5527.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 5528.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 5528.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 5529.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 5529.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 wavelength (A) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='1 Flux Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Mg i 5528Å region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The Mg-rich spectrum of Tar- get 1 (black solid line) is compared with the standard VMP star HD122563 (black dashed line, [Fe/H] ∼ −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='7, [Mg/Fe] ∼ +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='3 Lind et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Kielty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2021) and three synthetic spec- tra with [Mg/Fe] = +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5, +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='8, +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 (light blue, yellow, and pink shaded areas, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Synthetic spectra have been gener- ated using the synth mode in MOOG (Sneden 1973) with the line list from linemake (Placco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Model atmosphere are from MARCS (Gustafsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Plez 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The syn- thetics are created at the same resolution of GRACES and with the stellar parameters and metallicity as Target 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' via the slow-neutron capture processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The only neutron- capture process element present in our GRACES spectra is Ba, with two Ba ii lines (λλ6141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='73, 6496.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='91 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' To infer the A(Ba ii), MOOG has been run with the synthetic config- uration to account for the hyperfine structure corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Bottom right panel of Figure 3 displays [Ba/Fe] (LTE and NLTE) as a function of [Fe/H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 NLTE corrections The elemental abundances in the atmospheres of very metal- poor stars are affected by departures from Local Thermody- namic Equilibrium (LTE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Thus, the statistical equilibrium solutions need to be corrected for radiative effects (non-LTE effects, or “NLTE”), which can be large for some species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' To correct for NLTE effects in Fe (Bergemann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2012) and Na i (Lind et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2012), we adopted the results compiled in the INSPECT9 database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The NLTE corrections for Mg i (Bergemann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2017), Ca i (Mashonkina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2017), Ti i and Ti ii (Bergemann 2011), and Cr i (Bergemann & Ces- cutti 2010) are from the MPIA webtool database10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' For Ba ii lines, we adopted the NLTE corrections from Mashonkina & Belyaev (2019), also available online11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 9 http://inspect-stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='com 10 http://nlte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='mpia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='de 11 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='inasan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='ru/~lima/pristine/ba2/ Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Chemical abundances of Target 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The LTE and NLTE ratios are reported together with the σ and the number of lines for each measured species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' For Fe and ti we report the number of lines relative to both the neutral and the single-ionised states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Ratio LTE σ Nlines NLTE (dex) (dex) (dex) [Fe/H] −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='09 29+3 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='98 [Mg/Fe] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='20 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='75 [Ca/Fe] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='11 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='07 [Ti/Fe] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='12 12+9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='27 [Na/Fe] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='24 2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='82 [K/Fe] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='10 2 −− [Sc/Fe] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='10 1 −− [Cr/Fe] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='24 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='14 [Ni/Fe] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='18 4 −− [Ba/Fe] −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='15 2 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='6 Uncertainty on the chemical abundances The uncertainty on element X is given by σA(X) = δA(X)/√NX if the number of the measured spectral lines is NX > 5, or σA(X) = δA(Fe i)/√NX otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Given the SNR across the observed combined spectrum of Target 1, the uncertainty on the chemical abundance ratios is in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='10 ≤ σ[X/Fe] ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' This range for the uncertainty is compatible with the ones measured by Kielty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2021) and Waller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2023), in which they use a similar obser- vational setup with GRACES to study chemical abundances of very metal-poor giant stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='7 Elemental abundance compilation from the literature UMi is an interesting and nearby dwarf galaxy that has had extensive observations of stars in its inner regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' We have gathered the elemental abundance results from optical high- resolution observations of stars in Ursa Minor from the lit- erature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' This compilation is composed of 21 stars in total, including Shetrone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2001, 4 stars), Sadakane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2004, 3 stars), Cohen & Huang (2010, 10 stars), Kirby & Cohen (2012, 1 star), and Ural et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2015, 3 stars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' All of these studies provide 1D LTE chemical abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' We also compare the chemistry of the stars in UMi with those in the MW halo from a compilation including Aoki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Yong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Kielty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Buder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The stars from Buder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2021) are from the third data release of GALactic Archaeology with HERMES (GALAH, De Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Buder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2021) collab- oration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' We select GALAH stars to be in the halo, with reliable metallicities (flag_fe = 0), chemical abundances (flag_X_fe = 0), and stellar parameters (flag_sp = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' MNRAS 000, 1–18 (2023) Extreme outskirt of Ursa Minor 9 6 METALLICITIES FROM THE NIR CA ii T LINES For our UMi Targets 2–5 observed in low-SNR mode, metal- licities are derived from the NIR Ca ii T lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' We follow the method described in Starkenburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2010) with some mi- nor modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Starting with their Equation A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='1: [Fe/H] = a+b·MV +c·EW2+3 +d·EW−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 2+3 +e·EW2+3 ·MV, (1) where MV is the absolute V magnitude of the star, EW2+3 is the sum of the equivalent width of the Ca ii λλ8542.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='09, 8662.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='14 Å lines, and a, b, c, d are the coefficients listed in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='1 of Starkenburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' MV is de- rived converting the Gaia EDR3 magnitudes to the Johnson- Cousin filter following the relation from Riello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2021, see their Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='2 for the coefficients) and adopting a helio- centric distance of 76 ± 10 kpc (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', McConnachie 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Our minor modification is due to the fact that the third component of our Ca ii T spectra is contaminated by sky lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Therefore, EW2+3 is derived assuming that the EW ratio between the second and the third Ca ii T lines is EW8542/EW8662 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='03, in agreement with Starken- burg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2010, see their Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The EW of the Ca ii 8542 Å line is measured using the splot routine in IRAF (Tody 1986, 1993), fitting the line with multiple profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The median and the standard deviation have been adopted as final values for the EW and its uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' We perform a Monte Carlo test with 106 randomisations on the heliocen- tric distance, the EW8542, the EW8542/EW8662 ratio, and the de-reddened magnitudes assuming a Gaussian distribu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The final [Fe/H] and its uncertainty are the median and the standard deviation from the randomisations, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Although Starkenburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2010) proved that this metallicity calibration is reliable and compatible with high- resolution studies, we use Target 1 to check for a possible offset in [Fe/H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Given the different SNR between Target 1 (∼ 35 at Ca ii T) and the other targets (∼ 8 − 15 at Ca ii T), the spectrum of Target 1 has been degraded to match the SNR of the other targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Its metallicity from Ca ii T is [Fe/H]CaT = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='26, compatible within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='9σ with the metallicity inferred from Fe lines ([Fe/H] = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='09).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The SNR of the Ca ii T region in the observed spectra is sufficient to obtain an uncertainty on the metallicity in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='08 ≤ σ[Fe/H] ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Table 4 reports the inferred metallicities together with the stellar parameters and radial velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Figure 5 dis- plays the metallicities and radial velocities of our targets and known UMi members (Spencer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Pace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2020, APOGEE DR17) as a function of their elliptical distances (left panels);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' the [Fe/H] vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' RV space and their histograms (central and right panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The five targets have metallicities and radial velocities compatible with the UMi distributions, therefore we identify them as new members of UMi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 7 ORBITAL PARAMETERS In this section, we want to test the gravitational potential so far used for kinematical studies in the disk and the halo of the Milky Way (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', Sestito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2019, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Lucchesi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' We make use of Galpy12 (Bovy 2015) to infer the pericentric, apocentric, and galactocentric distances of Ursa Minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The choice on the isolated gravitational potential and on all the other assumptions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', distance and motion of the Sun etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' ), the orbital integration time, and the deriva- tion of the uncertainties mirror the method fully described in Sestito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The code is run on the sample of stars from Spencer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2018), Pace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2020), and our five new targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The system’s orbital parameters are ob- tained from the median of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The uncertainties on the system parameters are derived dividing the dispersion by the square root of the number or stars in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The inferred quantities are compared with the values from the literature (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Battaglia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Pace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2022), in which a variety of MW gravitational potentials were adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' In particular, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2021) make use of four isolated MW gravitational potential, one NFW dark mat- ter halo (PNFW) and three with Einasto profiles (PEHM, PEIM, and PELM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Battaglia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2022) adopted two iso- lated MW profiles (LMW and HMW) and one perturbed by the the passage of the Large Magellanic Cloud (PMW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Pace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2022) used two gravitational potentials, one in which the MW is isolated (MW), and the other perturbed by the LMC (MW+LMC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Both Battaglia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2022) and Pace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2022) make use of NFW dark matter profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Figure 6 displays our results for the apocentric (red shaded area), pericentric (green shaded area), and Galac- tocentric (blue shaded area) distances in comparison with values from the aforementioned gravitational potentials from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' For the literature, we report the pericentric (green diamonds) and apocentric (red diamonds) distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The Galactocentric position of UMi is closer to the apoc- entre, while the blue arrow indicates the system is moving towards its pericentre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The inferred orbital parameters are in broad agreement with the results from the variety of gravitational potentials adopted in the literature so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' In particular, the apocentre (Rapo = 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='67+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='17 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='41 kpc) is similar to the ones inferred as- suming a more massive dark matter halo, such as the PEHM from Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2021), HMW from Battaglia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2022), or MW+LMC and MW from Pace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' While the pericentric distance (Rperi = 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='23+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='48 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='83 kpc) is very differ- ent from the one inferred with the PMW from Battaglia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2022), PEIM and PELM from Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The pericentre variation is narrower among different potentials, although we can observe our inference is much less in agree- ment with HMW and PMW from Battaglia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 8 DISCUSSION In this section, we discuss the membership of the five targets observed with GRACES and the chemo-dynamical proper- ties of the dwarf galaxy, Ursa Minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 12 http://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='com/jobovy/galpy MNRAS 000, 1–18 (2023) 10 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Sestito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 0 2 4 6 8 10 12 290 270 250 230 210 RV (km s 1) 0 2 4 6 8 10 12 R/rh 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 [Fe/H] 300 280 260 240 220 RV (km s 1) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 [Fe/H] 10 30 50 70 90 110 130 10 30 50 70 90 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Distribution of UMi stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Left panels: Radial velocities (top) and metallicities (bottom) as a function of the elliptical distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Central panel: distribution of UMi stars in the [Fe/H] vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' RV space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Corner plots: histograms of the RV (top) and metallicities (right) distributions of UMi star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Target 1 is marked with a red diamond, while Target 2–5 are displayed with black diamonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Magenta dots are the compilation of stars from Spencer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2018) and Pace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Blue squares are UMi members selected from APOGEE DR17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='1 The stellar halo of a dwarf: five new far outlying members of UMi Radial velocities and metallicities for five new targets were measured above, where [Fe/H] is from Fe i and Fe ii lines in case of Target 1 (see Section 5), yet [Fe/H] is inferred through the NIR Ca ii Triplet lines for Target 2–5 (see Sec- tion 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Figure 5 clearly shows that these five stars lie within the distributions in metallicity and radial velocity of Ursa Minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The chemical properties of Target 1 shown in Fig- ure 3 support its membership to UMi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The slightly higher [Mg/Fe] and the low [Ba/Fe] are interesting, however their values are in agreement with at least one other member of UMi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' To further exclude the possibility that these 5 targets are halo interlopers, we run the Besançon simulation (Robin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2003, 2017) of the MW halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' We select all the stellar particles in the direction of UMi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Out of the 300 stellar par- ticles produced, only 21 inhabit the same proper motion region as in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Within this sample, only 4 stellar particles lie in the highest RV range of UMi, −250 < RV < −210 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' All of them have [Fe/H] > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5, and 2 with [Fe/H] > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The latter 2 stellar particles are outside the metallicity range of UMi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' While the former two stellar particles, however, have a photometry that differs by 2 mag- nitudes in the G band from UMi stars at the same colour BP − RP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Therefore, none of our target, or more in general, known UMi members are reproduced by the Besançon MW halo simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' This is another indication that Target 1–5 are not foreground stars, but rather new UMi members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Previously, it was shown that UMi is more elongated (ϵUMi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='55) than other classical satellites (ϵ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='45, Muñoz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The most distant member had been located near ∼ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5rh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' With our results, Ursa Minor extends out to a projected elliptical distance of ∼ 12rh, or ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 kpc (projected) from its centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' This distance is close to the tidal radius inferred by Pace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2020), 5 − 6 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Errani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2022) analysed the dynamical properties of many satellites of the MW in terms of their dark matter content and distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The authors show that the dynam- ical properties of UMi are compatible with Λ−CDM model if tidal stripping effects are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The find- ing of a member at ∼ 12rh the multiple apocentric and pericentric passages reinforce the idea that UMi is strongly dominated by tidal stripping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' In fact, as shown in the left panel of Figure 1, the proper motion of UMi is almost par- allel to the semi-major axis of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' In addition, su- pernovae feedback can play a role in pushing members to the extreme outskirt of their host galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' These scenarios have also been proposed to explain the extended structure of Tucana II ultra-faint dwarf galaxy (Chiti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The authors discuss a third possible scenario which involves mergers of UFDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' We discuss and rule out the merger hy- pothesis for UMi in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' MNRAS 000, 1–18 (2023) Extreme outskirt of Ursa Minor 11 20 40 60 80 100 120 140 160 180 d (kpc) PEHM L21 PNFW L21 PEIM L21 PELM L21 LMW B22 HMW B22 PMW B22 MW+LMC P22 MW P22 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Orbital parameters for Ursa minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The green, red, and blue vertical bands are the pericentric (Rperi = 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='23+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='48 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='83 kpc), apocentric (Rapo = 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='67+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='17 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='41 kpc), and Galactocentric distances (RGC = 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='55+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='03 kpc) inferred in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' To infer the orbital parameters, we use the Spencer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Pace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2020) compilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Vertical lines are their median values, while shaded area are the interval between the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='16 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='84 quantiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The blue horizontal arrow departing from the verti- cal line of the Galactocentric distance represents the direction of the Galactocentric radial velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Pericentric and apocentric distances from the literature are represented by green and red points, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Tick labels in the y axis indicate the studies from which the parameters have been taken: the L21 potentials are from Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2021), the B22 are from Battaglia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2022), and the P22 are from Pace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='2 Contributions from Supernovae Type Ia The contribution of SNe Ia in UMi is still under debate (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', Ural et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2015, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The flat distribu- tion in the α− and Fe−peak elements shown in Figure 3 are consistent with no contributions from SN Ia, with the exception for the most metal-rich star, COS171 (Cohen & Huang 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' While this lone star might draw the eye to the conclusion of a possible α−knee, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', the rapid change in the slope of the α−elements from a plateau to a steep decrease, it is really the [Na, Ni/Fe] (and likely [Ti, Sc/Fe]) ratios that favour the steep decrease, and suggest contri- butions from SN Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' In support, McWilliam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2018) re-analysed COS171 showing that its [Mn, Ni/Fe] ratios do indicate SN Ia contributions, but from sub-Chandrasekhar- mass degenerate stars, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='95 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Alternatively, one of the more metal-rich star, COS347 ([Fe/H] = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='63, Sadakane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2004), is slightly enriched in Mg, Ca, Ti, and Na compared to the stars at the same metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' This may suggest that at higher metallicities there is a large scatter in chemical abundance ratios, rather than a decrease with metallicity as expected from enrich- ment by SN Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' To investigate more thoroughly the contribution of SNe Ia above [Fe/H] ≳ −2, we explore APOGEE DR17 (Ab- durro’uf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The selection of UMi members from this dataset is described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' We choose Mg and O as amongst the most reliable species13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Spectral lines of O are well-measured in the infrared (APOGEE) spectra, while in the optical they are hard to measure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', weak lines, [O i] λλ6300, 6363 Å) or strong lines also suffer from large NLTE effects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', the O i T λλ7772, 7774, 7775 Å).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The optical and APOGEE chemical abundance results are shown in Figure 7, and compared with MW halo stars from APOGEE and GALAH (optical, Buder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' With the addition of reliable [O/Fe] from APOGEE, the presence of a plateau to [Fe/H] ≲ −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='1 and then a steeper decrease, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', a knee, is more clearly seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' This decrease, now ob- served in several α-elements, indicates contributions from SNe Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' A deeper analysis of the APOGEE spectra in terms of the chemo-dynamical analyses of dwarf galaxies is cur- rently under investigation, Shetrone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2023, in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' This study will also quantify any offsets between optical and infrared measurements, as seen in Figure 7 for [Mg/Fe].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The metallicity at which the knee occurs ([Fe/H]knee), is correlated with the time when SNe Ia begin to contribute to the chemical evolution of a galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' This time is also de- pendent on the star formation efficiency, which is expected to be lower in dwarf galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', Matteucci 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Tolstoy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Recently, Theler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2020) discussed that the slope of the knee-decrease is governed by the balance between the amount of metals ejected by SNe Ia vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' SNe II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Therefore, a smaller slope indicates an extended star forma- tion rather than a sharply quenching galaxy (Theler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' On the theoretical side, Revaz & Jablonka (2018) de- veloped cosmological zoom-in simulations that are able to reproduce most of the observable quantities of dwarf galax- ies, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', velocity dispersion profiles, star formation histories, stellar metallicity distributions, and [Mg/Fe] abundance ra- tios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Similarly, the FIRE simulations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', Hopkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2014) have been used to (a) reproduce the star formation histories of the MW satellites (Escala et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2018), and (b) reproduce the properties and numbers of ultra-faint dwarf galaxies (Wheeler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' These models suggest that a higher [Fe/H]knee is attained when the star formation is more efficient and the system can retain the metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Given the value of [Fe/H]knee ∼ −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='1, then the low star formation efficiency of UMi appears to be similar to measurements in other dwarf galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', Reichert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Tolstoy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Simon 2019), and much less efficient than in the MW, where [Fe/H]knee ∼ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5, (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', Venn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Haywood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Buder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Recio-Blanco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='3 Presence of rapid- and slow-neutron capture processes To examine the contributions from SNe II in UMi, we exam- ine the distribution in [Ba/Mg] vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' [Mg/H] in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' At very low-metallicities, if Ba is produced by the r-processes (see the review by Cowan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2021, and references therein), then a tight and flat distribution will be visible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', a Ba- floor, also shown in Mashonkina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' This seems to be the case for UMi stars with [Mg/H]< −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0, including Tar- get 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' A spread in [Ba/Mg] that is significantly larger than a 3σ error, and subsequent rise from a presumed Ba-floor, 13 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='sdss4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='org/dr17/irspec/abundances MNRAS 000, 1–18 (2023) 12 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Sestito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 [Fe/H] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='00 [Mg/Fe] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 [Fe/H] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='00 [O/Fe] Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' UMi chemical abundances from APOGEE DR17 (Ab- durro’uf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Blue squares are stars from APOGEE with high SNR (> 70) and very likely to be UMi members (Psat> 70 percent) according to our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' UMi stars from the literature are marked with magenta squares, while magenta triangles denote their upper limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Target 1 is marked with a red (LTE) and or- ange (NLTE) diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Cyan open circles are MW stars from APOGEE with high SNR (> 70) and good Gaia EDR3 paral- lax measurements (ϖ/δϖ > 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Grey open circles are MW stars from GALAH (Buder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2021) selected as in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Typi- cal uncertainties are denoted with blue and magenta crosses for APOGEE (infrared NLTE) and literature stars (high-resolution optical LTE), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' An offset in [Mg/Fe] between the opti- cal LTE and infrared NLTE measurements is under investigation by the APOGEE team (Shetrone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2023, in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' is interpreted as Ba contributions from metal-poor asymp- totic giant branch stars (AGB), via slow neutron-captures (s-process, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', Pignatari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Cescutti & Chiappini 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' This chemical behaviour is also visible in the bottom panel of Figure 8, in which we report the [Ba/Fe] vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' [Fe/H] (as in Figure 3) as a check that our interpretation is not biased by measurements of Mg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Based on an overabundance of [Y/Ba] observed in UMi stars at very low metallicities, [Fe/H]< −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5, Ural et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2015) have suggested that there are also contributions from spinstars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', Cescutti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2013) at the earliest epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Spinstars are fast rotating massive stars (25–40 M⊙) that produce s-process elements from neutron rich isotopes in their atmospheres (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', Cescutti & Chiappini 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Un- fortunately, our GRACES spectra are insufficient (SNR too low for the weak Y ii lines) to determine an abundance for [Y/Ba], including our spectrum of Target 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='4 No trace of pair-instability supernovae Pair-instability supernovae (PISNe) are produced by very metal-poor, very massive stars (> 120 M⊙), predicted to be amongst the first stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' PISN produce a strong odd-even ef- fect in the yields, with no neutron-capture process elements above the mass cut (Takahashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The odd-even ef- fect leads to a high [Ca/Mg] and low [Na/Mg] (green shaded area in Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Yields of PISNe, coupled with other SNe II predicted from a normal initial mass function, have been estimated by Salvadori et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2019), and are shown by a slightly higher [Na/Mg] ratio (red shaded area Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' There is no trace of PISNe, nor PISNe + SNe II, yields in Ursa Minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 The Chemistry of Target 1 The detailed chemistry of Target 1 may provide a glimpse into the early star formation events in UMi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' It stands out in [Ba/Mg] with unusually low Ba for a stars in UMi or the MW (see Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' It also appears to be lower in [Na/Mg] and [Ca/Mg] than the other stars in UMi and the MW;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' see Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' This is partially due to the higher [Mg/Fe] com- pared to other UMi stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' These low abundances relative to Mg in combination with the little amount of Ba even at rel- atively higher metallicities ([Fe/H]∼ −2) have been found in some stars of Coma Berenices (Frebel & Bromm 2012), Segue 1 (Frebel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2014), Hercules (Koch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2008, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' François et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2016), and in the Milky Way (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', Sit- nova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Kielty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Sestito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' This particular chemical pattern has been interpreted as contri- bution from only one or a few low-mass core-collapse SNe II (CCSNe), known as the "one-shot" model (Frebel & Bromm 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' We explore a variety of core collapse supernovae yields to compare to our chemical abundances in Target 1 to test this "one shot" model hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Various yields of SNe II are on the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' We choose to compare the chemistry of Target 1 against the widely used faint SNe II yields from Nomoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2013) and the recent ones from Ebinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' We included this ad- ditional comparison as the yields from Nomoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2013) are predicted only up to proton number 32, whereas the yields from Ebinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2020) reach heavier elements up MNRAS 000, 1–18 (2023) Extreme outskirt of Ursa Minor 13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 [Mg/H] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 [Ba/Mg] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 [Fe/H] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 [Ba/Fe] Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Top panel: [Ba/Mg] vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' [Mg/H] space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Bottom panel: [Ba/Fe] vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' [Fe/H] as in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Target 1 is denoted with a red (LTE) and a orange (NLTE) diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Literature UMi stars (magenta diamonds) are from Shetrone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2001), Sadakane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2004), Cohen & Huang (2010), Kirby & Cohen (2012), and Ural et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Literature MW halo compilation (grey open circles) from Aoki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2013), Yong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2013), Kielty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2021), and Buder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The black cross at the upper left corner represents the typical uncertainty on the UMi literature chemical abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='00 [Ca/Mg] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 [Na/Mg] PISNe + SNe PISNe Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' PISNe yields space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Target 1 is marked with a red and a orange diamond when LTE and NLTE, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The green band is the region of stars polluted by PISNe alone (Taka- hashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The red zone is the locus in which the stars would have been polluted by PISNe and SN II as in Salvadori et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' For the latter case, we show the yields relative to a PISNe to SN II ratio between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='9 (see Figure 6 from Salvadori et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Literature UMi stars (magenta diamonds) from Shetrone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2001), Sadakane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2004), Cohen & Huang (2010), Kirby & Cohen (2012), and Ural et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Literature MW halo compilation (grey open circles) from Aoki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2013), Yong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2013), Kielty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2021), and Buder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The black cross at the corner represents the typical uncertainty on the UMi literature chemical abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' to proton number 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Another difference is how the energy of the supernovae explosion is parametrized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' While Nomoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2013) fixed the energy to the value of 1051 erg, this is treated as a free parameter by Ebinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2020), in which it spans from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='2 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 ×1051 erg, and varies with the pro- genitor mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Both of them uses non-rotating models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The spatial symmetry of the explosion is also modelled differ- ently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Nomoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2013) employed the so-called mixing and fallback model, which implies the presence of polar jets and fallback materials around the equatorial plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' On the other hand, Ebinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2020) adopted spherical symme- try.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' When comparing the yields from Nomoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2013) with Target 1, the chemistry of this star is well described by pollution from a low-mass faint CCSNe (∼ 30 M⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Al- ternatively, we are not able to reproduce the chemistry of Target 1 when comparing to the yields from Ebinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Their predictions at all masses are higher than our observations for the majority of elements, and we cannot re- produce their strong odd-even effect, with the exception of [Ba/Mg].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' This is the only ratio we can reproduce adopting a progenitor mass 25 ≤ Mprog ≤ 30 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' As Target 1 is very far from the UMi central body, we suggest it may have formed just after the contributions from MNRAS 000, 1–18 (2023) 14 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Sestito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' low-mass SN II and was exiled by supernova feedback and/or tidal forces by pericentric passage(s) with the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' A deeper analysis of chemistry (heavy elements) of the newly discovered members in the APOGEE survey, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', those lo- cated between the central body and Target 1 and, more generally the kinematical characterisation of the UMi halo, could help to clarify this picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='6 Outside-in star formation vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' late-time merger Pace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2020) measured radial velocities and metallici- ties of likely UMi members selected from Gaia DR2 within 2 half-light radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' They interpreted the spatial distribution of the stars as composed of two populations with different chemo-dynamical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' A more metal-rich ([Fe/H] = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='03) kinematically colder (σRV = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='8 km s−1) and centrally concentrated (rh = 221 ± 17 pc) population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' And a metal-poor hotter and more extended ([Fe/H] = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='05, σRV = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='9 km s−1, rh = 374 ± 49 pc) population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Pace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2020) discussed that the two metal- licity distributions in UMi are much closer than in other dwarf spheroidal galaxies (dSphs) found so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Benítez-Llambay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2016) and Genina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2019) proposed that dwarf-dwarf mergers are the cause of the mul- tiple populations in dSphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Therefore, Pace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2020) concluded that UMi underwent a late-time merger event between two dwarfs with very similar chemical and phys- ical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' However, Genina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2019) also pointed out that kinematic and spatial information alone are insuf- ficient to disentangle the formation mechanisms of multi- populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Additional evidence from precise chemical abundances and star formation histories are needed, data that was not included in the study by Pace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' In this paper, we propose an alternative scenario to ex- plain the chemo-dynamical properties of the two populations in Ursa Minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' An outside-in star formation history can also be used to describe the properties of low mass systems, such as dwarf galaxies (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Briefly, the extended metal-poor population ([Fe/H] ≲ −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0) formed everywhere in the dwarf, such that the relatively younger stars popu- late the centre of the galaxy at times when SNe Ia begin to contribute (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', Hidalgo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Benítez-Llambay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' This enhances the metallicity only in the central re- gion, giving the galaxy a non-linear metallicity gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' In support of our simpler interpretation, the distribu- tions in the chemical elements over a wide range in metal- licity suggests a common path amongst the stars in UMi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' UMi stars are polluted by low mass CCSNe (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=', their low [Ba/Fe, Mg] and [Na, Ca/Mg]), they show a SNe Ia knee at [Fe/H] ∼ −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='1 and a contribution from AGB is also visible in the more metal-rich stars, and they display a low dispersion in [Ca/Mg] from star to star over 2 dex in metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Furthermore, Revaz & Jablonka (2018) used a cosmo- logical zoom-in simulation to show that the kinematics in UMi are consistent with secular heating in the central region of the satellite without invoking late-time mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Thus, a more simple scenario of outside-in star formation is consis- tent with the chemical, structural, and kinematic properties of UMi, and we suggest these do not necessarily require a late-time merger event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 10 15 20 25 30 35 Proton number 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 [X/Mg] [Fe/H]=-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 CCSNe (Nomoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2013) M13 M15 M18 M20 M25 M30 M40 10 15 20 25 30 35 Proton number 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 [X/Mg] [Fe/H]=-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 CCSNe (Ebinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2020) u11 u12 u13 u14 u15 u16 u17 u18 u19 u20 u24 u25 u26 u27 u28 u30 10 20 30 40 50 60 70 Proton number 2 1 0 1 2 3 [X/Mg] [Fe/H]=-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0 CCSNe (Ebinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2020) u11 u12 u13 u14 u15 u16 u17 u18 u19 u20 u24 u25 u26 u27 u28 u30 Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Chemistry of Target 1 in the CCSne yields space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Top panel: EMP ([Fe/H] = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0) CCSNe yields from Nomoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Central panel: UMP ([Fe/H] = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='0) CCSNe from Ebinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2020) in the proton number range as top panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Bottom panel: same as the central panel but for all the species pre- dicted by Ebinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The legend indicates the model’s name, in which the number is the progenitor’s mass in M⊙ at its ZAMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The darker the line, the heavier the mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Progenitor masses for models from Ebinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2020) are predicted up to 30 M⊙, while Nomoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (2013) modeled the yields up to 100 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' MNRAS 000, 1–18 (2023) Extreme outskirt of Ursa Minor 15 9 CONCLUSIONS A new Bayesian algorithm was used to find new members in the very extreme outskirts of the ultra faint dwarf galaxy, Ursa Minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Five targets were selected for high-resolution spectroscopy with GRACES at Gemini North.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' For all five stars, we determine precise radial velocities and metallicities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' for the brightest and farthest target in projection (Target 1), the higher SNR of our GRACES spectrum also permitted a detailed chemical abundance analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' With the use of data from th eliterature and APOGEE DR17, we find that: (i) The Bayesian algorithm is very efficient in finding new members, even at very large elliptical distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' All five can- didates are new members of UMi, according to their radial velocities and metallicities (see Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (ii) Ursa Minor extends at least out to a projected ellip- tical distance of ∼ 12rh, which corresponds to ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='5 kpc for an adopted distance of 76 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (iii) The orbital properties of UMi indicate that the sys- tem has recently passed apocentre and it is moving towards pericentre (see Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Tidal stripping is one scenario that can explain UMi’s elongated shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (iv) The chemical properties of Target 1 (see Figure 3), the most distant member discovered so far, are compatible with the overall distribution of the known UMi members from high-resolution spectral analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (v) The low [Ca, Na/Mg] and the low [Ba/Fe] of Tar- get 1 suggest that the star formed in an environment pol- luted by low-mass supernovae type II (Mprog ∼ 30 M⊙, see Figures 9 and 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The star is likely exiled by supernovae feedback or tidal forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (vi) Looking at all the UMi stars with high-resolution chemical analyses, including those from APOGEE DR17, we conclude there is evidence of pollution by supernovae type Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' There is a knee at [Fe/H]knee ∼ −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='1 in the [Mg, O, Na, Ni/Fe] distributions (see Figures 3 and 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (vii) Ursa Minor is also clearly polluted by supernovae type II and AGB stars given the distribution of [Ba/Mg, Fe] as a function of [Mg, Fe/H] (see Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (viii) There is no trace of yields from pair-instability su- pernovae, either alone or combined with type II (see Fig- ure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (ix) The chemo-dynamical properties of UMi can be ex- plained by an outside-in star formation and the following SNe Ia enrichment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' We propose this as a simpler scenario than a late-time merger event between two very similar sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' (x) We have found two new UMi members at a distance of ∼ 7rh in APOGEE DR17 (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='1 and Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' As their metallicities are at the edge of the APOGEE grid (∼ −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='4), their true [Fe/H] may be lower and their chemical ratios might be affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' In the very near future, the Gemini High resolution Op- tical SpecTrograph (GHOST, Pazder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2016) will be operative at Gemini South.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' It will cover a wider spectral re- gion than GRACES, especially towards the blue where many spectral lines of heavy elements are found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' In synergy with Gaia satellite and the powerful Bayesian algorithm for tar- get selections, it should be possible to discover a plethora of new members in the centre and extreme outskirts of this and many other ultra-faint and classical dwarf galaxies to study their star formation histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' This will be a giant leap forward for detailed studies of low mass systems, and both observational and theoretical near field cosmological investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We acknowledge and respect the l@IJkw@ŋ@n peoples on whose traditional territory the University of Victoria stands and the Songhees, Esquimalt and ¯WSÁNEĆ peoples whose his- torical relationships with the land continue to this day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Maunakea has always had within the Native Hawaiian com- munity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' We are very fortunate to have had the opportunity to conduct observations from this mountain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' We want to thank the supporter astronomers, Joel Roediger and Hyewon Suh, for their help during Phase II and the observational runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' FS thanks the Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Margaret "Marmie" Perkins Hess postdoctoral fellowship for funding his work at the Univer- sity of Victoria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' KAV, LDA, and JG thank the National Sciences and Engineering Research Council of Canada for funding through the Discovery Grants and CREATE pro- grams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' DZ thanks the Mitacs Globalink program for sum- mer funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The authors thanks the International Space Science Institute (ISSI) in Bern, Switzerland, for funding the "The Early Milky Way" Team led by Else Starkenburg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Based on observations obtained through the Gem- ini Remote Access to CFHT ESPaDOnS Spectrograph (GRACES), as part of the Gemini Program GN-2022A-Q- 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' ESPaDOnS is located at the Canada-France-Hawaii Telescope (CFHT), which is operated by the National Re- search Council of Canada, the Institut National des Sci- ences de l’Univers of the Centre National de la Recherche Scientifique of France, and the University of Hawai’i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' ES- PaDOnS is a collaborative project funded by France (CNRS, MENESR, OMP, LATT), Canada (NSERC), CFHT and ESA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' ESPaDOnS was remotely controlled from the inter- national Gemini Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' a program of NSF’s NOIR- Lab,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' which is managed by the Association of Universi- ties for Research in Astronomy (AURA) under a coop- erative agreement with the National Science Foundation on behalf of the Gemini partnership: the National Science Foundation (United States),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' the National Research Coun- cil (Canada),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Agencia Nacional de Investigación y Desar- rollo (Chile),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Ministerio de Ciencia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Tecnología e Innovación (Argentina),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Ministério da Ciência,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Tecnologia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Inovações e Comunicações (Brazil),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' and Korea Astronomy and Space Science Institute (Republic of Korea).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' int/web/gaia/dpac/consortium).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agree- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Sloan Foundation, the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' De- MNRAS 000, 1–18 (2023) 16 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Sestito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' partment of Energy Office of Science, and the Participating Institutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' SDSS-IV acknowledges support and resources from the Center for High Performance Computing at the University of Utah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The SDSS website is www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='sdss4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' SDSS-IV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' the Carnegie Institution for Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Carnegie Mellon Uni- versity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Center for Astrophysics | Harvard & Smithsonian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' the Chilean Participation Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' the French Participation Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Instituto de Astrofísica de Canarias,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The Johns Hop- kins University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Kavli Institute for the Physics and Math- ematics of the Universe (IPMU) / University of Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' the Korean Participation Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Lawrence Berkeley Na- tional Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Leibniz Institut für Astrophysik Potsdam (AIP),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Max-Planck-Institut für Astronomie (MPIA Heidel- berg),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Max-Planck-Institut für Astrophysik (MPA Garch- ing),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Max-Planck-Institut für Extraterrestrische Physik (MPE),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' National Astronomical Observatories of China,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' New Mexico State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' New York University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' University of Notre Dame,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Observatário Nacional / MCTI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The Ohio State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Pennsylvania State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Shanghai Astronomical Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' United Kingdom Participation Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Universidad Nacional Autónoma de México,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Univer- sity of Arizona,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' University of Colorado Boulder,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' University of Oxford,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' University of Portsmouth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' University of Utah,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' University of Virginia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' University of Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' University of Wisconsin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' Vanderbilt University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' and Yale University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France (Wenger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' This work made extensive use of TOPCAT (Taylor 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' DATA AVAILABILITY GRACES spectra will be available at the Gemini Archive web page https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='gemini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content='edu/searchform after the proprietary time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' The data underlying this article are available in the article and in its online supplementary ma- terial.' metadata={'source': 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This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} +page_content=' MNRAS 000, 1–18 (2023)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdFPT4oBgHgl3EQf-DWo/content/2301.13214v1.pdf'} diff --git a/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf b/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..cd8c8af1e73c60995639a289613b1ea88a62b061 --- /dev/null +++ b/vtFPT4oBgHgl3EQf-TW6/content/2301.13215v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c9e81ad7de38964b4fa6d87648580ce4424467666587ddab2bb32268f3311ec6 +size 717159 diff --git a/vtFPT4oBgHgl3EQf-TW6/vector_store/index.faiss b/vtFPT4oBgHgl3EQf-TW6/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..ba0d820c97c12d34ed0fdc2f86b64fbad50f9d8f 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sha256:a8291d65e5a8b9af5c930f80e3d0fa5495240d5e0006f01957408d3e381ef7e3 +size 3538989 diff --git a/wtE0T4oBgHgl3EQfcADo/content/tmp_files/2301.02358v1.pdf.txt b/wtE0T4oBgHgl3EQfcADo/content/tmp_files/2301.02358v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1821b91847525144adb3448977cd80206116c6db --- /dev/null +++ b/wtE0T4oBgHgl3EQfcADo/content/tmp_files/2301.02358v1.pdf.txt @@ -0,0 +1,694 @@ +Berry curvature induced valley Hall effect in non-encapsulated hBN/Bilayer graphene +heterostructure aligned with near-zero twist angle +Teppei Shintaku∗,1 Afsal Kareekunnan∗,†,1 Masashi Akabori,1 +Kenji Watanabe,2 Takashi Taniguchi,2 and Hiroshi Mizuta1 +1Japan Advanced Institute of Science and Technology, 1-1 Asahidai, Nomi, 923-1292, Japan +2National Institute for Materials Science, 1-1 Namiki, Tsukuba, 305-0044, Japan +(Dated: January 9, 2023) +Valley Hall effect has been observed in asymmetric single-layer graphene and bilayer graphene +systems. +In single-layer graphene systems, asymmetry is introduced by aligning graphene with +hexagonal boron nitride (hBN) with a near-zero twist angle, thereby breaking the sub-lattice sym- +metry. Although a similar approach has been used in bilayer graphene to break the layer symmetry +and thereby observe the valley Hall effect, the bilayer graphene was sandwiched with hBN on both +sides in those studies. It has been shown theoretically that having hBN on one side and both sides +affect the symmetry of the system differently. Thus, here we show the valley Hall effect through +non-local resistance measurement (RNL) in non-encapsulated hBN/bilayer graphene heterostructure +where the hBN and bilayer graphene crystallographic axes are aligned. We also show that the RNL +can be controlled by applying a displacement field across the heterostructure. Furthermore, the +electronic band structure and Berry curvature calculations validate the experimental observations. +PACS numbers: +With the introduction of two-dimensional materials, +the valley degree of freedom of carriers has gained much +prominence in recent years [1–3]. Materials like graphene +and MoS2 have two in-equivalent valleys at the K and K′ +high symmetry points of their Brillouin zone, which can +be interpreted as valley-up and valley-down, much like +the spin degree of freedom of carriers. However, the fun- +damental criterion for a material to be valleytronic is to +have a broken inversion symmetry. +While single-layer +graphene is symmetric, aligning graphene with hexag- +onal boron nitride (hBN) with a near-zero twist angle +has proved to create a moire superlattice that breaks the +sub-lattice symmetry of the graphene layer [4–7]. Such a +system has exhibited the valley Hall effect (VHE) due to +the emergence of Berry curvature at the valley as a re- +sult of broken inversion symmetry [8–10]. As for bilayer +graphene, the asymmetry can be introduced by either ap- +plying an out-of-plane electric field across the layers or +by aligning with an hBN layer, both of which break the +layer symmetry of the system as they introduce differ- +ent potentials between the top and bottom layers of the +bilayer [11–15]. +Both methods have been employed to +observe VHE in bilayer graphene in recent years [16–18]. +In the case of single-layer graphene, it has been shown +that both encapsulated and non-encapsulated graphene +exhibit the valley Hall effect, provided either of the hBN +(top or bottom) is oriented with graphene [8]. +More- +over, it has also been shown that although aligning both +the top and bottom hBN with graphene and aligning ei- +ther the hBN with the graphene breaks the symmetry +[*] These authors contributed equally to this work +[†] afsal@jaist.ac.jp +of the system, they show entirely different valleytronic +behavior. +Non-local resistance (RNL) associated with +the VHE has a strong signal when the hBN at the top +and bottom are aligned with the graphene in between +[10]. This observation is also consistent with the theo- +retical study, which shows a strong Berry curvature at K +and K′ valleys when the hBN is commensurately aligned +above and below graphene [19]. As for bilayer graphene, +although it has been shown theoretically that the con- +figuration (whether encapsulated or not), as well as the +orientation of the hBN, has an immense impact on the +asymmetry of the system [19], experimental studies in +this aspect is lacking. Thus, in this study, we explore +the VHE in non-encapsulated hBN/bilayer graphene het- +erostructure (hBN/bilayer graphene/SiO2) where the top +hBN is aligned with the bilayer graphene with near zero +twist angle. We show that the non-encapsulated bilayer +graphene shows a strong VHE signal at the primary Dirac +point, which can be further manipulated by applying a +displacement electric field across the layers. To substan- +tiate our arguments, we also performed ab initio calcula- +tions, where we show that aligned hBN/bilayer graphene +heterostructure has an intrinsic band gap and a non-zero +Berry curvature. The band gap and the Berry curvature +can be manipulated with an out-of-plane electric field +applied across the layers. +The hBN/bilayer graphene heterostructure is fabri- +cated following the dry transfer method [20]. Figure 1a +shows the optical image of the heterostructure, where +the hBN and graphene edges are aligned with a near- +zero twist angle. The dotted line outlines the graphene +area. After etching the heterostructure into a Hall bar, +edge contacts were fabricated [20]. Later a passivation +hBN layer is transferred on top of the heterostructure, +above which the top gate electrode is fabricated. Fig- +ure 1a inset and Figure 1b shows the optical image and +arXiv:2301.02358v1 [cond-mat.mes-hall] 6 Jan 2023 + +2 +schematic diagram of the final device. +The local and +non-local electrical measurements are performed follow- +ing the standard four-terminal method using KEITHLEY +4200 semiconductor parameter analyzer with very high in- +put impedance (1013 Ω) at the voltage terminals. A separate +source meter (KEITHLEY 2400) is used to apply gate volt- +age. Figure 1c compares the local (RL) and non-local (RNL) +resistance measurement results for the device at 10 K. For +RL, a current is applied between terminal 2 and 3 and the +voltage drop between terminals 9 and 8 is detected giving RL += V9,8/I2,3. For RNL measurement, the current is applied at +the local terminals 3 and 8, and the voltage drop at terminals +2 and 9 is measured, giving RNL = V2,9/I3,8. The length and +width of the Hall bar in the measured region are 2.5 µm and +1 µm, respectively. A strong RNL signal is detected around +the charge neutrality point (CNP) with zero electric or mag- +netic field applied across the layers. +The peak of the RNL +generally appears at the CNP. The shift in the RNL peak +is attributed to the in-homogeneity in the bilayer graphene +channel, especially since the graphene is on SiO2 substrate. +To rule out the possibility of diffusive charge contribution to +the measured non-local signal, we also calculated the Ohmic +contribution using the formula ROhm +NL += RL +� W +πL +� +exp +� +− πL +W +� +[21]. The calculated Ohmic contribution is at least one order +of magnitude less than that of the measured RNL (Fig. 1c), +thereby ruling out the possibility of diffusive charge contri- +bution. One possible origin of the observed RNL would be +the VHE. The Berry curvature induced VHE and the resul- +tant transverse valley Hall conductivity (σVH +xy ) is related to +the measured RNL as +RNL = 1 +2 +� +σVH +xy +σxx +�2 +W +σxxlv exp +� +− L +lv +� +(1) +where L and W are the length and width of the device, lv is +the valley diffusion length and σxx = 1/ρ is the conductivity. +In the small valley Hall angle regime (σVH +xy /σxx) ≪ 1), RNL +and ρ holds a cubic scaling relation (RNL ∝ ρ3). Thus we +plotted RNL as a function of ρ as shown in figure 1d, which +exhibits a clear cubic relation implying that the measured +RNL indeed originates from the VHE. +Next, we investigate the effect of an electric displacement +field applied across the bilayer graphene on both RL and RNL. +Figure 2a and b show the heat map of the RL and RNL respec- +tively as a function of both top-gate and bottom-gate voltage. +It can be seen that the RNL is narrower than RL indicating +the difference in the physical origin of both peaks. Applying +voltages on the top and bottom gates allows the independent +control of carrier concentration and the displacement field. +The displacement fields related to the top and bottom gates +(VTG and VBG) are defined as: +DT G = −ϵT G(VT G − V 0 +T G)/dT G +DBG = −ϵBG(VBG − V 0 +BG)/dBG +(2) +where ϵT G(ϵBG) and dT G(dBG) are the dielectric constant +and thickness of the top(bottom) layer. V 0 +T G,BG is the voltage +offset due to initial doping. The difference between the two +displacement fields gives the carrier doping, and the average +of the two is the net displacement fields. +Figure 2c shows +the evolution of RL and RNL as a function of the displace- +ment field. Here RL and RNL are plotted as a function of top +gate voltage with the back gate fixed at different values. As +TABLE I: Band gap extracted from the 1/ρmax +L +versus 1/T +plot for different VBG. E⊥ is the electric field corresponding +to each VBG at ρmax +L +. +VBG (V) +E⊥ (V/nm) +Band gap (meV) +-20 +-0.363 +45.5 +-10 +-0.24 +35 +0 +0 +25 +10 +0.0043 +17.5 +mentioned earlier, at zero electric field (VBG = 0V ), a clear +non-local signal is observed, suggesting an in-built asymme- +try present in the heterostructure. Application of a negative +electric field (VBG = -20V and -10V) increases the intensity of +both RL and RNL. This implies that a negative electric field +widens the band gap and enhances the asymmetry between +the layers. Whereas the application of a small positive elec- +tric field (VBG = 10V ) reduces the intensity of both peaks, +suggesting a reduction in the band gap and the asymmetry. +However, the application of a strong positive electric field yet +again increases the intensity of both RL and RNL, implying an +increase in band gap and asymmetry of the heterostructure. +To validate the above hypothesis, we measured the tem- +perature dependence of the RL at different gate voltages to +calculate the band gap. The maximum of the local resistivity +(ρmax +L +) at the high-temperature regime is related to the band +gap as: +1 +ρmax +L += 1 +ρL exp +� +− EL +kBT +� +(3) +where ρL is the local resistivity, EL is the activation energy, +kB is the Boltzmann constant and T is the temperature. The +band gap Eg, defined as 2EL, can be extracted by plotting +1/ρmax +L +as a function of 1/T as shown in figure 3. The dotted +line is the fit to equation 3 at the high-temperature regime. +Table I shows the extracted band gap values at different VBG. +The heterostructure has an intrinsic band gap of 25 meV (at +VBG = 0 V). Application of a negative displacement field en- +hances the band gap (35 meV at VBG= -10 V and 45.5 meV +at VBG= -20 V). At the same time, the application of a pos- +itive displacement field reduces the band gap (17.5 meV at +VBG= 10 V). This is consistent with the previous observation +of increase(decrease) in the RL and RNL peak intensity at +negative(positive) displacement field. +We have also performed ab initio calculations to substanti- +ate the experimental observations (see [22] for calculation de- +tails). Figure 4a-e shows the electronic band structure calcu- +lated for the hBN/bilayer graphene heterostructure at differ- +ent electric fields applied across the layers. The heterostruc- +ture has an intrinsic band gap of 36 meV (Fig. 4c), implying +asymmetry between the layers. As the low energy bands are +constituted by the non-dimer atoms in the bottom and top +layers of the bilayer graphene, the hBN induces different po- +tentials between them, which opens a band gap. Applying a +negative electric field (Fig. 4a-b) introduces additional asym- +metry between the layers, enhancing the band gap (75 meV +for -0.25 V/nm and 92 meV for -0.5 V/nm). However, apply- +ing a positive electric field initially works against the inbuilt +asymmetry between the layers, reducing the band gap to 21 +meV (Fig. 4d). At higher positive electric fields, the electric +potential surpasses the intrinsic potential difference between + +3 +the layers and widens the band gap further (40 meV for 1.25 +V/nm), as shown in figure 4e. The results of the band struc- +ture calculation strongly agree with the experimental obser- +vations. Next, we look at the Berry curvature calculated for +the heterostructure. Berry curvature for an electronic band +is defined as +Ωn(k) = i ℏ2 +m2 +� +n̸=n′ +⟨un,k|ˆp|un′ ,k⟩ × ⟨un′ ,k|ˆp|un,k⟩ +(εn − εn′ )2 +(4) +where |un,k⟩ is the periodic part of the Bloch function, ˆp is +the momentum operator, εn is the energy of the nth band and +εn′ represents the energy of all other bands. The total Berry +curvature is the sum of the individual occupied band’s Berry +curvature (Ω(k) = � +n fnΩn(k)). The Wannier interpolation +scheme dictates that a pair of bands that are either occu- +pied or unoccupied have a negligible contribution to the total +Berry curvature [26]. +The major contribution to the total +Berry curvature comes from a pair of bands where one is oc- +cupied and another unoccupied, such as the low energy bands +in the hBN/bilayer graphene heterostructure. +In addition, +the denominator of equation 4 suggests that Berry curvature +value varies as the square of the energy difference between +two adjacent bands. Thus in our case, the Berry curvature +changes with the electric field as the band gap changes. The +heterostructure has an intrinsic non-zero Berry curvature as +shown in figure 4h. A negative electric field reduces the mag- +nitude of the Berry curvature as it widens the gap between +the low energy bands (Fig. 4f-g). On the other hand, a small +positive electric field (0.5 V/nm) enhances the magnitude of +the Berry curvature due to the narrow band gap (Fig. 4i). +However, at a higher positive electric field two key differences +could be observed (Fig. 4j). One, the magnitude of the Berry +curvature reduces owing to the widening of the band gap. +The second is the change in the polarity of the Berry curva- +ture, which implies that the polarity of the layer asymmetry +switches direction at higher positive electric fields. +In conclusion, we have observed Berry curvature induced +VHE in non-encapsulated hBN/bilayer graphene heterostruc- +ture where the hBN and bilayer graphene are aligned with +near-zero twist angle. The VHE is detected as a non-local re- +sistance near the CNP. The cubic relation observed between +RNL and ρ validates that the measured RNL indeed origi- +nates from the VHE. The measured RNL could be manipu- +lated with the application of a displacement field across the +layers, which is attributed to the change in the electronic band +structure and the asymmetry of the bilayer graphene under a +displacement field. The intrinsic band gap of the heterostruc- +ture and its evolution under the displacement field is con- +firmed from the temperature-dependent RL measurement in +the high-temperature regime. The experimental observations +were substantiated with ab initio calculations which showed +that the heterostructure has an intrinsic band gap and a non- +zero Berry curvature, both of which could be controlled by a +perpendicular electric field. +Acknowledgements: Part of this research was supported +by Toshiba electronic devices & storage corporation’s aca- +demic encouragement program. +[1] John R. Schaibley, Hongyi Yu, Genevieve Clark, Pasqual +Rivera, Jason S. Ross, Kyle L. Seyler, Wang Yao, and +Xiaodong Xu, Valleytronics in 2D materials. 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The dotted line shows +the graphene region. The inset shows the optical image of the final device. (b) Schematic diagram of the heterostructure, which +shows the passivation hBN layer and the top gate electrode. (C) Measured local (RL) and non-local (RNL) resistance for the +heterostructure at 10K. The blue line is the calculated Ohmic contribution to the non-local resistance. (d) RNL and ρ follows +a cubic scaling relation (RNL ∝ ρ3) indicating that the measured RNL originates from VHE. + +(a) +(b) +Au/Cr +Au/Cr +hBN +hBN +BLG +BLG +SiO, +10 μm +Si +6 +1098 +(c) +(d) +100 +RNL +10 K + Exp. data +6 +ROhm +NL +R, +(U>) +50 +R +3 +R +0 +0 +0 +30 +4.5 +5 +-5 +0 +5 +p (kΩ) +VTG (V)6 +FIG. 2: +Heat map of the (a) RL and (b) RNL as a function of top-gate and bottom-gate. (c) RL and RNL measured as a +function of top-gate voltage with back-gate fixed at different values. All the measurements are performed at 10 K. + +(b) +(a) +R, (kQ) +(2) +15 +18 +15 +160 +14 +120 +M +0 +0 +M +BG +9 +80 +V +V +-15 +-15 +5 +40 +-30 +-30 +0 +0 +0 +2 +4 +6 +0 +2 +6 +4 +VTG (V) +VTG (V) +(c) +16 +VBG = -20 V +VBG = -10 V +VBG= O V +VBG = 10 V +VBG = 30 V +160 +12 +120 +(>) +8 +R +80 +RM +4 - +40 +0 +0 +2 +-2 +2 +-2 +0 +2 +9- +-4 +-2 +0 +2 +4 +9 +0 +6 +0 +4 +.4 +0 +VTG (V)7 +FIG. 3: +1/ρmax +L +as a function of 1/T plot for different VBG showing the temperature dependence of ρmax +L +. The dotted line +indicates the fitting to equation 3 in the high-temperature regime from which the band gap of the heterostructure at different +VBG is extracted. +FIG. 4: +Electronic band structure calculated for the hBN/bilayer graphene heterostructure at electric fields of magnitude (a) +-0.5 V/nm, (b) -0.25 V/nm, (c) 0 V/nm, (d) 0.5 V/nm and (e) 1.25 V/nm. The path of the band structure calculation is M +→ K → Γ. Berry curvature calculated at K high symmetry point for the heterostructure at electric fields of magnitude (f) +-0.5 V/nm, (g) -0.25 V/nm, (h) 0 V/nm, (i) 0.5 V/nm and (j) 1.25 V/nm. The path of the Berry curvature calculation is the +same as the band structure calculation. + +=-20 V +10-3 +=-10V +BG +=V +BG +(1/Q2) +1/pmax +0.00 +0.05 +0.10 +1/T (K-1)0.2 +-0.5 V/nm +0.25 V/nm +0 V/nm +(d) +0.5 V/nm +1.25 V/nm +(a +(c) +(eV) +0.1 +E +0.0 +E +-0.1 +-0.2 +0.80 +0.88 +0.80 +0.88 +0.80 +0.88 +0.80 +0.88 +0.80 +0.88 +K +K +(i) +(i) +(f) +(g) +(h) +-0.5 V/nm +-0.25 V/nm +0 V/nm +60 +0.5 V/nm +1.25 V/nm +εOTX) +0 +-60 +0.6 +1.0 +0.6 +1.0 +0.6 +1.0 +0.6 +1.0 +0.6 +1.0 +K +K +K +K +K \ No newline at end of file diff --git a/wtE0T4oBgHgl3EQfcADo/content/tmp_files/load_file.txt b/wtE0T4oBgHgl3EQfcADo/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0c872fe924d654a2caaa897dbe1e3191706f0e54 --- /dev/null +++ b/wtE0T4oBgHgl3EQfcADo/content/tmp_files/load_file.txt @@ -0,0 +1,433 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf,len=432 +page_content='Berry curvature induced valley Hall effect in non-encapsulated hBN/Bilayer graphene heterostructure aligned with near-zero twist angle Teppei Shintaku∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content='1 Afsal Kareekunnan∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content='†,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content='1 Masashi Akabori,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content='1 Kenji Watanabe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content='2 Takashi Taniguchi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content='2 and Hiroshi Mizuta1 1Japan Advanced Institute of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' 1-1 Asahidai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Nomi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' 923-1292,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Japan 2National Institute for Materials Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' 1-1 Namiki,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Tsukuba,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' 305-0044,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Japan (Dated: January 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' 2023) Valley Hall effect has been observed in asymmetric single-layer graphene and bilayer graphene systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' In single-layer graphene systems, asymmetry is introduced by aligning graphene with hexagonal boron nitride (hBN) with a near-zero twist angle, thereby breaking the sub-lattice sym- metry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Although a similar approach has been used in bilayer graphene to break the layer symmetry and thereby observe the valley Hall effect, the bilayer graphene was sandwiched with hBN on both sides in those studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' It has been shown theoretically that having hBN on one side and both sides affect the symmetry of the system differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Thus, here we show the valley Hall effect through non-local resistance measurement (RNL) in non-encapsulated hBN/bilayer graphene heterostructure where the hBN and bilayer graphene crystallographic axes are aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' We also show that the RNL can be controlled by applying a displacement field across the heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Furthermore, the electronic band structure and Berry curvature calculations validate the experimental observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' PACS numbers: With the introduction of two-dimensional materials, the valley degree of freedom of carriers has gained much prominence in recent years [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Materials like graphene and MoS2 have two in-equivalent valleys at the K and K′ high symmetry points of their Brillouin zone, which can be interpreted as valley-up and valley-down, much like the spin degree of freedom of carriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' However, the fun- damental criterion for a material to be valleytronic is to have a broken inversion symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' While single-layer graphene is symmetric, aligning graphene with hexag- onal boron nitride (hBN) with a near-zero twist angle has proved to create a moire superlattice that breaks the sub-lattice symmetry of the graphene layer [4–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Such a system has exhibited the valley Hall effect (VHE) due to the emergence of Berry curvature at the valley as a re- sult of broken inversion symmetry [8–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' As for bilayer graphene, the asymmetry can be introduced by either ap- plying an out-of-plane electric field across the layers or by aligning with an hBN layer, both of which break the layer symmetry of the system as they introduce differ- ent potentials between the top and bottom layers of the bilayer [11–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Both methods have been employed to observe VHE in bilayer graphene in recent years [16–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' In the case of single-layer graphene, it has been shown that both encapsulated and non-encapsulated graphene exhibit the valley Hall effect, provided either of the hBN (top or bottom) is oriented with graphene [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' More- over, it has also been shown that although aligning both the top and bottom hBN with graphene and aligning ei- ther the hBN with the graphene breaks the symmetry [*] These authors contributed equally to this work [†] afsal@jaist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content='jp of the system, they show entirely different valleytronic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Non-local resistance (RNL) associated with the VHE has a strong signal when the hBN at the top and bottom are aligned with the graphene in between [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' This observation is also consistent with the theo- retical study, which shows a strong Berry curvature at K and K′ valleys when the hBN is commensurately aligned above and below graphene [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' As for bilayer graphene, although it has been shown theoretically that the con- figuration (whether encapsulated or not), as well as the orientation of the hBN, has an immense impact on the asymmetry of the system [19], experimental studies in this aspect is lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Thus, in this study, we explore the VHE in non-encapsulated hBN/bilayer graphene het- erostructure (hBN/bilayer graphene/SiO2) where the top hBN is aligned with the bilayer graphene with near zero twist angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' We show that the non-encapsulated bilayer graphene shows a strong VHE signal at the primary Dirac point, which can be further manipulated by applying a displacement electric field across the layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' To substan- tiate our arguments, we also performed ab initio calcula- tions, where we show that aligned hBN/bilayer graphene heterostructure has an intrinsic band gap and a non-zero Berry curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' The band gap and the Berry curvature can be manipulated with an out-of-plane electric field applied across the layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' The hBN/bilayer graphene heterostructure is fabri- cated following the dry transfer method [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Figure 1a shows the optical image of the heterostructure, where the hBN and graphene edges are aligned with a near- zero twist angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' The dotted line outlines the graphene area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' After etching the heterostructure into a Hall bar, edge contacts were fabricated [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Later a passivation hBN layer is transferred on top of the heterostructure, above which the top gate electrode is fabricated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Fig- ure 1a inset and Figure 1b shows the optical image and arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content='02358v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content='mes-hall] 6 Jan 2023 2 schematic diagram of the final device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' The local and non-local electrical measurements are performed follow- ing the standard four-terminal method using KEITHLEY 4200 semiconductor parameter analyzer with very high in- put impedance (1013 Ω) at the voltage terminals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' A separate source meter (KEITHLEY 2400) is used to apply gate volt- age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Figure 1c compares the local (RL) and non-local (RNL) resistance measurement results for the device at 10 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' For RL, a current is applied between terminal 2 and 3 and the voltage drop between terminals 9 and 8 is detected giving RL = V9,8/I2,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' For RNL measurement, the current is applied at the local terminals 3 and 8, and the voltage drop at terminals 2 and 9 is measured, giving RNL = V2,9/I3,8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' The length and width of the Hall bar in the measured region are 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content='5 µm and 1 µm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' A strong RNL signal is detected around the charge neutrality point (CNP) with zero electric or mag- netic field applied across the layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' The peak of the RNL generally appears at the CNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' The shift in the RNL peak is attributed to the in-homogeneity in the bilayer graphene channel, especially since the graphene is on SiO2 substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' To rule out the possibility of diffusive charge contribution to the measured non-local signal, we also calculated the Ohmic contribution using the formula ROhm NL = RL � W πL � exp � − πL W � [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' The calculated Ohmic contribution is at least one order of magnitude less than that of the measured RNL (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' 1c), thereby ruling out the possibility of diffusive charge contri- bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' One possible origin of the observed RNL would be the VHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' The Berry curvature induced VHE and the resul- tant transverse valley Hall conductivity (σVH xy ) is related to the measured RNL as RNL = 1 2 � σVH xy σxx �2 W σxxlv exp � − L lv � (1) where L and W are the length and width of the device, lv is the valley diffusion length and σxx = 1/ρ is the conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' In the small valley Hall angle regime (σVH xy /σxx) ≪ 1), RNL and ρ holds a cubic scaling relation (RNL ∝ ρ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Thus we plotted RNL as a function of ρ as shown in figure 1d, which exhibits a clear cubic relation implying that the measured RNL indeed originates from the VHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Next, we investigate the effect of an electric displacement field applied across the bilayer graphene on both RL and RNL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Figure 2a and b show the heat map of the RL and RNL respec- tively as a function of both top-gate and bottom-gate voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' It can be seen that the RNL is narrower than RL indicating the difference in the physical origin of both peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Applying voltages on the top and bottom gates allows the independent control of carrier concentration and the displacement field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' The displacement fields related to the top and bottom gates (VTG and VBG) are defined as: DT G = −ϵT G(VT G − V 0 T G)/dT G DBG = −ϵBG(VBG − V 0 BG)/dBG (2) where ϵT G(ϵBG) and dT G(dBG) are the dielectric constant and thickness of the top(bottom) layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' V 0 T G,BG is the voltage offset due to initial doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' The difference between the two displacement fields gives the carrier doping, and the average of the two is the net displacement fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Figure 2c shows the evolution of RL and RNL as a function of the displace- ment field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Here RL and RNL are plotted as a function of top gate voltage with the back gate fixed at different values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' As TABLE I: Band gap extracted from the 1/ρmax L versus 1/T plot for different VBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' E⊥ is the electric field corresponding to each VBG at ρmax L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' VBG (V) E⊥ (V/nm) Band gap (meV) 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content='363 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content='5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content='24 35 0 0 25 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content='0043 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content='5 mentioned earlier, at zero electric field (VBG = 0V ), a clear non-local signal is observed, suggesting an in-built asymme- try present in the heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Application of a negative electric field (VBG = -20V and -10V) increases the intensity of both RL and RNL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' This implies that a negative electric field widens the band gap and enhances the asymmetry between the layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Whereas the application of a small positive elec- tric field (VBG = 10V ) reduces the intensity of both peaks, suggesting a reduction in the band gap and the asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' However, the application of a strong positive electric field yet again increases the intensity of both RL and RNL, implying an increase in band gap and asymmetry of the heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' To validate the above hypothesis, we measured the tem- perature dependence of the RL at different gate voltages to calculate the band gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' The maximum of the local resistivity (ρmax L ) at the high-temperature regime is related to the band gap as: 1 ρmax L = 1 ρL exp � − EL kBT � (3) where ρL is the local resistivity, EL is the activation energy, kB is the Boltzmann constant and T is the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' The band gap Eg, defined as 2EL, can be extracted by plotting 1/ρmax L as a function of 1/T as shown in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' The dotted line is the fit to equation 3 at the high-temperature regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Table I shows the extracted band gap values at different VBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' The heterostructure has an intrinsic band gap of 25 meV (at VBG = 0 V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Application of a negative displacement field en- hances the band gap (35 meV at VBG= -10 V and 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content='5 meV at VBG= -20 V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' At the same time, the application of a pos- itive displacement field reduces the band gap (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content='5 meV at VBG= 10 V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' This is consistent with the previous observation of increase(decrease) in the RL and RNL peak intensity at negative(positive) displacement field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' We have also performed ab initio calculations to substanti- ate the experimental observations (see [22] for calculation de- tails).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Figure 4a-e shows the electronic band structure calcu- lated for the hBN/bilayer graphene heterostructure at differ- ent electric fields applied across the layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' The heterostruc- ture has an intrinsic band gap of 36 meV (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' 4c), implying asymmetry between the layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' As the low energy bands are constituted by the non-dimer atoms in the bottom and top layers of the bilayer graphene, the hBN induces different po- tentials between them, which opens a band gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Applying a negative electric field (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' 4a-b) introduces additional asym- metry between the layers, enhancing the band gap (75 meV for -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content='25 V/nm and 92 meV for -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content='5 V/nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' However, apply- ing a positive electric field initially works against the inbuilt asymmetry between the layers, reducing the band gap to 21 meV (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' 4d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' At higher positive electric fields, the electric potential surpasses the intrinsic potential difference between 3 the layers and widens the band gap further (40 meV for 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content='25 V/nm), as shown in figure 4e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' The results of the band struc- ture calculation strongly agree with the experimental obser- vations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Next, we look at the Berry curvature calculated for the heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Berry curvature for an electronic band is defined as Ωn(k) = i ℏ2 m2 � n̸=n′ ⟨un,k|ˆp|un′ ,k⟩ × ⟨un′ ,k|ˆp|un,k⟩ (εn − εn′ )2 (4) where |un,k⟩ is the periodic part of the Bloch function, ˆp is the momentum operator, εn is the energy of the nth band and εn′ represents the energy of all other bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' The total Berry curvature is the sum of the individual occupied band’s Berry curvature (Ω(k) = � n fnΩn(k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' The Wannier interpolation scheme dictates that a pair of bands that are either occu- pied or unoccupied have a negligible contribution to the total Berry curvature [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' The major contribution to the total Berry curvature comes from a pair of bands where one is oc- cupied and another unoccupied, such as the low energy bands in the hBN/bilayer graphene heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' In addition, the denominator of equation 4 suggests that Berry curvature value varies as the square of the energy difference between two adjacent bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Thus in our case, the Berry curvature changes with the electric field as the band gap changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' The heterostructure has an intrinsic non-zero Berry curvature as shown in figure 4h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' A negative electric field reduces the mag- nitude of the Berry curvature as it widens the gap between the low energy bands (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' 4f-g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' On the other hand, a small positive electric field (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content='5 V/nm) enhances the magnitude of the Berry curvature due to the narrow band gap (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' 4i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' However, at a higher positive electric field two key differences could be observed (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' 4j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' One, the magnitude of the Berry curvature reduces owing to the widening of the band gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' The second is the change in the polarity of the Berry curva- ture, which implies that the polarity of the layer asymmetry switches direction at higher positive electric fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' In conclusion, we have observed Berry curvature induced VHE in non-encapsulated hBN/bilayer graphene heterostruc- ture where the hBN and bilayer graphene are aligned with near-zero twist angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' The VHE is detected as a non-local re- sistance near the CNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' The cubic relation observed between RNL and ρ validates that the measured RNL indeed origi- nates from the VHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' The measured RNL could be manipu- lated with the application of a displacement field across the layers, which is attributed to the change in the electronic band structure and the asymmetry of the bilayer graphene under a displacement field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' The intrinsic band gap of the heterostruc- ture and its evolution under the displacement field is con- firmed from the temperature-dependent RL measurement in the high-temperature regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' The experimental observations were substantiated with ab initio calculations which showed that the heterostructure has an intrinsic band gap and a non- zero Berry curvature, both of which could be controlled by a perpendicular electric field.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Dean, One-Dimensional Electrical Contact to a Two- Dimensional Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Science 342, 614-617 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' [21] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Abanin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Morozov, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Ponomarenko, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Gorbachev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Mayorov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Katsnelson, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Watan- abe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Taniguchi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Novoselov, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Levitov, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Geim, Giant Nonlocality Near the Dirac Point in Graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Science 332, 328-330 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' [22] The ab initio calculations are performed using the SIESTA package, which uses the linear combination of atomic orbitals (LCAO) basis sets [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' To maintain sim- ilar interlayer distance between the layers as the experi- mental values, van der Waals exchange-correlation func- tionals were used [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' A vacuum layer of 25 ˚Awas used to reduce the interaction between adjacent structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' For all the calculations, a Monkhorst-Pack grid of 40 × 40 × 1 and an energy cutoff of 500 Ry were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Structure op- timization was performed until the force between atoms was less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content='01 eV/˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' The Berry curvature calculation was performed using the WANNIER90 package [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' A total of 15 Wannier functions were used on a Monkhorst- Pack grid of 40 × 40 × 1 to calculate the Berry curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' [23] Jos´e M Soler, Emilio Artacho, Julian D Gale, Alberto Garc´ıa, Javier Junquera, Pablo Ordej´on and Daniel S´anchez-Portal, The SIESTA method for ab initio order- N materials simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' : Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Matter 14, 2745 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' [24] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Dion, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Rydberg, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Schr¨oder, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Langreth and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Lundqvist, Van der Waals Density Functional for General Geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' 92, 246401 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' [25] Giovanni Pizzi et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=', Wannier90 as a community code: new features and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' : Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Matter 32, 165902 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' [26] Xinjie Wang, Jonathan R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Yates, Ivo Souza, and David Vanderbilt, Ab initio calculation of the anomalous Hall conductivity by Wannier interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' B 74, 195118 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' 5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' 1: (a) Optical image of hBN/bilayer graphene heterostructure aligned with near-zero twist angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' The dotted line shows the graphene region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' The inset shows the optical image of the final device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' (b) Schematic diagram of the heterostructure, which shows the passivation hBN layer and the top gate electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' (C) Measured local (RL) and non-local (RNL) resistance for the heterostructure at 10K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' The blue line is the calculated Ohmic contribution to the non-local resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' (d) RNL and ρ follows a cubic scaling relation (RNL ∝ ρ3) indicating that the measured RNL originates from VHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' (a) (b) Au/Cr Au/Cr hBN hBN BLG BLG SiO, 10 μm Si 6 1098 (c) (d) 100 RNL 10 K Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' data 6 ROhm NL R, (U>) 50 R 3 R 0 0 0 30 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content='5 5 5 0 5 p (kΩ) VTG (V)6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' 2: Heat map of the (a) RL and (b) RNL as a function of top-gate and bottom-gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' (c) RL and RNL measured as a function of top-gate voltage with back-gate fixed at different values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' All the measurements are performed at 10 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' (b) (a) R, (kQ) (2) 15 18 15 160 14 120 M 0 0 M BG 9 80 V V 15 15 5 40 30 30 0 0 0 2 4 6 0 2 6 4 VTG (V) VTG (V) (c) 16 VBG = -20 V VBG = -10 V VBG= O V VBG = 10 V VBG = 30 V 160 12 120 (>) 8 R 80 RM 4 - 40 0 0 2 2 2 2 0 2 9- 4 2 0 2 4 9 0 6 0 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content='4 0 VTG (V)7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' 3: 1/ρmax L as a function of 1/T plot for different VBG showing the temperature dependence of ρmax L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' The dotted line indicates the fitting to equation 3 in the high-temperature regime from which the band gap of the heterostructure at different VBG is extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' 4: Electronic band structure calculated for the hBN/bilayer graphene heterostructure at electric fields of magnitude (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content='5 V/nm, (b) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content='25 V/nm, (c) 0 V/nm, (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content='5 V/nm and (e) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content='25 V/nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' The path of the band structure calculation is M → K → Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' Berry curvature calculated at K high symmetry point for the heterostructure at electric fields of magnitude (f) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content='5 V/nm, (g) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content='25 V/nm, (h) 0 V/nm, (i) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content='5 V/nm and (j) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content='25 V/nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' The path of the Berry curvature calculation is the same as the band structure calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content=' =-20 V 10-3 =-10V BG =V BG (1/Q2) 1/pmax 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content='10 1/T (K-1)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content='5 V/nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE0T4oBgHgl3EQfcADo/content/2301.02358v1.pdf'} +page_content='25 V/nm 0 V/nm (d) 0.' metadata={'source': 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